Spin/Pin-Structures and Real Enumerative Geometry 9789811278532, 9789811278549, 9789811278556

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Spin/Pin-Structures and Real Enumerative Geometry

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Spin/Pin-Structures and Real Enumerative Geometry Xujia Chen Harvard University, USA

Aleksey Zinger Stony Brook University, USA

World Scientific NEW JERSEY

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LONDON

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SINGAPORE

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BEIJING

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SHANGHAI

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HONG KONG

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TAIPEI

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CHENNAI

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TOKYO

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Names: Chen, Xujia, author. | Zinger, Aleksey, author. Title: Spin/pin-structures and real enumerative geometry / Xujia Chen, Harvard University, USA, Aleksey Zinger, Stony Brook University, USA. Description: New Jersey : World Scientific, [2024] | Includes bibliographical references and index. Identifiers: LCCN 2023023573 | ISBN 9789811278532 (hardcover) | ISBN 9789811278549 (ebook for institutions) | ISBN 9789811278556 (ebook for individuals) Subjects: LCSH: Vector bundles. | Geometry, Enumerative. | Elliptic operators. | Symplectic manifolds. Classification: LCC QA612.63 .C44 2024 | DDC 516.3/5--dc23/eng/20230927 LC record available at https://lccn.loc.gov/2023023573 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

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For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/13476#t=suppl Desk Editors: Soundararajan Raghuraman/Nijia Liu Typeset by Stallion Press Email: [email protected] Printed in Singapore

Preface

Spin- and Pin-structures on vector bundles have long played an important role in differential geometry. Since the mid-1990s, they have also been central to the open and real sectors of Gromov–Witten theory, mirror symmetry, and enumerative geometry. Spin- and Pinstructures in the classical perspective are (equivalence classes of) principal Spin- and Pin-bundles that doubly cover the oriented orthonormal frame bundles of oriented vector bundles and the orthonormal frame bundles of vector bundles, respectively, over paracompact (and usually locally contractible) topological spaces; this perspective motivates the terminology. In the standard modern perspective of symplectic topology, Spin- and Pin-structures are homotopy classes of trivializations of vector bundles over 2-skeleta of CW complexes. In the perspective more recently introduced in [16], Spin- and Pin-structures are collections of homotopy classes of trivializations of vector bundles over loops that respect cobordisms between the loops. This last perspective applies to any topological space, is completely intrinsic, and connects directly with the usage of Spinand Pin-structures in symplectic topology. This monograph provides an accessible introduction to Spin- and Pin-structures in general, demonstrates their role in the orientability considerations in symplectic topology, and exhibits their applications in enumerative geometry. Part I contains a systematic treatment of Spin- and Pin-structures in all three perspectives. We in particular verify that these structures satisfy a collection of succinctly formulated properties and describe natural correspondences between the three perspectives. v

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Spin/Pin-Structures & Real Enumerative Geometry

We also recall the notions of relative Spin- and Pin-structures originating in the early versions of [12] in a CW perspective and introduce an alternative perspective on them in the spirit of the perspective of [16] on Spin- and Pin-structures. In the latter perspective, relative Spin- and Pin-structures are collections of homotopy classes of trivializations over boundaries of compact bordered surfaces that respect certain cobordisms. We conclude Part I by verifying that the relative Spin- and Pin-structures in both perspectives satisfy a collection of succinctly formulated properties and describe a natural correspondence between the two perspectives. Real Cauchy–Riemann operators are first-order elliptic operators that arise in symplectic topology as linearizations of the pseudoholomorphic map equation; the latter has been central to the field since Gromov’s seminal work [23]. Spin- and Pin-structures and their relative counterparts induce orientations of the determinants of such operators on vector bundle pairs over bordered surfaces and on real bundle pairs over symmetric surfaces. We detail constructions of these orientations, which are central to the open and real sectors of Gromov–Witten theory, in Part II. We also verify that these orientations satisfy a collection of succinctly formulated properties, many of which are associated with the properties of relative Spin- and Pin-structures established in Part I. The intrinsic perspective on these structures introduced in Part I fits perfectly with all considerations in Part II. Part III applies some of the key results of Parts I and II to real enumerative geometry. Welschinger’s invariants [44–46] are signed counts of real (involution-invariant) holomorphic curves on K¨ ahler surfaces and threefolds with anti-holomorphic involutions and more generally of pseudoholomorphic curves in symplectic fourfolds and sixfolds with anti-symplectic involutions. The interpretation of these invariants, up to a topological sign, in [39] as signed counts of pseudoholomorphic maps from disks brought the potential of applying the machinery of (complex) Gromov–Witten theory to study Welschinger’s invariants. In Part III, we describe a further reinterpretation of these invariants in [6, 8] as the degrees of relatively orientable pseudocycles from moduli spaces of real pseudoholomorphic maps with signed marked points introduced in [14]. We also determine the exact topological signs relating different versions of Welschinger’s invariants, using the classical perspective on the

Preface

vii

Spin- and Pin-structures in addition to the properties of orientations of real Cauchy–Riemann operators established in Part II. The examples of Sections 14.2 and 14.3 precisely match computations of these invariants in various perspectives. The first author’s pseudocycle re-interpretation of Welschinger’s invariants underpins her proof in [6] of Solomon’s relations [40] for these invariants envisioned almost 12 years earlier and has led in [8] to similar relations for Welschinger’s invariants of some real symplectic sixfolds. As illustrated in [9], these relations provide an effective way of computing the invariants of [44–46] for many real symplectic fourfolds. We include two appendices containing standard statements to simplify the presentation in the main body of this monograph. ˇ Appendix A recalls some relations between the Cech cohomology of sheaves over paracompact spaces, singular cohomology, vector bundles, and their characteristic classes. Appendix B relates the algebra and topology of covering projections that are Lie group homomorphisms. Many statements appearing in this monograph are well known, either available somewhere in the literature in some form or believed to be true. Some of them are gathered among the properties of Spinand Pin-structures, relative Spin- and Pin-structures, and the orientations of the determinants of real Cauchy–Riemann operators in Sections 1.2, 6.2, 7.2, and 7.3, ready for immediate use and fully established in later sections. The more delicate and technical statements are new and motivated by modern developments in symplectic topology and real enumerative geometry; they come with complete proofs as well. We hope this monograph overall will facilitate access to and will further progress in these interrelated fields. The authors were partially supported by the NSF grants DMS 1500875 and 1901979 and the Simons Collaboration grant 587036.

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About the Authors

Xujia Chen is a 2021–2024 Junior Fellow of the Society of Fellows at Harvard University. Her doctoral dissertation work, completed at Stony Brook University, focused on the topological foundations of real Gromov– Witten theory and its applications to real enumerative geometry.

Aleksey Zinger is a Professor of Mathematics at Stony Brook University. His research interests lie in symplectic topology and enumerative geometry and are inspired by mathematical predictions of string theory. His research has been funded by the National Science Foundation, the Simons Foundation, and the Sloan Foundation. He has published over 50 peer-reviewed research articles in symplectic topology and enumerative geometry.

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Contents

Preface

v

About the Authors

ix

Part I: Spin- and Pin-Structures

1

1. Main Results and Examples of Part I

3

1.1 1.2 1.3 1.4

Definitions and Main Theorem . . . . . Properties of Spin- and Pin-Structures Basic Examples . . . . . . . . . . . . . Further Examples . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. 3 . 8 . 15 . 20

2. The Lie Groups Spin(n) and Pin˘ (n) 2.1 2.2

The Groups SOpnq and Spinpnq . . . . . . . . . . . . 28 The groups O(n) and Pin˘ (n) . . . . . . . . . . . . 34

3. Proof of Theorem 1.4(1): Classical Perspective ˇ 3.1 Spin/Pin-Structures and Cech Cohomology . . ˘ 3.2 The Sets P pV q and SppV, oq . . . . . . . . . . 3.3 Correspondences and Obstructions to Existence 3.4 Short Exact Sequences . . . . . . . . . . . . . . 4. Proof of Theorem 1.4(1): Trivializations Perspectives 4.1

27

43 . . . .

. . . .

. . . .

43 48 56 62 73

Topological Preliminaries . . . . . . . . . . . . . . . . 73

xi

xii

Spin/Pin-Structures & Real Enumerative Geometry

4.2 4.3

The Spin- and Pin-Structures of Definition 1.3 . . . . 79 The Spin- and Pin-Structures of Definition 1.2 . . . . 90

5. Equivalence of Definitions 1.1–1.3 5.1 5.2

99

Proof of Theorem 1.4(2) . . . . . . . . . . . . . . . . 100 Proof of Theorem 1.4(3) . . . . . . . . . . . . . . . . 104

6. Relative Spin- and Pin-Structures 6.1 6.2 6.3 6.4 6.5 6.6

111

Definitions and Main Theorem . . . . . . . . . . Properties of Relative Spin- and Pin-Structures Proof of Theorem 6.4(1): Definition 6.3 Perspective . . . . . . . . . . . . . . . . . . . . Topological Preliminaries . . . . . . . . . . . . . Proof of Theorem 6.4(1): Definition 6.1 Perspective . . . . . . . . . . . . . . . . . . . . Equivalence of Definitions 6.1 and 6.3 . . . . . .

. . . 111 . . . 116 . . . 121 . . . 132 . . . 139 . . . 145

Part II: Orientations for Real CR-Operators

153

7. Main Results and Applications of Part II

155

7.1 7.2 7.3 7.4 7.5

Definitions and Main Theorem . . . . . . . . Properties of Orientations: Smooth Surfaces Properties of Orientations: Degenerations . Some Implications . . . . . . . . . . . . . . . Orientations and Evaluation Isomorphisms .

. . . . .

. . . . .

8. Base Cases 8.1 8.2 8.3 8.4 8.5

. . . . .

. . . . .

155 160 171 180 184 193

pS 2 , τ q:

Construction Line Bundles over and Properties . . . . . . . . . . . . . . . . . . . Line Bundles over C Degenerations of pS 2 , τ q . Line Bundles over H3 Degenerations of pS 2 , τ q . Even-Degree Bundles over pS 2 , τ q: Construction and Properties . . . . . . . . . . . . . . . . . . . Even-Degree Bundles over Degenerations of pS 2 , τ q and Exact Triples . . . . . . . . . . .

9. Intermediate Cases 9.1 9.2 9.3

. . . . .

. . . 194 . . . 199 . . . 208 . . . 216 . . . 224 231

Orientations for Line Bundle Pairs . . . . . . . . . . 231 Proofs of Propositions 9.1(1), 9.2, and 9.3 . . . . . . 240 Orientations from OSpin-Structures . . . . . . . . . . 254

Contents

xiii

10. Orientations for Twisted Determinants 10.1 10.2 10.3 10.4

Orientations of the Twisting Target Orientations of Real CR-Operators Degenerations and Exact Triples . . Properties of Twisted Orientations .

. . . .

267 . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

268 273 277 288

Part III: Real Enumerative Geometry

293

11. Pin-Structures and Immersions

297

11.1 11.2 11.3

Main Statements and Examples . . . . . . . . . . . 298 Admissible Immersions into Surfaces . . . . . . . . . 302 Proofs of Lemmas 11.9–11.12 . . . . . . . . . . . . . 308

12. Counts of Rational Curves on Surfaces 12.1 12.2 12.3 12.4 12.5 12.6

P2

. . Complex Low-Degree Curves in Real Low-Degree Curves in P2 . . . . Welschinger’s Invariants in Dimension Moduli Spaces of Real Maps . . . . . Proof of Theorem 12.1 . . . . . . . . Proof of Proposition 12.5 . . . . . . .

315 . . 4 . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

13. Counts of Stable Real Rational Maps 13.1 13.2 13.3 13.4 13.5 13.6 13.7

Invariance and Properties . Orienting Moduli Spaces of Orienting Moduli Spaces of Definition of Curve Signs . Proof of Invariance . . . . Signs at Immersions . . . . Proof of Lemma 13.10 . . .

. . . Real Real . . . . . . . . . . . .

355

. . . . . Curves . Maps . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

14. Counts of Real Rational Curves vs. Maps 14.1 14.2 14.3 14.4

Comparison Theorems . . . . . . Basic Examples: Fourfolds . . . . Basic Examples: Sixfolds . . . . . Proofs of Theorems 14.1 and 14.2

. . . .

315 320 323 328 333 338

. . . .

. . . .

356 362 366 371 376 384 390 393

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

393 396 399 405

Appendices

409

ˇ A Cech Cohomology

409

A.1

Identification with singular cohomology . . . . . . . . 409

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Spin/Pin-Structures & Real Enumerative Geometry

A.2 A.3 A.4 A.5

Sheaves of groups . . . . . . . . . . . . Sheaves determined by Lie groups . . . Relation with principal bundles . . . . Orientable vector bundle over surfaces

B Lie Group Covers B.1 B.2 B.3

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

413 421 425 429 433

Terminology and summary . . . . . . . . . . . . . . . 433 Proof of Lemma B.1 . . . . . . . . . . . . . . . . . . 436 Disconnected Lie groups . . . . . . . . . . . . . . . . 438

Bibliography

445

Index of Terms

449

Index of Notation

451

Part I

Spin- and Pin-Structures

The purpose of this part is to provide a comprehensive introduction to Spin- and Pin-structures in the classical perspective, based on the Lie groups Spinpnq and Pin˘ pnq, and in the two modern perspectives of symplectic topology, based on trivializations of vector bundles. Chapter 1 formally presents Spin- and Pin-structures in the three perspectives, collects their properties in a ready-to-use format, and gives several explicit examples of these structures that are used in Parts II and III. This chapter also contains one of the two theorems of Part I, Theorem 1.4, identifying the three perspectives on Spinand Pin-structures. We review the definitions and key properties of the Lie groups Spinpnq and Pin˘ pnq in Chapter 2 from purely topological considerations. Explicit constructions of these Lie groups in terms of Clifford algebras appear in [3, §3], in the first part of [28, Section 1], and in [29, Sections I.1, I.2]; in the notation of [29], Pin` pnq and Pin´ pnq are Pinn,0 and Pin0,n , respectively. However, these explicit constructions are not necessary for many purposes. Chapter 2 is needed for the classical perspective on Spin- and Pin-structures but is irrelevant for the other two perspectives of Chapter 1 and for formulating key properties of these structures. Chapter 3 details the classical perspective on Spin- and Pinstructures, provides further examples of these structures, and relates ˇ them to Cech cohomology. It also establishes the properties of 1

2

Spin/Pin-Structures & Real Enumerative Geometry

Spin- and Pin-structures listed in Chapter 1 in the classical perspective. Chapter 4 establishes the same properties in the two modern perspectives of symplectic topology. In Chapter 5, we show that the three perspectives on Spin- and Pin-structures are equivalent, when restricted to the appropriate categories of topological spaces, by constructing natural correspondences between these structures in the classical perspective and in each of the two modern perspectives; these correspondences respect the properties listed in Chapter 1. Chapter 5 thus establishes Theorem 1.4. Relative Spin- and Pin-structures trace their origins to the early versions of [12], where they appeared as natural topological data for orienting the determinants of real Cauchy–Riemann operators on vector bundle pairs over the unit disk D2` Ă C. In Chapter 6, we recall the original perspective on relative Spin- and Pin-structures, which involves an auxiliary choice of a CW structure on the underlying pair of topological spaces Y Ă X, introduce completely intrinsic perspective on them, which fits directly with how these structures are used to orient real Cauchy–Riemann operators, and collect important properties of relative Spin- and Pin-structures in a ready-touse format. We also verify these properties in both perspectives and describe a natural correspondence between the two perspectives which respects the stated properties of relative Spin- and Pinstructures. This establishes the other theorem of Part I, Theorem 6.4, that the two perspectives on relative Spin- and Pin-structures are equivalent when restricted to CW complexes (or topological spaces homeomorphic to a CW complex). In retrospect, the CW perspectives on Spin- and Pin-structures and their relative counterparts are rather artificial, in terms of both their formulations and the intended applications. The former is immediately apparent from the need to choose a CW structure on a topological space and is further reflected in the verification of some of the properties of these structures. The latter is demonstrated by the need in the construction of orientations of the determinants of real Cauchy–Riemann operators in [12, Section 8.1] to first choose a CW structure on the topological spaces Y Ă X involved, a vector bundle over the 3-skeleton X3 Ă X, and a homotopy between a given continuous map from pD2` , S 1 q to pX, Y q and a continuous map to pX3 , Y2 q and then to show that the induced orientation does not depend on all these choices. We bypass these complications in Part II by using the intrinsic perspective on relative Spin- and Pin-structures introduced in Chapter 6.

Chapter 1

Main Results and Examples of Part I

We define Spin- and Pin-structures in three perspectives and state the first theorem of Part I, that these perspectives are essentially equivalent, in Section 1.1. The properties of these structures described in Section 1.2 include the obstructions to the existence of these structures and compatibility with short exact sequences of vector bundles. Basic examples of Spin- and Pin-structures appear in Section 1.3; additional examples with more nuanced considerations appear in Section 1.4. 1.1

Definitions and Main Theorem

For a topological space Y , let τY ” Y ˆR ÝÑ Y be the trivial line bundle over Y and oY be its standard orientation. For a vector bundle V over Y , we denote by λpV q ” Λtop R V ÝÑ Y its top exterior power and by OpV q the set of the orientations on V . For each o P OpV q, we denote by λpV, oq the real vector bundle λpV q with the orientation λpoq induced by o and by ˘ ` (1.1) StpV, oq ” τY ‘V, StV poq 3

4

Spin/Pin-Structures & Real Enumerative Geometry

the real vector bundle StpV q ” τY ‘V with the induced orientation. We make the canonical identifications λpτY ‘ V q ” τY b λpV q “ λpV q, ` ˘ λ StpV, oq ” pτY , oY q b λpV, oq “ λpV, oq.

(1.2)

For n P Z` , the Lie groups Spinpnq and Pin˘ pnq are double covers of the n-th special orthogonal group SOpnq and the n-th orthogonal group Opnq, qn : Spinpnq ÝÑ SOpnq

and

qn˘ : Pin˘ pnq ÝÑ Opnq;

(1.3)

see Chapter 2. For n ě 2, qn is the unique connected double cover of SOpnq. The double covers qn˘ restrict to qn over SOpnq; they are the same topologically but have different group structures. The preimages of an order 2 element of Opnq under qn˘ generate a subgroup of Pin˘ pnq of order 4. The subgroup generated by the preimages of an order 2 element of Opnq with precisely one p´1q-eigenvalue is Z22 in the case of Pin` pnq and Z4 in the case of Pin´ pnq. A rank n vector bundle V over a paracompact space Y determines a principal Opnq-bundle OpV q of orthonormal frames; see Section 3.1. Each orientation o on V determines a principal SOpnq-bundle SOpV, oq of oriented orthonormal frames. In the classical perspective, Pin˘ - and Spin-structures are liftings of these frame bundles over the projection maps qn˘ and qn . Definition 1.1. Let Y be a paracompact space and V be a rank n real vector bundle over Y . (a) A Pin˘ -structure p on V is a principal Pin˘ pnq-bundle Pin˘ pV q over Y with a 2 : 1 covering map qV : Pin˘ pV q ÝÑ OpV q which commutes with the projections to Y and is equivariant with respect to the group homomorphism qn˘ . (b) If o P OpV q, a Spin-structure s on pV, oq is a principal Spinpnqbundle SpinpV, oq over Y with a 2 : 1 covering map qV : SpinpV, oq ÝÑ SOpV, oq which commutes with the projections to Y and is equivariant with respect to the group homomorphism qn .

Main Results and Examples of Part I

5

We call a pair os ” po, sq consisting of an orientation o on a vector bundle V and a Spin-structure s on pV, oq an OSpin-structure on V . Two Spin-structures qV : SpinpV, oq ÝÑ SOpV, oq

and

qV1 : Spin1 pV, oq ÝÑ SOpV, oq (1.4)

are equivalent if there exists a Spinpnq-equivariant isomorphism r : SpinpV, oq ÝÑ Spin1 pV, oq Ψ

s.t.

r qV “ qV1 ˝ Ψ.

The analogous notions of equivalence apply to Pin˘ and OSpinstructures. When there is no ambiguity, we will not distinguish between the Spin- and Pin-structures of Definition 1.1 and their equivalence classes. An OSpin-structure on the trivial rank n vector bundle nτY ” Y ˆRn ÝÑ Y with its canonical orientation is given as follows: ` ˘ qV “ idY ˆqn : Spin nτY , noY ” Y ˆSpinpnq ÝÑ Y ˆSpinpnq (1.5) ˘ ” SOpnτY , noY . An orientation o on a line bundle V over a paracompact space Y determines a homotopy class of isomorphisms pV, oq « pτY , oY q. A split of an oriented vector bundle pV, oq into n-oriented line bundles thus determines a homotopy class of trivializations of the principal SOpnq-bundle SOpV, oq and thus an OSpin-structure os0 pV, oq on V . In general, os0 pV, oq depends on the choice of the split. Once its summands are chosen, the induced orientation on V depends on their ordering and on the orientations of the individual summands. However, the Spin-structures induced by orderings giving rise to different orientations are identified under the natural correspondence (1.14). For typical applications in symplectic topology, it is more convenient to view these structures in terms of trivializations of vector bundles. The standard variation of this perspective, captured by Definition 1.2, goes back at least to the mid-1990s. The second variation, captured by Definition 1.3, appears in [16, Section 5.1].

6

Spin/Pin-Structures & Real Enumerative Geometry

It is completely intrinsic and connects directly with its usage in symplectic topology. Definition 1.2. Let pV, oq be an oriented vector bundle over a CW complex Y with rkR V ě 3. A Spin-structure s on pV, oq is a homotopy class of trivializations of pV, oq over the 2-skeleton Y2 of Y . A loop in a topological space Y is a continuous map α : S 1 ÝÑ Y . We denote the collection of all loops in Y by LpY q. We call a compact two-dimensional topological manifold Σ with the boundary BΣ possibly non-empty a bordered surface. A closed surface is a bordered surface Σ with BΣ “ H. Definition 1.3. Let pV, oq be an oriented vector bundle over a topological space Y with rkR V ě 3. A Spin-structure s on pV, oq is a collection psα qαPLpY q of homotopy classes sα of trivializations of α˚ pV, oq such that for every continuous map F : Σ ÝÑ Y from a bordered surface, the vector bundle F ˚pV, oq over Σ admits a trivialization whose restriction to each component Br Σ of BΣ lies in su|Br Σ under any identification of Br Σ with S 1 . The existence of a Spin-structure s in the sense of Definition 1.3 on an oriented vector bundle pV, oq with rkR V ě 3 explicitly requires the bundle F ˚ V to be trivializable for every continuous map F : Σ ÝÑ Y from a closed surface. In both variations of the trivializations perspective, a Spin structure s on an oriented vector bundle pV, oq over Y with rkR V “ 1, 2 is a Spin-structure on the vector bundle 2τY ‘ V with the induced orientation in the first case and on τY ‘ V in the second. A Pin˘ structure p on a real vector bundle V over Y is a Spin-structure on the canonically oriented vector bundle V˘ ” V ‘p2˘1qλpV q ;

(1.6)

see the paragraph containing (1.18). An OSpin-structure on V is again a pair os ” po, sq consisting of an orientation o on V and a Spinstructure s on pV, oq. Analogously to the perspective of Definition 1.1, a split of an oriented vector bundle pV, oq into n-oriented line bundles determines an OSpin-structure os0 pV, oq on V . For a vector bundle V over a topological space Y satisfying the appropriate conditions, we denote by P˘ pV q and OSppV q the sets

Main Results and Examples of Part I

7

of Pin˘ -structures and OSpin-structures, respectively, on V in any of the three perspectives (up to equivalence in the perspective of Definition 1.1). For o P OpV q, we denote by SppV, oq the set of Spinstructures on pV, oq. We identity SppV, oq with a subset of OSppV q in the obvious way. By Theorem 1.4, there is no fundamental ambiguity in the definitions of the sets P˘ pV q, OSppV q, and OSppV, oq when Y is a CW complex (and so the perspectives of Definitions 1.1, 1.2, and 1.3 apply) or more generally Y is a paracompact locally contractible space (and the perspectives of Definitions 1.1 and 1.3 apply). The local contractability restriction can in fact be weakened to the local H 2 p¨; Z2 q-simplicity of Definition 3.2. Theorem 1.4. Let Y be a topological space and V be a vector bundle over Y . (1) The OSpin- and Pin-structures on V in the perspectives of Definitions 1.1 if Y is paracompact and locally contractible, 1.2 if Y is a CW complex, and 1.3 satisfy all properties of Section 1.2. (2) If Y is a paracompact locally contractible space, there are canonical identifications of the sets OSppV q in the perspectives of Definitions 1.1 and 1.3 and of the sets P˘ pV q in the two perspectives for every vector bundle V over Y . These identifications associate the distinguished elements s0 pV, oq in the two perspectives with each other for all oriented vector bundles pV, oq split into oriented line bundles and respect all structures and correspondences of Section 1.2. (3) If Y is a CW complex, the same statements apply to the OSpinand Pin-structures on V in the perspectives of Definitions 1.1 and 1.2. This theorem is fundamentally a consequence of |π0 pSOpnqq|, |π0 pSpinpnqq|,

|π2 pSOpnqq| “ 1, |π1 pSpinpnqq|,

|π1 pSOpnqq| “ 2,

(1.7)

|π2 pSpinpnqq| “ 1

(1.8)

for n ě 3 for the following reasons: Suppose Y is a CW complex of dimension at most 2 and V is an oriented vector bundle over Y of rank n ě 3. By (1.8), every bundle SpinpV, oq as in Definition 1.1(b) is trivializable as a principal bundle and thus admits a section sr;

8

Spin/Pin-Structures & Real Enumerative Geometry

any two such sections are homotopic. The section qV ˝r s of SOpV, oq then determines a trivialization of the vector bundle V over Y ; the trivializations of V determined by different sections of SpinpV, oq are homotopic. The above implies that a Spin-structure in the sense of Definition 1.1 determines a Spin-structure in the sense of Definition 1.2 and a Spin-structure in the sense of Definition 1.3, under appropriate topological conditions on the underlying topological space Y . It is immediate that equivalent Spin-structures in the sense of Definition 1.1 determine the same Spin-structures in the perspectives of Definitions 1.2 and 1.3 in the above construction. The next observation ensures that this leads to a correspondence between the three notions of Spin-structure. Lemma 1.5. Let pV, oq be an oriented vector bundle over S 1 with rk V ě 3. The homotopy classes of trivializations of V determined by different equivalence classes of Spin-structures on pV, oq in the sense of Definition 1.1 are different. Theorem 1.4 and Lemma 1.5 are proved in Chapter 5.

1.2

Properties of Spin- and Pin-Structures

Let Y be a topological space. An isomorphism Ψ : V 1 ÝÑ V of vector bundles over Y induces bijections Ψ˚ : OSppV q ÝÑ OSppV 1 q and Ψ˚ : P˘ pV q ÝÑ P˘ pV 1 q

(1.9)

between the OSpin-structures on V and V 1 and the Pin˘ -structures on V and V 1 in the perspectives of Definitions 1.1 if Y is paracompact, 1.2 if Y is a CW complex, and 1.3. If the ranks of V and V 1 are at least 3, the OSpin- and Pin˘ -structures on V 1 in the last two perspectives are obtained from the same types of structures on V by pre-composing the relevant trivializations with Ψ. If Y is paracompact, Ψ induces an isomorphism OpΨq : OpV 1 q ÝÑ OpV q of principal Oprk V q-bundles over Y . If qV is a Pin˘ -structure on V in the sense of Definition 1.1(a), then OpΨq˚ qV : OpΨq˚ PinpV q ÝÑ OpV 1 q,

pp1 , prq ÝÑ p1 ,

Main Results and Examples of Part I

9

is a Pin˘ -structure on V 1 . If in addition o P OpV q and Ψ˚ o is the orientation on V 1 induced by o via Ψ, then Ψ also induces an isomorphism SOpΨ, oq : SOpV 1 , Ψ˚ oq ÝÑ SOpV, oq of principal SOprk V q-bundles over Y . If qV is a Spin-structure on pV, oq in the sense of Definition 1.1(b), then SOpΨ, oq˚ qV : SOpΨ, oq˚ SpinpV, oq ÝÑ SOpV 1 , Ψ˚ oq,

pp1 , prq ÝÑ p1 ,

is a Spin-structure on pV 1 , Ψ˚ oq. Let V be a vector bundle over a topological space Y and Y 1 be another topological space. A continuous map f : Y 1 ÝÑ Y induces maps f ˚ : OSppV q ÝÑ OSppf ˚ V q

and f ˚ : P˘ pV q ÝÑ P˘ pf ˚ V q (1.10)

in the perspectives of Definitions 1.1 if Y and Y 1 are paracompact, 1.2 if f is a map of CW complexes, and 1.3. In the last two perspectives, the OSpin- and Pin˘ -structures on f ˚ V are obtained from the same types of structures on V by pre-composing the relevant trivializations with the map frV : f ˚ V ÝÑ V,

py 1 , vq ÝÑ v,

(1.11)

covering f . If Y is paracompact, this map induces an Oprk V qequivariant map OpfrV q : Opf ˚ V q ÝÑ OpV q covering f . If qV is a Pin˘ -structure on V in the sense of Definition 1.1(a), then ` 1 ˘ p , pr ÝÑ p1 , OpfrV q˚ qV : OpfrV q˚ PinpV q ÝÑ Opf ˚ V q, is a Pin˘ -structure on f ˚ V . If in addition o P OpV q and f ˚ o is the orientation on f ˚ V induced by o via frV , then frV also induces an SOprk V q-equivariant map ` ˘ SOpfrV , oq : SO f ˚ V, f ˚ o ÝÑ SOpV, oq

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Spin/Pin-Structures & Real Enumerative Geometry

covering f . If qV is a Spin-structure on pV, oq in the sense of Definition 1.1(b), then SOpfrV , oq˚ qV : SOpfrV , oq˚ SpinpV, oq ÝÑ SOpf ˚ V, f ˚ oq, pp1 , prq ÝÑ p1 , is a Spin-structure on pf ˚ V 1 , f ˚ oq. If an oriented vector bundle pV, oq is split as a direct sum of oriented line bundles, we take the induced splitting of pV, oq to be the splitting obtained from the splitting of pV, oq by negating the last component. We take the induced splitting of StV pV, oq to be the splitting obtained by combining the canonical trivialization of τY as the first component with the splitting of pV, oq. The following SpinPin properties apply in any of the three perspectives of Definitions 1.1–1.3 on Pin˘ -, Spin-, and OSpinstructures, provided the topological space Y appearing in these statements satisfies the appropriate restrictions: ‚ Y is a paracompact locally contractible space for the perspective of Definition 1.1. ‚ Y is a CW complex for the perspective of Definition 1.2. The following naturality properties of the group actions and correspondences refer to the commutativity with the pullbacks (1.9) induced by isomorphisms of vector bundles over Y and the pullbacks (1.10) induced by the admissible continuous maps, as appropriate: SpinPin 1 (Obstruction to existence). (a) A real vector bundle V over Y admits a Pin´ -structure (resp. Pin` -structure) if and only if w2 pV q “ w12 pV q (resp. w2 pV q “ 0). (b) An oriented vector bundle pV, oq over Y admits a Spin-structure if and only if w2 pV q “ 0. SpinPin 2 (Affine structure). Let V be a real vector bundle over Y . (a) If V admits a Pin˘ -structure, then the group H 1 pY ; Z2 q acts naturally, freely, and transitively on the set P˘ pV q. (b) If o P OpV q and pV, oq admits a Spin-structure, then the group H 1 pY ; Z2 q acts naturally, freely, and transitively on the set SppV, oq.

Main Results and Examples of Part I

11

If V 1 and V 2 are vector bundles over Y , with at least one of them of positive rank, then the action of the automorphism Ψ : V 1 ‘V 2 ÝÑ V 1 ‘V 2 ,

Ψpv 1 , v 2 q “ p´v 1 , v 2 q,

on P˘ pV 1 ‘V 2 q is given by ` ˘ Ψ˚ p “ prk V 1 ´1qw1 pV 1 q`prk V 1 qw1 pV 2 q ¨p

(1.12)

@ p P P˘ pV 1 ‘V 2 q. (1.13)

SpinPin 3 (Orientation reversal). Let V be a real vector bundle over Y . There is a natural H 1 pY ; Z2 q-equivariant involution

OSppV q ÝÑ OSppV q,

os ÝÑ os,

(1.14)

which maps SppV, oq bijectively onto SppV, oq for every o P OpV q and sends os0 pV, oq to os0 pV, oq for every oriented vector bundle pV, oq split as a direct sum of oriented line bundles. SpinPin 4 (Reduction). Let V be a real vector bundle over Y . For every o P OpV q, there are natural H 1 pY ; Z2 q-equivariant bijections ˘ R˘ o : P pV q ÝÑ SppV, oq

(1.15)

˘ so that R˘ o p¨q “ Ro p¨q.

SpinPin 5 (Stability). Let V be a real vector bundle over Y . There are natural H 1 pY ; Z2 q-equivariant bijections ˘ ˘ ` ` ˘ ˘ : P pV q ÝÑ P ‘V τ StV : OSppV q ÝÑ OSp τY ‘V , St˘ Y V (1.16) so that StV pos0 pV, oqq “ os0 pStV pV, oqq for every oriented vector bundle pV, oq split as a direct sum of oriented line bundles, StV posq “ StV posq @ os P OSppV q, ˘ ˘ StV ˝R˘ o “ RStV poq ˝StV

@ o P OpV q.

(1.17)

If V is a real vector bundle over Y , then the real vector bundles (1.6) over Y have canonical orientations, which we denote by o´ V and o` V . They are described as follows. The orientations on these

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Spin/Pin-Structures & Real Enumerative Geometry

bundles canonically correspond to the orientations on the real line bundles λpV´ q “ λpV qbλpV q and

λpV` q “ λpV qbλpV qbλpV qbλpV q, (1.18)

respectively. We orient the fiber of the first line bundle over a point y P Y by choosing an orientation on the first factor λpVy q and using the same orientation on the second factor. The resulting orientation on the product does not depend on the choice of the orientation on the first factor. We thus obtain an orientation on λpV´ q. The same reasoning applies to λpV` q as well. The identifications (1.2) induce identifications ˘ ` ˘ ˘ ` ` ˘ (1.19) pτY ‘V q˘ , oτ˘Y ‘V “ τY ‘V˘ , StV˘ po˘ V q ” St V˘ , oV . SpinPin 6 (Correspondences). Let V be a real vector bundle over Y . There are natural H 1 pY ; Z2 q-equivariant bijections ` ˘ ˘ ˘ (1.20) Co˘ V : P pV q ÝÑ Sp V˘ , oV ˘ ˘ so that Co˘ τY ‘V ˝StV “ StV˘ ˝CoV .

Suppose j

0 ÝÑ V 1 ÝÑ V ÝÑ V 2 ÝÑ 0 ι

(1.21)

is a short exact sequence of vector bundles over Y . Orientations o1 on V 1 and o2 on V 2 determine an orientation o1eo2 on V as follows. 1 is an oriented basis for pV 1 , o1 q, then If y P Y and v11 , . . . , vm y 1 q, vm`1 , . . . , vn P Vy ιpv11 q, . . . , ιpvm

is an oriented basis for pVy , o1eo2 q if and only if jpvm`1 q, . . . , jpvn q is an oriented basis for pVy2 , o2 q. Whenever the vector bundles V 1 and V 2 are specified, we denote the canonical, direct short exact sequence j

0 ÝÑ V 1 ÝÑ V 1 ‘V 2 ÝÑ V 2 ÝÑ 0 ι

(1.22)

by ‘. If o1 and o2 are orientations on V 1 and o2 on V 2 , respectively, we denote by o1 o2 the induced orientations on V .

Main Results and Examples of Part I

13

If e is the short exact sequence (1.21) and o1 is an orientation on V 1 , we denote by eo˘1 any short exact sequence pι,0q

j˘1

o V˘2 ÝÑ 0 0 ÝÑ V 1 ÝÑ V˘ ÝÑ

(1.23)

obtained by combining (1.21) with the isomorphism λpV q « λpV 1 qbλpV 2 q « λpV 2 q induced by an orientation-preserving trivialization of pV 1 , o1 q. If Y is paracompact (as in the perspectives of Definitions 1.1 and 1.2), o1 determines a homotopy class of trivializations of V 1 and thus a homotopy class of short exact sequences eo˘1 of vector bundles (1.23) over Y . For the purposes of Definition 1.3, it is most natural to define a short exact sequence of vector bundles V 1 , V, V 2 over Y as a collection of homotopy classes of short exact sequences j

α α α˚ V ÝÑ α˚ V 2 ÝÑ 0 0 ÝÑ α˚ V 1 ÝÑ

ι

over S 1 in the usual sense, one homotopy class for each loop α in Y , which extend to short exact sequences j

F F F ˚ V ÝÑ F ˚ V 2 ÝÑ 0 0 ÝÑ F ˚ V 1 ÝÑ

ι

over bordered surfaces Σ in the usual sense for each continuous map F : Σ ÝÑ Y as in Definition 1.3. An orientation o1 on V 1 then still determines a short exact sequence eo˘1 of vector bundles (1.23). SpinPin 7 (Short exact sequences). Every short exact sequence e of vector bundles over Y as in (1.21) determines natural H 1 pY ; Z2 q-biequivariant maps xx¨, ¨yy e : OSppV 1 qˆ OSppV 2 q ÝÑ OSppV q, xx¨, ¨yy e : OSppV 1 qˆ P˘ pV 2 q ÝÑ P˘ pV q

(1.24)

so that the following properties hold: (ses1) If oriented vector bundles pV 1 , o1 q and pV 2 , o2 q over Y are split as direct sums of oriented line bundles, then ˘ ` xxos0 pV 1 , o1 q, os0 pV 2 , o2 qyy‘ “ os0 pV 1 , o1 q‘pV 2 , o2 q .

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Spin/Pin-Structures & Real Enumerative Geometry

(ses2) If e is as in (1.21), o1 P OpV 1 q, and o2 P OpV 2 q, then xxos1 , os2 yy e P SppV, o1eo2 q @ os1 P SppV 1 , o1 q, os2 P SppV 2 , o2 q, (1.25) ` ˘ @@ 1 ˘ 2 DD 1 2 R˘ o1 o2 xxos , p yy e “ os , Ro2 pp q e e

@ os1 P SppV 1 , o1 q, p2 P P˘ pV 2 q.

(1.26)

(ses3) If e is as in (1.21), o1 P opV 1 q, os1 P SppV 1 , o1 q, and p2 P P˘ pV 2 q, then xxos1 , p2 yy e “ w1 pV 2 q¨xxos1 , p2 yy e, DD ` ˘ @@ 1 1 2 ˘ 2 Co˘ V xxos , p yy e “ os , CoV 2 pp q e˘ . o1

(ses4) If V11 , V21 , V 2 are vector bundles over Y , os11 P OSppV11 q, os12 P OSppV21 q, and p2 P P˘ pV 2 q, then @@ DD DD os11 , xxos12 , p2 yy‘ ‘ “ xxos11 , os12 yy‘ , p2 ‘ .

@@

(ses5) If V is a vector bundle over Y and p P P˘ pV q, then St˘ V ppq “ xxos0 pτY , oY q, pyy‘ . (ses6) If V is a vector bundle over Y , o P OpV q, and p P P˘ pV q, then @@ ˘ DD Co˘ V ppq “ Ro ppq, os0 pp2˘1qλpV, oqq ‘ . By SpinPin 4, (1.26), and the first statement in SpinPin 7(ses3), xxos1 , os2 yy e “ xxos1 , os2 yy e “ xxos1 , os2 yy e

(1.27)

for every short exact sequence e of vector bundles as in (1.21), os1 P OSppV 1 q, and os2 P OSppV 2 q. By SpinPin 4 and 7(ses4) and (1.26), @@

@@ DD DD os11 , xxos12 , os2 yy‘ ‘ “ xxos11 , os12 yy‘ , os2 ‘

(1.28)

Main Results and Examples of Part I

15

for all vector bundles V11 , V21 , V 2 over Y, os11 P OSppV11 q, os12 P OSppV21 q, and os2 P OSppV 2 q. By SpinPin 4, 5, and 7(ses5) and (1.26), StV posq “ xxos0 pτY , oY q, osyy‘

(1.29)

for every vector bundle V over Y and os P OSppV q. Combining the second map in (1.24) with the canonical isomorphism of V 1 ‘ V 2 with V 2 ‘ V 1 , we obtain a natural H 1 pY ; Z2 qbiequivariant map xx¨, ¨yy e : P˘ pV 1 qˆ OSppV 2 q ÝÑ P˘ pV q.

(1.30)

By the SpinPin 7 property, this map satisfies the obvious analogs of (1.26), the first statement in SpinPin (ses3), and SpinPin (ses4) and (ses6). Remark 1.6. The first statement of the SpinPin 7(ses3) property corrects [39, Lemma 8.1], which suggests that the induced Pin˘ structure on V 1 ‘V 2 does not depend on the orientation of V 1 . 1.3

Basic Examples

The first two cases of the first homomorphism (1.3) are given by q1 : Spinp1q ” Z2 ÝÑ SOp1q “ t1u, q2 : Spinp2q “ S 1 ÝÑ SOp2q “ S 1 ,

u ÝÑ u2 .

(1.31)

The groups Pin` p1q and Pin´ p1q are Z22 and Z4 , respectively, and q1´ : Pin´ p1q “ Z4 ÝÑ Op1q “ t˘1u « Z2 ” Z{2Z

(1.32)

is the unique surjective homomorphism. For concreteness, we take ` ˘ rI1;1 “ 1Z ” 1`4Z P Pin´ p1q, rI1;1 “ 0, 1Z P Pin` p1q, 4 2 ` ˘ pI1 “ 1Z , 0 P Pin` p1q. (1.33) 2 The last choice implies that q1`“ pr2 : Pin` p1q “ Z2 ‘Z2 ÝÑ Op1q “ t˘1u « Z2 ” Z{2Z

(1.34)

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Spin/Pin-Structures & Real Enumerative Geometry

is the projection to the second component. We next explicitly describe Spin- and Pin˘ -structures on some rank 1 and 2 vector bundles. Example 1.7. Let V be a real line bundle over a connected space Y . Every automorphism Ψ of V is homotopic to either the identity or the multiplication by ´1. An automorphism Ψ of V homotopic to the identity clearly acts trivially on the sets P˘ pV q of Pin˘ ´ and OSppV q of OSpin-structures on V in either of the three perspectives of Definitions 1.1–1.3. By (1.13) with V 2 of rank 0, an automorphism Ψ of V homotopic to the multiplication by ´1 also acts trivially on P˘ pV q. Thus, every automorphism Ψ of V acts trivially on P˘ pV q. If o P OpV q, then every orientation-preserving automorphism of pV, oq acts trivially on the set SppV, oq of Spin-structures on pV, oq. Example 1.8. The Spin-structures on an oriented line bundle pV, oq over a path-connected paracompact space Y in the perspective of Definition 1.1 are the topological double covers qV : Yr ÝÑ Y . The distinguished element s0 pV, oq of SppV, oq is the disconnected double cover of Y . Examples 1.9, 1.10, and 1.12 are the base inputs in the proof of the SpinPin 1 property for the perspective of Definition 1.1. This property for the perspective of Definition 1.2 is a direct consequence of its validity in the perspective of Definition 1.1. The proof of the SpinPin 1 property for the perspective of Definition 1.3 is fundamentally different. Example 1.9. The real tautological line bundle γR ÝÑ RP8 admits two Pin` -structures in the sense of Definition 1.1, up to equivalence. Since OpγR q “ S 8 is simply connected, the domain Pin` pV q of qV is Z2 ˆS 8 and qV : Pin` pV q “ Z2 ˆS 8 ÝÑ OpγR q “ S 8 is the projection to the second factor. This map is equivariant with respect to the homomorphism (1.34) if the first Z2 factor in Pin` p1q acts by addition on the first factor of Pin` pV q, leaving the second factor of Pin` pV q fixed, and the non-trivial element 1Z2 of the second Z2 factor in Pin` p1q acts by the antipodal map on the second factor

Main Results and Examples of Part I

17

of Pin` pV q, either leaving the first factor fixed or interchanging its two elements. Since any automorphism of the bundle Pin` pV q “ Z2 ˆS 8 ÝÑ RP8 is equivariant with respect to either Pin` p1q-action, the two Pin` structures are not equivalent. Example 1.10. The line bundle γR ÝÑ RP8 does not admit a Pin´ structure in the sense of Definition 1.1. Since OpγR q “ S 8 is simply connected, the domain Pin´ pV q of qV would have to be Z2ˆS 8 . The restriction of qV to each connected component of Pin´ pV q would then have to be a homeomorphism onto S 8 and equivariant with respect to the Z2 subgroup of Pin´ p1q « Z4 preserving each connected component of Pin´ pV q. This is impossible because this subgroup lies in the kernel of the homomorphism (1.32). Example 1.11. The real tautological line bundle γR;1 ÝÑ RP1 admits two Pin´ -structures in the sense of Definition 1.1. By the reasoning in Example 1.10, the projection qV is the connected double cover ` ˘ qV : Pin´ γR;1 “ S 1 ÝÑ OpγR;1 q “ S 1 , qV pwq “ w2 @ w P S 1 Ă C. This map is equivariant with respect to the homomorphism (1.32) if 1Z4 P Z4 acts by the multiplication by either eiπ{2 or e´iπ{2 . Since any automorphism of the bundle ` ˘ Pin´ γR;1 “ S 1 ÝÑ RP1 is a multiplication by a power of eiπ{2 , the two Pin´ -structures are ´ not equivalent. We denote by p´ 0 pγR;1 q the Pin -structure on γR;1 in which 1Z4 P Z4 acts on Pin´ pγR;1 q by the multiplication by eiπ{2 and ´ by p´ 1 pγR;1 q the other Pin -structure on γR;1 . Example 1.12. The complex tautological line bundle γC ÝÑ CP8 does not admit a Spin-structure in the sense of Definition 1.1. Since Spinp2q “ S 1 and SOpγC q “ S 8 are path-connected, the domain SpinpV q of qV would have to be path-connected. However, this is impossible because SOpγC q is simply connected and thus does not admit any path-connected double covers.

18

Spin/Pin-Structures & Real Enumerative Geometry

The perspectives of Definitions 1.2 and 1.3 on Spin- and Pin˘ structures make it feasible to specify them explicitly in some cases central to the open and real sectors of Gromov–Witten theory, mirror symmetry, and enumerative geometry. This is indicated by the usage of the following examples later in this monograph. Specifications of topological data on a target manifold analogous to the description of Example 1.16 lead to explicit determinations of orientations of moduli spaces of open and real maps and signed counts of curves in [11, Sections 5,6] and [21, Section 5]; such computations are notoriously difficult to carry out correctly. Example 1.13. Let V be a vector bundle over a topological space Y : (a) The vector bundle V ‘V has a canonical orientation. Its restriction to the fiber Vy ‘Vy of V ‘V over a point y P Y is obtained by choosing any orientation in the first copy of Vy and then taking the same orientation in the second copy. (b) If V is oriented, then the oriented vector bundle V ‘ V has a canonical Spin-structure in the sense of Definition 1.3. The canonical homotopy class of trivializations of V ‘V over a loop α in Y is obtained by choosing any trivialization of the first copy of α˚ V and then taking the same trivialization of the second copy. The standard metric on Rn`1 determines an identification of the real tautological line bundle ` ˘ ( γR;n ” , v P RPn ˆRn`1 : v P  Ă Rn`1 ˚ . This identification and Euler’s exact over RPn with its dual γR;n sequence, f

g

˚ ÝÑ T pRPn q ÝÑ 0, 0 ÝÑ τRPn ÝÑ pn`1qγR;n

(1.35)

of vector bundles over RPn determine a canonical homotopy class of isomorphisms τRPn ‘ T pRPn q « pn`1qγR;n

(1.36)

of vector bundles over RPn . The bundle homomorphisms in (1.35) can be described explicitly as in the proof of [21, Lemma 2.1]. Example 1.14. We denote by Bθ the vector field on RP1 ” S 1 {Z2 induced by the vector field p´y, xq on R2 and by oT RP1 the

Main Results and Examples of Part I

19

induced orientation of RP1 . Under the standard identification of γR;1 ˚ , the homomorphisms in the n “ 1 case of (1.35) are given by with γR;1 ˙ ˆ ` ˘ xpx, yq ypx, yq f rx, ys, a “ rx, ys, a 2 2 , a 2 2 x `y x `y ` ˘ 1 2 2 P RP ˆR ˆR @ rx, ys, a P RP1 ˆR, ˇ ˘ ` xpxx1 `yy1 q´ypxx0 `yy0 q ˇˇ Bθ ˇ . g rx, ys, px0 , y0 q, px1 , y1 q “ x2 `y 2 rx,ys In particular,

` ˘ ` ˘ f r1, 0s, 1 “ r1, 0s, p1, 0q, p0, 0q ˇ ` ˘ g r1, 0s, p0, 0q, p1, 0q “ Bθ ˇr1,0s .

and

Thus, the orientation on 2γR;1 induced by the orientations oRP1 of τRP1 and oT RP1 of RP1 is the canonical orientation o´ γR;1 provided by Example 1.13(a). Example 1.15. Under the standard identification R2 “ C, the bundle homomorphism ` ˘ ` ˘ , c1 , c2 ÝÑ , c1 `ic2 , (1.37) Φ0 : 2γR;1 ÝÑ RP1 ˆC, is a trivialization of the vector bundle 2γR;1 over RP1 ; this is the trivialization [19, (5.11)]. The homotopy class of the trivialization id‘Φ0 ´ of τRP1 ‘2γR;1 is a Spin-structure os0 p2γR;1 , o´ γR;1 q on p2γR;1 , oγR;1 q in the perspective of Definition 1.3 and a Pin´ -structure on γR;1 in this perspective. By the SpinPin 2(b) property, the group H 1 pRP1 ; Z2 q acts freely and transitively on the set Spp2γR;1 , o´ γR;1 q of Spinq. There is thus one other Spin-structure structures on p2γR;1 , o´ γR;1 q on this oriented vector bundle. Example 5.1 idenos1 p2γR;1 , o´ γR;1 ´ ´ tifies os0 p2γR;1 , oγR;1 q and os1 p2γR;1 , o´ γR;1 q with the Pin -structures ´ p´ 0 pγR;1 q and p1 pγR;1 q, respectively, of Example 1.11. Example 1.16. By the SpinPin 5 and 6 properties and n “ 2 case of (1.36), the Pin´ -structures on (the tangent bundle of) RP2 correspond to the Spin-structures on the canonically oriented vector bundle ˘ ` ˘ ` b3 . StpT pRP2 qq ´ ” τRP2 ‘T pRP2 q ‘ λ τRP2 ‘T pRP2 q « 3γR;2 ‘ γR;2

20

Spin/Pin-Structures & Real Enumerative Geometry

b2 Since the line bundle γR;2 has a canonical trivialization, there is thus a canonical homotopy class of isomorphisms ` ˘ τRP2 ‘T pRP2 q ‘ λ τRP2 ‘T pRP2 q « 4γR;2 .

By the conclusion of Example 1.14, the orientation on 4γR;2 induced via this homotopy class of isomorphism is the orientation o` γR;2 provided by Example 1.13(a). Example 1.13(b) endows this oriented vector bundle with a canonical OSpin-structure os0 p4γR;2 , o` γR;2 q in the sense of Definition 1.3. By the SpinPin 2 property, p4γR;2 , o` γR;2 q ` admits precisely one other Spin-structure os1 p4γR;2 , oγR;2 q. The associated homotopy class of trivializations over a loop α in RP2 is obtained by combining trivializations of the two copies of α˚ p2γR;2 q in α˚ p4γR;2 q that differ by one rotation. By Corollary 11.14, the ´ 2 2 2 Pin´ -structures p´ 0 pRP q and p1 pRP q on T RP associated with ` ` os0 p4γR;2 , oγR;2 q and os1 p4γR;2 , oγR;2 q, respectively, via (1.20) are fixed by every automorphism of RP2 . Example 1.17. We denote by os0 p4γR;3 , o` γR;3 q the canonical OSpinstructure on the vector bundle 4γR;3 over RP3 in the sense of Definition 1.3 provided by Example 1.13(b). By the SpinPin 2 property, p4γR;3 , o` γR;3 q admits precisely one other Spin-structure ` os1 p4γR;3 , oγR;3 q. By the n “ 3 case of (1.36), these two OSpinstructures correspond to OSpin-structures on RP3 ; we denote the latter by os0 pRP3 q and os1 pRP3 q, respectively. 1.4

Further Examples

We now combine Examples 1.7–1.12 with the properties of Spin- and Pin-structures collected in Section 1.2 to explore these structures on real line bundles more systematically. As described in Section 1.1, an orientation o P OpV q on a line bundle V over a topological space Y determines an element s0 pV, oq P ˘ SppV, oq. We denote its preimage under (1.15) by p˘ 0 pV q P P pV q. ˘ By the identity after (1.15), p0 pV q does not depend on the choice of o. By Example 1.9, the real tautological line bundle γR ÝÑ RP8 admits two Pin` -structures. By [34, Theorem 5.6], for every real line bundle V over a paracompact space Y , there exists a continuous

Main Results and Examples of Part I

21

map f : Y ÝÑ RP8 such that V « f ˚ γR ; this map is unique up to homotopy. By (1.13), the Pin` -structure on V induced from a Pin` structure on γR does not depend on the choice of this isomorphism. Lemma 1.18. Let V be a line bundle over a topological space Y and o P OpV q. If f : Y ÝÑ RP8 is a continuous map such that V « f ˚ γR , then the Pin` -structure p` 0 pV q on V corresponding to the canonical Spin-structure s0 pV, oq on pV, oq is the pullback by f of either Pin` structure on γR . Proof. Let q : S 8 ÝÑ RP8 be the standard quotient projection. Since S 8 is simply connected, τS 8 ” S 8 ˆR “ q ˚ γR ÝÑ S 8 . Since V is orientable, there exists a continuous map fr: Y ÝÑ S 8 so that q˝ fr is homotopic to f and ` ˘ pV, oq « fr˚ τS 8 , oS 8 ÝÑ Y as oriented line bundles over Y . The canonical Spin-structure s0 pV, oq on V is the pullback by fr of s0 pτS 8 , oS 8 q. By the SpinPin 2(b) property (or Example 1.8), there are no other Spin-structures on τS 8 . By the SpinPin 4 property, the pullback by q of either Pin` -structure on γR is thus the Pin` -structure p` 0 pτR q on τS 8 corresponding to s0 pτS 8 , oS 8 q. Since the pullback by f of a Pin` -structure p on γR is the pullback by fr of the pullback of q of p, the claim now follows from the naturality of the correspondence of the SpinPin 4 property  with respect to continuous maps. Lemma 1.19. A real line bundle V over a topological space Y carries a canonical Pin` -structure p` 0 pV q. For every unorientable line bundle V over an oriented circle Y , there is a natural bijection

P´ pV q ÝÑ Z2 ;

(1.38)

reversing the orientation of Y flips this bijection. Proof. By the SpinPin 1(a) and 2(a) properties (or Example 1.9), there are two Pin` -structure on the real tautological line bundle γR over RP8 . In the perspective of Definition 1.1 and Example 1.9, we take the one in which the non-trivial element 1Z2 of the second Z2

22

Spin/Pin-Structures & Real Enumerative Geometry

factor in Pin` p1q fixes the topological components of Pin` pV q to be the canonical Pin` -structure p` 0 pγR q on γR . Given a real line bundle V over a paracompact space Y , let f : Y ÝÑ RP8 be a continuous map such that V « f ˚ γR ; this map is unique up to homotopy. We ` take the canonical equivalence class p` 0 pV q of Pin -structures on V ` to be the pullback by f of p0 pγR q in the perspective of Definition 1.1. By (1.13), p` 0 pV q does not depend on the choice of isomorphism of V with f ˚ γR . This also specifies p` 0 pγR q in the perspective of Definition 1.3 for an arbitrary topological space Y . We view RP1 as the quotient of the unit circle S 1 Ă C by the antipodal map. By the SpinPin 1(a) and 2(a) properties (or Example 1.11), there are two Pin´ -structures on the real tautological line bundle γR;1 over RP1 . An oriented circle Y can be identified with RP1 as oriented manifold; such an identification is unique up to homotopy. An unorientable line bundle V over RP1 is isomorphic to γR;1 ; every two such isomorphisms differ by an automorphism of γR;1 . By (1.13), it thus remains to show that the orientation-reversing map f : RP1 ÝÑ RP1 ,

` ˘ “ ‰ f rzs “ z @ z P S 1 ,

interchanges the two Pin´ -structures on γR;1 . Under the standard identifications RP1 “ S 1 {Z2 , S 1 Ă C, and γR;1 Ă RP1 ˆC,  ( f ˚ γR;1 “ przs, vq P RP1ˆC : v P Rz , Opf ˚ γR;1 q “ f ˚ OpγR;1 q  ( “ przs, vq P RP1ˆC : v P t˘zu ,  ( π2˚ Pin´ pγR;1 q “ przs, v, wq P RP1ˆCˆS 1 : v P t˘zu, w2 “ v , where π2 : f ˚ OpγR;1 q ÝÑ OpγR;1 q is the projection to the second component. The map F : f ˚ γR;1 ÝÑ γR;1 ,

` ˘ ` ˘ F rzs, v “ rzs, v ,

is an isomorphism of line bundles over RP1 and ( (˚ ` ˘   F |Opf ˚ γR;1 q Pin´ γR;1 ” przs, v, wq P RP1ˆCˆS 1: v P t˘zu, w2 “ v .

Main Results and Examples of Part I

23

The map (˚ ` ˘  Ψ : π2˚ Pin´ pγR;1 q ÝÑ F |Opf ˚ γR;1 q Pin´ γR;1 , ˘ ` ˘ ` Ψ rzs, v, w “ rzs, v, w , is an isomorphism of Z4 -bundles over RP1 . This map is Z4 equivariant if 1Z4 P Z4 acts on Pin´ pγR;1 q in the domain of Ψ by the multiplication by eiπ{2 and on Pin´ pγR;1 q in the target of Ψ by the multiplication by e´iπ{2 . Thus, the pullback by f interchanges the two  Pin´ -structures on γR;1 . Corollary 1.20. Let V be an unorientable line bundle over S 1 . If f : S 1 ÝÑ S 1 is a continuous map of even degree, then the pullbacks by f of the two Pin´ -structures on V are the same. If the degree of f ˚ is divisible by 4, then these pullbacks are the Pin´ -structure p´ 0 pf V q ˚ ˚ corresponding to s0 pf V, oq for any o P Opf V q. If the degree of f is even, but not divisible by 4, then this is not the case. Proof. Let f : S 1 ÝÑ S 1 be a continuous map of even degree. We can assume that it is given by f : S 1 ÝÑ S 1 ,

z ÝÑ z 2d ,

for some d P Z. By Example 1.7, we can assume that V “ γR;1 . By Examples 1.11 and 1.8, the pullback by f of a Pin´ -structure on V ˚ is p´ 0 pf V q if and only if the total space of the cover ` ˘  ( f ˚ Pin´ γR;1 ” pz, wq P S 1 ˆS 1 : z 2d “ w4 ÝÑ S 1 , pz, wq ÝÑ z, has four connected components. This is the case if and only if d is  even. Remark 1.21. By the first statement of Lemma 1.19 and the definition above Lemma 1.18, the canonical equivalence classes of Pin` structures on real line bundles and of Pin´ - and Spin-structures on orientable line bundles are preserved by the pullbacks by continuous maps. By the second statement of Lemma 1.19 and Corollary 1.20, a continuous map f from S 1 to itself need not pull back the canonical ´ equivalence class p´ 0 pV q of Pin -structures on an unorientable line ˚ bundle V (i.e. the class corresponding to 0 P Z2 ) to p´ 0 pf V q.

24

Spin/Pin-Structures & Real Enumerative Geometry

Remark 1.22. By the second statement of Lemma 1.19, the second claim of [39, Lemma 8.2] is not correct in the Pin´ -case: there is no canonical way to choose a Pin´ -structure on an unorientable real line bundle V over a paracompact space Y that admits such a structure. The existence of such a way would have implied that every real line bundle admits a Pin´ -structure; this is not the case by Example 1.10. However, there is a canonical way to choose a Pin´ -structure on an unorientable real line bundle V over an oriented circle Y . This is analogous to the phenomenon described by [19, Corollary 5.6]. Example 1.23. The projection ` ˘L πV : V “ r0, 1sˆR „ÝÑ RP1 “ S 1 {Z2 , ` ˘ “ ‰ p0, ´aq „ p1, aq, πV rt, as “ eπit , defines a real line bundle over RP1 . The associated orthogonal frame bundle is ` ˘L ˘ “ ‰ ` OpV q “ r0, 1sˆZ2 „ÝÑ RP1 , 0, a`1Z2 „ p1, aq, rt, as ÝÑ eπit . The two Pin` -structures on V in the perspective of Definition 1.1 are ` ˘L 2 qV`;0 : Pin` 0 pV q ” r0, 1sˆZ2 „ÝÑ OpV q, ` ˘ ` ˘ ` ˘ 0, pa, b`1q „ 1, pa, bq , qV`;0 rt, pa, bqs “ rt, bs, ˘L ` 2 qV`;1 : Pin` 1 pV q ” r0, 1sˆZ2 „ÝÑ OpV q, ` ˘ ` ˘ ` ˘ 0, pa`1, b`1q „ 1, pa, bq , qV`;1 rt, pa, bqs “ rt, bs. The group Pin` p1q acts on the two bundles via the addition on the Z22 -factor, making the projections qV`;0 and qV`;1 above equivariant with respect to the homomorphism (1.34). Since the action of rI1;1 ” p0, 1Z2 q preserves the connected components of Pin` 0 pV q, this bundle ` ` provides the canonical Pin -structure p0 pV q on V of Lemma 1.19. Example 1.24. Let V be as in Example 1.23. The two Pin´ structures on V in the perspective of Definition 1.1 are ` ˘L ` ˘ qV´;0 : Pin´ 0 pV q ” r0, 1sˆZ4 „ÝÑ OpV q, 0, a`1Z4 „ p1, aq, ` ˘ qV´;0 rt, as “ rt, as,

Main Results and Examples of Part I

25

` ˘L ` ˘ qV´;1 : Pin´ 1 pV q ” r0, 1sˆZ4 „ÝÑ OpV q, 0, a´1Z4 „ p1, aq, ` ˘ qV´;1 rt, as “ rt, as. (1.39) The group Pin´ p1q acts on Pin´ pV q via the addition on the Z4 factor, making the projections qV´;0 and qV´;1 above equivariant with respect to the homomorphism (1.32). The Pin´ -structures on the real tautological line bundle γR;1 over RP1 are described in Example 1.11. The map ` ˘ ` ˘ F rt, as “ reπit s, aeπit , (1.40) F : V ÝÑ γR;1 , is an isomorphism of line bundles over RP1 . The induced isomorphisms between the orthogonal frame bundles and Pin´ -structures of V and γR;1 are given by ˘ ` ˘ ` ´ OpV q ÝÑ O γR;1 , Pin´ 0 pV q ÝÑ Pin0 γR;1 , ` ˘ ´ Pin´ 1 pV q ÝÑ Pin1 γR;1 , rt, as ÝÑ p´1qa eπit ,

rt, as ÝÑ eπia{2 eπit{2 ,

rt, as ÝÑ e´πia{2 eπit{2 . The middle and last maps above are equivariant with respect to the Z4 -actions on the right-hand sides with rI1;1 ” 1Z4 acting by the multiplication by eiπ{2 and e´iπ{2 , respectively. Thus, the first Pin´ structure on V in (1.39) corresponds to 0 P Z2 in the sense of Lemma 1.19. Example 1.25. Let Y be the infinite Mobius band, i.e. the total space of the tautological real line bundle γR;1 over RP1 , and SγR;1 Ă Y be its unit circle bundle. The bundle projection π : Y ÝÑ RP1 induces an exact sequence dπ

0 ÝÑ π ˚ γR;1 ÝÑ T Y ÝÑ π ˚ T RP1 ÝÑ 0

(1.41)

of vector bundles over Y and lifts an orientation oT RP1 on RP1 to an orientation oT SγR;1 of SγR;1 . The inclusion of SγR;1 into Y induces an exact sequence ˇ ÝÑ 0 . (1.42) 0 ÝÑ T pSγR;1 q ÝÑ T Y |Sγ ÝÑ π ˚ γR;1 ˇ R;1

SγR;1

We denote by os0 pRP1 , oT RP1 q the distinguished OSpin-structure on T RP1 determined by oT RP1 as in Example 1.8 and by

26

Spin/Pin-Structures & Real Enumerative Geometry

os0 pSγR;1 , oT SγR;1 q the distinguished OSpin-structure on T pSγR;1 q determined by oT SγR;1 . Let @@ ˚ ` DD ˚ 1 p` 0 ” π p0 pγR;1 q, π os0 pRP , oT RP1 q p1.41q be the Pin` -structure on T Y induced by p` 0 pγR;1 q and os0 pRP1 , oT RP1 q via the exact sequence (1.41). By the naturality of the Spin-structure s0 pV, oq of Example 1.8 and the Pin` structure p` 0 pV q of Lemma 1.19, ˘ˇ ` ˘ ` os0 SγR;1 , oT SγR;1 “ π ˚ os0 pRP1 , oT RP1 q ˇSγ and R;1 ˘ˇ ` ˘ ` ˚ ˚ ` ˇ p` 0 π γR;1 |SγR;1 “ π p0 pγR;1 q Sγ . R;1

Along with the definition and naturality of the map (1.30), this gives ˇ @@ ` ˚ ˘DD ˇ “ os0 pSγR;1 , oSγR;1 q, p` . (1.43) p` 0 Sγ 0 π γR;1 |SγR;1 p1.42q R;1

Chapter 2

The Lie Groups Spin(n) and Pin˘(n)

In this chapter, we describe the Lie groups Spinpnq and Pin˘ pnq and key homomorphisms between them from purely topological considerations. These homomorphisms are used to establish the properties of Section 1.2 for the perspective of Definition 1.1 in Chapter 3. Let m, n P Z` with m ă n. The standard identification of Rm ˆ n´m with Rn induces inclusions R ιn;m : SOpmqˆSOpn´mq ÝÑ SOpnq and ιn;m : SOpmqˆOpn´mq ÝÑ Opnq

(2.1)

so that the diagram

commutes; the homomorphisms to Z2 ” Z{2Z above are given by the sign of the determinant. We denote by ι1n;m : SOpmq ÝÑ SOpnq,

ι2n;n´m : SOpn´mq ÝÑ SOpnq,

ι2n;n´m : Opn´mq ÝÑ Opnq, the compositions of the maps in (2.1) with the canonical inclusions of SOpmq, SOpn´mq, and Opn´mq in the domains of these maps.

27

28

Spin/Pin-Structures & Real Enumerative Geometry

We note that SOp1q “ t1u, 1

SOp2q “ S ,

Op1q “ t˘1u, SOp3q « RP3 ,

|π0 pSOpnqq|, |π2 pSOpnqq| “ 1 @ n ě 1, " Z, if n “ 2; π1 pSOpnqq “ Z2 , if n ě 3.

(2.2)

The π2 statement above and the n ě 3 case of the π1 statement are obtained by induction from the homotopy exact sequence for the fibration ι1n`1;n

SOpnq ÝÝÝÝÑ SOpn`1q ÝÑ S n .

(2.3)

The induced homomorphism ι1n`1;n ˚ : π1 pSOpnqq ÝÑ π1 pSOpn`1qq

(2.4)

is an isomorphism for n ě 3 and is a surjection for n “ 2. 2.1

The Groups SO(n) and Spin(n)

By the last statement in (2.2), for every n ě 2 there exists a unique 2 : 1 covering projection qn : Spinpnq ÝÑ SOpnq

(2.5)

from a connected Lie group. This defines the Lie group Spinpnq and determines an exact sequence qn

t1u ÝÑ Z2 ÝÑ Spinpnq ÝÑ SOpnq ÝÑ t1u

(2.6)

of Lie groups; see Lemma B.1(a). The preimage of the identity In P SOpnq under (2.5) consists of the identity rIn P Spinpnq and an element pIn P Spinpnq of order 2. Since the inclusion of Z2 in (2.6) sends its non-trivial element 1Z2 to pIn P Spinpnq, r“A rpIn pIn A

r P Spinpnq. @A

(2.7)

Example 2.1. Let SUpnq denote the n-th special unitary group. We show that q3 : Spinp3q “ SUp2q « S 3 ÝÑ SOp3q « RP3 .

(2.8)

The Lie Groups Spinpnq and Pin˘ pnq

29

The subspace S 3 Ă C2 inherits a Lie group structure from the quaternion multiplication on H “ C‘jC. The map ˙ ˆ a ´b 3 ÝÑ a ` jb, SUp2q ÝÑ S , a b is a Lie group isomorphism; it intertwines the standard action of SUp2q on C2 with the action of S 3 on H by the multiplication on the left. The map HˆH ÝÑ H,

pu, xq ÝÑ Adu pxq ” uxu,

is linear in the second input x and restricts to an action of S 3 Ă H by isometries on the subspace  ( Im H ” x P H : Re x “ 0 « R3 of H. The action of the differential of this restriction at u “ 1 P S 3 is the homomorphism ` ˘ ad : Im H ÝÑ EndR Im H , v ÝÑ adv , adv pxq “ 2 Impvxq. Since this homomorphism is injective, the Lie group homomorphism Ad : S 3 ÝÑ SOp3q induced by the action of S 3 on Im H is a covering projection. Let m P Z` with m ă n. By the last statement in (2.2) and the surjectivity of the homomorphism (2.4) for n ě 2, the composition qm ˆqn´m

ιn;m

SpinpmqˆSpinpn´mq ÝÝÝÝÝÝÑ SOpmqˆSOpn´mq ÝÝÝÑ SOpnq induces the trivial homomorphism on the fundamental groups. If m, n ´ m ě 2, Lemma B.1(b) implies that the first embedding in (2.1) lifts uniquely to a Lie group homomorphism r ιn;m so that the diagram

(2.9) commutes. Since the homomorphisms on the fundamental groups induced by ι1n;m and ι2n;n´m surject onto π1 pSOpnqq « Z2 in this case, ` ˘ ` ˘ (2.10) ιn;m pIm , rIn´m “ pIn . r ιn;m rIm , pIn´m , r

30

Spin/Pin-Structures & Real Enumerative Geometry

If m “ 1 or n ´ m “ 1, Lemma B.1(b) implies the existence of a lift r ιn;m of ιn;m to the identity component of SpinpmqˆSpinpn´mq. The condition (2.10) and the property (2.7) extend it over the remaining component(s). We denote by r ι 1n;m : Spinpmq ÝÑ Spinpnq

and r ι 2n;n´m : Spinpn´mq ÝÑ Spinpnq

the compositions of r ιn;m with the canonical inclusions of Spinpmq and Spinpn ´ mq in the domain of r ιn;m . By (2.10), these maps are embeddings. Let In;2 P SOpnq for n ě 2 and In;4 P SOpnq for n ě 4 be the diagonal matrices satisfying # # ˘ ˘ ` 1, if i ď n´2; ` 1, if i ď n´4; In;4 ii “ In;2 ii “ ´1, otherwise; ´1, otherwise. (2.11) We denote the two elements of the preimage of In;2 under (2.5) by rIn;2 and pIn;2 and the two elements of the preimage of In;4 by rIn;4 and pIn;4 . If m P Z` and m ă n, ˘  ` ( r ι 2n;m trIm;2 , pIm;2 u “ rIn;2 , pIn;2 if m ě 2, ˘  ` ( r ι 2n;m trIm;4 , pIm;4 u “ rIn;4 , pIn;4 if m ě 4. (2.12) Since the multiplication by pIn is the deck transformation of the double cover (2.5),  ( rI2 “ pI2 P rIn , pIn , pIn;2 “ pInrIn;2 “ rIn;2pIn , n;2 n;2 (  rI2 “ pI2 P rIn , pIn . pIn;4 “ pInrIn;4 “ rIn;4pIn , n;4

n;4

By the second statement in (1.31), rI22;2 “ pI2 . Along with (2.12) and (2.10), this implies that ` ˘2 ` ˘ rI2 “ r ι 2n;2 ppI2 q “ pIn @ n ě 2. ι 2n;2 rI2;2 “ r ι 2n;2 rI22;2 “ r n;2 The following lemma gives a direct proof of this statement which readily adapts to show that rI2n;4 “ rIn for all n ě 4. Lemma 2.2. With the notation as above, rI2 “ pI2 “ pIn n;2 n;2

@ ně2

and rI2n;4 “ pI2n;4 “ rIn

@ n ě 4. (2.13)

The Lie Groups Spinpnq and Pin˘ pnq

31

Proof. Let n ě 2, n´ “ n, and n` “ n ` 2. We define a path γ´ in SOpnq from In to In;2 and a path γ` in SOpn ` 2q from In`2 to In`2;4 by γ´ : r0, 1s ÝÑ SOpnq, γ` : r0, 1s ÝÑ SOpn`2q, ¨ ˛ In´2 cospπtq ´ sinpπtq‚, γ´ ptq “ ˝ sinpπtq cospπtq ¨ ˛ γ´ ptq cospπtq ´ sinpπtq‚. γ` ptq “ ˝ sinpπtq cospπtq The endpoints of the lifts r` : r0, 1s ÝÑ Spinpn`2q γ r´ : r0, 1s ÝÑ Spinpnq and γ of these paths such that γ r˘ p0q “ rIn˘ lie in trIn;2 , pIn;2 u and trIn`2;4 , pIn`2;4 u, respectively. Let ` ˘ α˘ ” γ˘ ˚ γ˘ p1qγ˘ : r0, 1s ÝÑ SOpn˘ q and ˘ ` r˘ ˚ γ r˘ p1qr γ˘ : r0, 1s ÝÑ Spinpn˘ q α r˘ ” γ r˘ with the be the concatenations (products) of the paths γ˘ and γ paths  ( γ˘ p1qγ˘ ptq “ γ˘ p1qγ˘ ptq, and γ˘ p1qγ˘ : r0, 1s ÝÑ SOpn˘ q, (  r˘ p1qr γ˘ : r0, 1s ÝÑ Spinpn˘ q, γ r˘ p1qr γ˘ ptq “ γ γ˘ ptq, γ˘ p1qr r respectively. In particular, rI2 “ pI2 “ α r´ p1q, n;2 n;2

rI2 p2 r` p1q. n`2;4 “ In`2;4 “ α

(2.14)

By the homotopy exact sequences for the fibrations (2.3), the path ˛ ¨ In´2 cosp2πtq ´sinp2πtq‚, α´ : r0, 1s ÝÑ SOpnq, α´ ptq “ ˝ sinp2πtq cosp2πtq r´ is not a loop and so is a loop generating π1 pSOpnqq. Thus, its lift α r α r´ p1q ‰ In . Along with the first equation in (2.14), this establishes the first claim in (2.13).

32

Spin/Pin-Structures & Real Enumerative Geometry

By the homotopy exact sequences for the fibrations (2.3), the path ¨ ˛ α´ ptq cosp2πtq ´sinp2πtq‚, α` : r0, 1s ÝÑ SOpn`2q, α` ptq “ ˝ sinp2πtq cosp2πtq is a loop representing twice the generator of π1 pSOpn ` 2qq « Z2 . Thus, its lift α r` is a loop and so α r` p1q “ rIn`2 . Along with the  second equation in (2.14), this establishes the second claim in (2.13). For concreteness, we take rIn;2 ” γ r´ p1q P Spinpnq,

rIn`2;4 ” γ r` p1q P Spinpn`2q,

(2.15)

with the notation as in the proof of Lemma 2.2. With these choices, ` ˘ ι 2n;m prIm;4 q “ rIn;4 if 4 ď m ď n. r ι 2n;m rIm;2 “ rIn;2 if 2 ď m ď n and r (2.16) Let m1 , m2 P Z` with m1 , m2 ď n In;pm1 ,m2 q P SOpnq the diagonal matrix # ˘ ` ´1, In;pm1 ,m2 q ii “ 1,

and m1 ‰ m2 . We denote by satisfying if i “ m1 , m2 ; otherwise;

(2.17)

and by rIn;pm1 ,m2 q , pIn;pm1 ,m2 q P Spinpnq its two preimages under (2.26). By symmetry and Lemma 2.2 (or directly by its proof), p2 p rI2 n;pm1 ,m2 q , In;pm1 ,m2 q “ In . If in addition m P Z` with m1 , m2 ď m ă n, then ˘  ` ( r ι 1n;m trIm;pm1 ,m2 q , pIm;pm1 ,m2 q u “ rIn;pm1 ,m2 q , pIn;pm1 ,m2 q .

(2.18)

(2.19)

For B P Opnq, let cpBq : SOpnq ÝÑ SOpnq,

cpBqA “ BAB ´1 ,

be the conjugation action of B on SOpnq. By Lemma B.1(b), it lifts uniquely to an isomorphism rcpBq : Spinpnq ÝÑ Spinpnq.

The Lie Groups Spinpnq and Pin˘ pnq

33

In particular, rcp´In q “ id : Spinpnq ÝÑ Spinpnq.

(2.20)

Z`

with m ď n, we denote by In;pmq P Opnq the diagonal For m P matrix satisfying # ˘ ` ´1, if i “ m; (2.21) In;pmq ii “ 1, otherwise. Let In;1 “ In;pnq . Corollary 2.3. If m, m1 , m2 P Z` with m, m1 , m2 ď n and m1 ‰ m2 , then # rI , if m R tm1 , m2 u; rcpIn;pmq qrIn;pm1 ,m2 q “ rIn;pm1 ,m2 q ¨ n (2.22) pIn , if m P tm1 , m2 u. Proof. By symmetry, we can assume that m1 “ m2 ´1 and m2 ď m “ n. If m2 ă n, the group homomorphisms ` ˘ ` ˘ r ,rcpIn;1 qr r , r ÝÑ r A ι 1n;m2 A ι 1n;m2 A Spinpm2 q ÝÑ Spinpnq, lift ι1n;m2 . Lemma B.1(b) thus implies that ` ˘ ` ˘ r “r r r P Spinpm2 q. rcpIn;1 qr ι 1n;m2 A ι 1n;m2 A @A

(2.23)

Along with (2.19), this implies that ` ˘ ` ˘ rcpIn;1 qrIn;pm1 ,m2 q “ rcpIn;1 qr ι 1n;m2 rIm2 ;pm1 ,m2 q ι 1n;m2 rIm2 ;pm1 ,m2 q “ r “ rIm2 ;pm1 ,m2 q . This establishes the first case in (2.22). Let γ´ : r0, 1s ÝÑ SOpnq be as in the proof of Lemma 2.2. The path cpIn;1 qγ´ : r0, 1s ÝÑ SOpnq, ¨ ˛ In´2 cospπtq sinpπtq ‚, cpIn;1 qγ´ ptq “ ˝ ´ sinpπtq cospπtq r´ of the loop is the reverse of the path γ´ p1qγ´ . Since the lift α α´ ” γ´ ˚ pγ´ p1qγ´ q to Spinpnq is not a loop, the endpoints of the lifts ` ˘ ` ˘ γ´ : r0, 1s, 0 ÝÑ Spinpnq, rIn γ r´ ,rcpIn;1 qr

34

Spin/Pin-Structures & Real Enumerative Geometry

of γ´ and cpIn;1 qγ´ are different points of the preimage trIn;2 , pIn;2 u  of In;2 under (2.5). This establishes the second case in (2.22). Let m P Z` with m ă n. Since Spinpmq is connected if m ě 2 and the group homomorphisms ` ` ˘ ˘ r , r , rcpIn;pmq qr r ÝÑ r ι 1n;m A Spinpmq ÝÑ Spinpnq, A ι 1n;m rcpIm;1 qA lift cpIn;pmq qι1n;m , they agree; Lemma B.1(b). Thus, ˘ ` ` ˘ r “ rcpIn;pmq qr r r P Spinpmq. r ι 1n;m rcpIm;1 qA ι 1n;m A @A

(2.24)

Since Spinpmq is connected for m ě 2 and the group homomorphisms ` ` ˘ ˘ r , r , rcpIn;1 qr r ÝÑ r ι 2n;m A Spinpmq ÝÑ Spinpnq, A ι 2n;m rcpIm;1 qA lift cpIn;1 qι2n;m , the two homomorphisms agree. Since the group homomorphisms ` ˘ ` ˘ r , rcpIn;pmq qr r , r ÝÑ r ι 2n;m A Spinpmq ÝÑ Spinpnq, A ι 2n;m A lift ι2n;m , they also agree. Thus, ` ` ˘ ˘ r , r “ rcpIn;1 qr ι 2n;m A r ι 2n;m rcpIm;1 qA ` ˘ ` ˘ r “ rcpIn;pmq qr r ι 2n;m A r ι 2n;m A

r P Spinpmq. (2.25) @A

The m “ 1 cases of (2.24) and (2.25) follow from (2.10). 2.2

The groups O(n) and Pin˘ (n)

The group Opnq is isomorphic to the semi-direct product SOpnq¸Z2 with the action of the non-trivial element 1Z2 of Z2 on SOpnq given by the conjugation by In;pmq , for any fixed choice of m, or by any other order 2 element of the non-identity component of Opnq. Example 2.4. As a pointed smooth manifold, pOp2q, I2 q is diffeomorphic to pS 1 ˆZ2 , 1ˆ0q. However, the group structure of Op2q is given by ˘ ` ˘ ` ˘ ` R{2πZˆZ2 ˆ R{2πZˆZ2 ÝÑ R{2πZ, Z2 , ` ˘ pθ1 , k1 q¨pθ2 , k2 q “ θ1 `p´1qk1 θ2 , k1 `k2 .

The Lie Groups Spinpnq and Pin˘ pnq

35

By the last statement in (2.2), for every n ě 2 there exists a topologically unique double cover qn˘ : Pin˘ pnq ÝÑ Opnq

(2.26)

with two path components that restricts to (2.5) over SOpnq Ă Opnq. However, there are two Lie group structures on Pinpnq, denoted by Pin˘ pnq, so that the diagram t1u t1u

/ Z2 

id

/ Z2

/ Spinpnq qn

/ SOpnq

/ t1u

 ˘ / Pin˘ pnq qn

 / Opnq

/ t1u

of Lie group homomorphisms commutes; the inclusions of Z2 above send its non-trivial element 1Z2 to pIn P Spinpnq Ă Pin˘ pnq. The two group structures are described in the following. We denote by rIn;n , pIn;n P Pin˘ pnq the two preimages of In;n ” ´In under (2.26). For m P Z` with m ď n, let rIn;pmq , pIn;pmq P Pin˘ pnq be the two preimages of In;pmq . In particular,  ( ´1 p2 r p rI2 , pI2 , rI2 n;n n;n n;pmq , In;pmq P qn pIn q ” In , In . If m1 , m2 P Z` with m1 , m2 ď n and m1 ‰ m2 , then  (  ( rIn;pm q ¨ rIn;pm ,m q , pIn;pm ,m q , rIn;pm ,m q , pIn;pm ,m q ¨ rIn;pm q 1 1 2 1 2 1 2 1 2 1  ( P rIn;pm2 q , pIn;pm2 q . (2.27) Let rIn;1 “ rIn;pnq and pIn;1 “ pIn;pnq . Since the multiplication by pIn is the deck transformation of the double cover (2.26), r“A rpIn pIn A

r P Pin˘ pnq. @A

(2.28)

Along with (2.27), (2.22), and (2.18), this implies that ` ˘2 ` ˘ p2 r r rI2 “ rcpIn;1 qrIn;pm,nq rI2n;1rIn;pm,nq n;pmq , In;pmq “ In;1 In;pm,nq “ rIn;pm,nqpInrI2n;1rIn;pm,nq “ rI2n;1 , pI2n;1

(2.29)

36

Spin/Pin-Structures & Real Enumerative Geometry

for all m P Z` with m ă n. Along with another application of (2.18), this in turn implies that ` ˘ rIn;1rIn;pmqrI´1 rI´1 “ rIn;1rIn;pmq 2 rI´2 rI´2 “ pIn , n;1 n;pmq n;1 n;pmq rIn;pmqrIn;n “ rIn;nrIn;pmqpI n´1 n

(2.30)

for all m P Z` with m ă n. Every element of Pin˘ pnq ´ Spinpnq can be written uniquely as rrIn;1 with A r P Spinpnq and A ` ` ˘ ˘ rIn;1 A r “ rIn;1 A rrI ´1 rIn;1 “ rcpIn;1 qA r rIn;1 n;1

r P Spinpnq. @A

r P Spinpnq is determined by the adjoint The element rcpIn;1 qA action cpIn;1 q of In;1 on SOpnq. The group multiplication in Pin˘ pnq is thus characterized by whether rI2n;1 equals rIn or pIn or equivalently whether the subgroup of Pin˘ pnq generated by rIn;1 and pIn;1 is Z22 or Z4 . The Lie group Pin` pnq is defined to be the version of Pin˘ pnq in the first case and Pin´ pnq is defined to be the version of Pin˘ pnq in the second case. Since the subspace of order 2 elements of Opnq with precisely one p´1q-eigenvalue is connected, the distinction between Pin` pnq and Pin´ pnq can be equivalently formulated in terms of any such element. Example B.4 provides a more algebraic reason for this. The group Pin` pnq is isomorphic to the semi-direct product Spinpnq ¸ Z2 with the action of the non-trivial element 1Z2 of Z2 on Spinpnq given by the conjugation by rIn;pmq for any m P Z` with m ď n. Example 2.5. With the identification of Op2q given by Example 2.4, the n “ 2 case of the cover (2.26) can be written as q2˘ : Pin˘ p2q “ R{2πZˆZ2 ÝÑ Op2q “ R{2πZˆZ2 ,

q2˘ pθ, kq “ p2θ, kq,

in the smooth manifold category (ignoring the group structures). The group structures are given by Pin` p2qˆPin` p2q ÝÑ Pin` p2q, ˘ ` pθ1 , k1 q¨pθ2 , k2 q “ θ1 `p´1qk1 θ2 , k1 `k2 ,

The Lie Groups Spinpnq and Pin˘ pnq

37

Pin´ p2qˆPin´ p2q ÝÑ Pin´ p2q, ` ˘ pθ1 , k1 q¨pθ2 , k2 q “ θ1 `p´1qk1 θ2 `k1 k2 π, k1 `k2 . In particular, Pin` p2q « Op2q. Since the square of every element in the non-identity component of Pin` p2q (resp. Pin´ p2q) is rI2 (resp. pI2 ), Pin` p2q and Pin´ p2q are not isomorphic as groups (even ignoring the projections to Op2q). Let m P Z` with m ă n. By (2.23) and the first statement in (2.25), ` ˘ ` ˘ rIn;1r r P Spinpmq, r “r r rIn;1 @ A ι 1n;m A ι 1n;m A ` ˘ ` ˘ ´1 rrI´1 r ι2 rr r r ι 2n;n´m rIn´m;1 A n;n´m A In;1 @ A P Spinpn´mq. n´m;1 “ In;1r (2.31) Thus, the homomorphism r ιn;m in (2.9) can be extended to a Lie group homomorphism r ιn;m lifting the second embedding in (2.1) so that the diagram

(2.32) commutes by fixing ` ˘  ( rιn;m rIm , rIn´m;1 P rIn;1 , pIn;1 .

(2.33)

By (2.24), the second statement in (2.25), and the first statement in (2.30), ` ˘ ` 1 ˘ ´1 ` 2˘ r2 “ rIn;pmqr r1rI´1 , A r rI r ι 1n;m A ι 2n;n´m A r ιn;m rIm;1 A m;1 n;pmqr ` 1 2 ˘ ´1 r rI r ,A ιn;m A “ rIn;pmqr n;pmq # 1 2 r ,A r q P SpinpmqˆSpinpn´mq; rIn , if pA ˆ p 1 r ,A r2 q P SpinpmqˆpPin˘ pn´mq´Spinpn´mqq. In , if pA (2.34)

38

Spin/Pin-Structures & Real Enumerative Geometry

There are two possible lifts r ιn;m in (2.32); they differ by the composition with the automorphism ρ : Pin˘ pnq ÝÑ Pin˘ pnq s.t. # r r P Spinpnq; if A r “ A, ρpAq p r P Pin˘ pnq´Spinpnq. r AIn , if A We fix one lift by requiring that ` ˘ r ιn;m rIm , rIn´m;1 “ rIn;1 .

(2.35) (2.36)

We denote by r ι 2n;n´m : Pin˘ pn´mq ÝÑ Pin˘ pnq the composition of r ιn;m with the canonical inclusion of Pin˘ pn´mq in the domain of r ιn;m . Suppose m1 , m2 P Z` with m1 `m2 ă n. By (2.36), ˘ ` ιn´m1 ;m2 prIm2 , rIn´m1 ´m2 ;1 q r ιn;m1 rIm1 , r ` ˘ ιm1 `m2 ;m1 prIm1 , rIm2 q, rIn´m1 ´m2 ;1 . (2.37) “ rιn;m1 `m2 r By Lemma B.1(b),  ( ιn´m1 ;m2 r ιn;m1 ˝ idSpinpm1 q ˆr  ( ιm1 `m2 ;m1 ˆidSpinpn´m1 ´m2 q : “r ιn;m1 `m2 ˝ r Spinpm1 qˆSpinpm2 qˆSpinpn´m1 ´m2 q ÝÑ Spinpnq (2.38) since these two homomorphisms lift the homomorphism (  ιn;m1 ˝ idSOpm1 q ˆιn´m1 ;m2  ( “ ιn;m1 `m2 ˝ ιm1 `m2 ;m1 ˆidSOpn´m1 ´m2 q : SOpm1 qˆSOpm2 qˆSOpn´m1 ´m2 q ÝÑ SOpnq. By (2.38) and (2.37),  ( ιn´m1 ;m2 r ιn;m1 ˝ idSpinpm1 q ˆr  ( ιm1 `m2 ;m1 ˆidPin˘ pn´m1 ´m2 q : “r ιn;m1 `m2 ˝ r Spinpm1 qˆSpinpm2 qˆPin˘ pn´m1 ´m2 q ÝÑ Pin˘ pnq. (2.39)

The Lie Groups Spinpnq and Pin˘ pnq

39

Remark 2.6. The natural embedding of OpmqˆOpn´mq into Opnq does not lift to a Lie group homomorphism r ι : Pin˘ pmqˆPin˘ pn´mq ÝÑ Pin˘ pnq. If such lift existed, it would satisfy ˘  ( ` r ι rIm , rIn´m;1 P rIn;1 , pIn;1 ,

` ˘  ( r ι rIm;1 , rIn´m P rIn;pmq , pIn;pmq .

ι, this Since prIm , rIn´m;1 q and prIm;1 , rIn´m q commute in the domain of r would contradict the first statement in (2.30). The embeddings ι´ n : Opnq ÝÑ SOpn`1q, ι´ n pAq “

ˆ

A

˙ det A

,

ι` n : Opnq ÝÑ SOpn`3q, ¨ A ˚ det A ` ιn pAq “ ˚ ˝ det A

˛ ‹ ‹, ‚ det A

also lift to Lie group homomorphisms. ` Corollary 2.7. The embeddings ι´ n and ιn lift to injective Lie group ´ ` ιn so that the diagrams homomorphisms r ιn and r

commute. These homomorphisms satisfy ` ˘ r ιn´ pIn “ pIn`1 , ` ˘ r ι ` pIn “ pIn`3 , n

` (˘  ( r ιn´ rIn;1 , pIn;1 “ rIn`1;2 , pIn`1;2 , ` (˘  ( r ι ` rIn;1 , pIn;1 “ rIn`3;4 , pIn`3;4 . n

(2.40)

40

Spin/Pin-Structures & Real Enumerative Geometry

Proof. Let qn be the topological double cover as in (2.5). The double covers ιn´ ˚ qn`1 : ιn´ ˚ Spinpn`1q ÝÑ Opnq and ιn` ˚ qn`3 : ιn` ˚ Spinpn`3q ÝÑ Opnq

(2.41)

are Lie group homomorphisms. The embeddings r ι 1n`1;n : Spinpnq ÝÑ Spinpn ` 1q

and

r ι 1n`3;n : Spinpnq ÝÑ Spinpn ` 3q induce Lie groups isomorphisms ˇ Spinpnq ÝÑ ιn´ ˚ Spinpn`1qˇSOpnq Ă SOpnqˆSpinpn`1q, ˘ ` r r r , r ÝÑ qn pAq, ι 1n`1;n pAq A ˇ Spinpnq ÝÑ ιn` ˚ Spinpn`3qˇSOpnq Ă SOpnqˆSpinpn`3q, ˘ ` r r r , r ÝÑ qn pAq, ι 1n`3;n pAq A commuting with the projections to SOpnq. By the discussion after (2.26), each of the two projections in (2.41) is thus isomorphic to either qn´ or qn` . By definition, rIn;1 is an element of Pin´ pnq of order 4 so that ˘ ` ´ ` ˘ ` (˘ ´ r p r ι´ n qn pIn;1 q “ ιn In;1 “ In`1;2 “ qn`1 In`1;2 , In`1;2 . By the first statement of Lemma 2.2 with n replaced by n`1, rIn`1;2 and pIn`1;2 are elements of Spinpn ` 1q of order 4. Thus, the first projection in (2.41) is isomorphic to qn´ ; this is equivalent to the Pin´ -case of the corollary. By definition, rIn;1 is an element of Pin` pnq of order 2 so that ˘ ` ` ˘ ` (˘ ιn` qn` prIn;1 q “ ιn` In;1 “ In`3;4 “ qn`3 rIn`3;4 , pIn`3;4 . By the second statement of Lemma 2.2 with n replaced by n ` 3, rIn`3;4 and pIn`3;4 are elements of Spinpn ` 3q of order 2. Thus, the

The Lie Groups Spinpnq and Pin˘ pnq

41

second projection in (2.41) is isomorphic to qn` ; this is equivalent to the Pin` -case of the corollary.  The extensions r ιn´ of r ι 1n`1;n and r ιn` of r ι 1n`3;n provided by Corollary 2.7 can be defined explicitly as follows. Let n´ “ n ` 1

and n` “ n`3.

By (2.24) with pn, mq replaced by pn˘ , nq, (2.23) with pn, m2 q replaced by pn˘ , nq, and three consecutive applications of these equations in the ` case, ` ˘ ` ˘ ´1 r rI rrI´1 “ rIn ;pnqr ι 1n˘ ;n A r ι 1n˘ ;n rIn;1 A n;1 ˘ n˘ ;pnq , ` ˘ ` ˘ ´1 1 r “ rIn ;1r rr r ι 1n´ ;n A ´ ι n´ ;n A In´ ;1 , ` ˘ ` ˘ 1 r “ rIn ;pn ´2qrIn ;pn ´1qrIn ;pn qr r A r ι 1n` ;n A ι ;n n ` ` ` ` ` ` ` r´1 r´1 ˆ rI´1 n´ ;pn` q In´ ;pn` ´1q In´ ;pn` ´2q r P Spinpnq. Since for all A  ( rIn ;pnqrIn ;1 P rIn ;2 , rIn ;2pIn ´ ´ ´ ´ ´

and

(  rIn ;pnqrIn ;pn ´2qrIn ;pn ´1qrIn ;pn q P rIn ;2 , rIn ;2pIn , ` ` ` ` ` ` ` ` ` `

the three equations above and (2.28) imply that ` ˘ ` ˘ ´1 1 r rI´1 “ rIn`1;2r rIn;1 A r rI ι 1n`1;n A r ιn`1;n n;1 n`1;2 ` ˘ ` ˘ ´1 1 1 ´1 r r r r r r ι n`3;n A In`3;4 r ι n`3;n In;1 A In;1 “ In`3;4r ι 1n`1;n The two possible extensions r ι´ n of r

rrIn;1 A

r P Spinpnq, @A

r P Spinpnq. @A (2.42) are given by

Pin´ pnq´Spinpnq ÝÑ Spinpn`1q, r rIn`1;2 , r r pIn`1;2 @ A r P Spinpnq. ÝÑ rι 1 pAq ι1 pAq n`1;n

n`1;n

Since rIn;1 is an element of Pin´ pnq of order 4 and rIn`1;2 and pIn`1;2 are elements of Spinpn ` 1q of order 4, the first equation in (2.42) and the second equation in (2.10) with pn, mq replaced by pn´ , nq

42

Spin/Pin-Structures & Real Enumerative Geometry

ensure that these extensions are Lie group homomorphisms. The two possible extensions r ι` ι 1n`3;n can similarly be defined by n of r r rIn;1 A

Pin` pnq ´ Spinpnq ÝÑ Spinpn ` 3q, r rIn`3;4 , r r pIn`3;4 @ AP r Spinpnq. ÝÑ r ιn`3;n pAq ιn`3;n pAq

Since rIn;1 is an element of Pin` pnq of order 2 and rIn`3;4 and pIn`3;4 are elements of Spinpn ` 3q of order 2, the second equation in (2.42) ensures that these extensions are Lie group homomorphisms. ιn` of ι` There are thus two lifts r ιn´ of ι´ n and two lifts r n ; they differ by the composition with the involution ρ in (2.35) on the right. For concreteness, we choose them so that ` ˘ r r (2.43) r ι˘ n In;1 “ In˘ ;3˘1 P Spinpn˘ q. Along with (2.36) and (2.16), this implies that ˘ ˘ ` ` r ιn˘ ;m rIm , r ι˘ ιn;m prIm , rIn´m;1 q “ r r ι˘ n r n´m pIn´m;1 q

(2.44)

for all m, n P Z` with m ă n. By Lemma B.1(b),  ι n;m “r ι n˘ ;m ˝ idSpinpmq r ι˘ n ˝r ( ˆr ι˘ n´m : SpinpmqˆSpinpn´mq ÝÑ Spinpn˘ q, (2.45) since these two homomorphisms lift the homomorphism  ι1n˘ ;n ˝ιn;m “ ιn˘ ;m ˝ idSOpmq ( ˆι1n˘ ´m;n´m : SOpmqˆSOpn´mq ÝÑ SOpn˘ q. By (2.45) and (2.44),  ι n;m “r ι n˘ ;m ˝ idSpinpmq r ι˘ n ˝r ( ˘ ˆr ι˘ n´m : SpinpmqˆPin pn´mq ÝÑ Spinpn˘ q. (2.46)

Chapter 3

Proof of Theorem 1.4(1): Classical Perspective

In this chapter, which relies heavily on the notation and notions introduced in Chapter 2, we establish the statements of Section 1.2 for the notions of Spin-structure and Pin˘ -structure arising from Definition 1.1 and give additional examples of these structures. Throughout this chapter, the terms Spin-structure and Pin˘ structure refer to these notions. In Section 3.1, we clarify the terminology around Definition 1.1 and then relate Spin- and Pin˘ ˇ structures to Cech cohomology with values in Lie groups. In ˇ Section 3.2, we use Cech cohomology to establish statements (a) and (b) of the SpinPin 2 property and the SpinPin 3 and 4 properties. We obtain the SpinPin 1, 5, and 6 properties in Section 3.3 and the SpinPin 7 property and the last statement of the SpinPin 2 property in Section 3.4. 3.1

ˇ Spin/Pin-Structures and Cech Cohomology

Every rank n real vector bundle V over a paracompact space Y admits a metric gV . A metric gV on V determines an orthonormal frame bundle ØpV q over Y ; the fiber of ØpV q over a point y P Y consists of ordered tuples pv1 , . . . , vn q so that tv1 , . . . , vn u is an orthonormal basis for the fiber Vy of V over y. This is a principal Øpnq-bundle over Y so that ` ˘L V “ ØpV q ˆØpnq Rn ” ØpV q ˆ Rn „, pp, cq „ ppA´1 , Acq @ A P Øpnq. 43

44

Spin/Pin-Structures & Real Enumerative Geometry

If o P OpV q is an orientation on V , then gV also determines an oriented orthonormal frame bundle SOpV, oq over Y ; it is the subspace of ØpV q consisting of the tuples pv1 , . . . , vn q that are oriented bases for the fibers of V with respect to o. This is a principal SOpnq-bundle over Y so that ` ˘L V “ SOpV, oqˆSOpnq Rn ” SOpV, oqˆRn „, pp, cq „ ppA´1 , Acq @ A P SOpnq. The inclusion SOpV, oq ÝÑ ØpV q is equivariant with respect to the inclusion SOpnq ÝÑ Øpnq. Since the space of metrics gV on a vector bundle V over a paracompact space Y is contractible, the isomorphism classes of the frame bundles ØpV q and SOpV, oq are independent of the choice of gV . Thus, the notions of Pin˘ -structure and Spin-structure of Definition 1.1 are also independent of the choice of gV . The following example illustrates the notion of equivalence of Spin-structures formulated after Definition 1.1. Along with the SpinPin 4 property, this example also implies the last statement of the SpinPin 2 property with V 1 orientable of even rank and rk V 2 “ 0. Example 3.1. Let V be a vector bundle over a paracompact space Y with n ” rk V even and Ψ : V ÝÑ V,

v ÝÑ ´v.

(3.1)

Suppose o P OpV q and s ” pSpinpV, oq, qV q is a Spin-structure on pV, oq as in Definition 1.1. Since n is even, rIn;n P Spinpnq and the map  ( pq ÝÑ SpinpV, oq, Ψ˚ SpinpV, oq ” pp, prq : p “ ´qV pr pp, prq ÝÑ pr ¨ rIn;n , is an isomorphism of Spinpnq-fiber bundles over Y . By (2.20), this map is Spinpnq-equivariant and is thus an equivalence of Spinstructures on pV, oq. We conclude that rΨ˚ ss “ rss in SppV, oq. Let Y be a topological space, P be a principal SOpnq-bundle over Y , and Pr be a principal Spinpnq-bundle over Y . Equivariant

Proof of Theorem 1.4(1): Classical Perspective

45

r of Pr over U Ă Y correspond to sections trivializations Φ of P and Φ ` ˘ ` ˘ s P Γ U; P and sr P Γ U ; Pr of the two bundles over U : ˘˘ ` ˘ ` ` s π1 Φppq ¨π2 Φppq “ p @ p P P |U , ` ` ˘˘ ` ˘ r pq ¨π2 Φpr r pq “ pr @ prP Pr|U , sr π1 Φpr

(3.2)

where π1 , π2 : U ˆSOpnq ÝÑ U, SOpnq and π1 , π2 : U ˆSpinpnq ÝÑ U, Spinpnq are the projection maps. If q : Pr ÝÑ P is a double cover satisfying the r conditions of Definition 1.1(b), a Spinpnq-equivariant trivialization Φ r of P over U thus induces an SOpnq-equivariant trivialization Φ of P over U so that the diagram Pr|U

r Φ

/ U ˆSpinpnq

q



P |U

Φ



idˆqn

/ U ˆSOpnq

(3.3)

commutes. Let tUα uαPA be an open cover of Y and q : Pr ÝÑ P be as above. For α0 , α1 , . . . , αp P A, we define Uα0 α1 ...αp “ Uα0 XUα1 X¨ ¨ ¨XUαp Ă Y. A collection ( r α : Pr|Uα ÝÑ Uα ˆSpinpnq Φ αPA



(3.4)

of Spinpnq-equivariant trivializations determines transition data (  s.t. grαβ : Uαβ ÝÑ Spinpnq α,βPA ` ˘ ˘ ` r β pr r β pr r β pr r α pr pq “ πβ;1 pΦ pqq, grαβ πβ;1 pΦ pqq πβ;2 pΦ pqq Φ @ prP Pr|Uαβ , α, β P A,

(3.5)

46

Spin/Pin-Structures & Real Enumerative Geometry

where πβ;1 , πβ;2 : Uβ ˆSpinpnq ÝÑ Uβ , Spinpnq are the projection maps. The collection (3.5) satisfies ˇ ˇ ˇ ´1 ˇ ˇ gαγ g r “ rIn @ α, β, γ P A. grβγ ˇU r αβ U U αβγ

αβγ

αβγ

(3.6)

The collection ( gαβ q : Uαβ ÝÑ SOpnq α,βPA gαβ “ qn pr



(3.7)

then consists of the transition data for the principal SOpnq-bundle P induced by the collection ˇ (  (3.8) Φα : P ˇUα ÝÑ Uα ˆSOpnq αPA of its trivializations determined by the collection (3.4). Every collection tr gαβ uα,βPA as in (3.5) satisfying (3.6) conversely determines a principal Spinpnq-bundle ˙M ˆğ “ ‰ r ÝÑ y, Uα ˆSpinpnq „ ÝÑ Y, y, A Pr ” αPA

` ˘ ` ˘ r „ y, A r P Uβ ˆSpinpnq gαβ pyqA Uα ˆSpinpnq Q y, r ` ˘ r P Uαβ ˆSpinpnq, α, β P A. @ y, A Every collection tgαβ uα,βPA as in (3.7) satisfying (3.6) with all tildes dropped similarly determines a principal SOpnq-bundle ˙M ˆğ “ ‰ Uα ˆSOpnq „ ÝÑ Y, y, A ÝÑ y, P ” αPA

` ˘ ` ˘ Uα ˆSOpnq Q y, gαβ pyqA „ y, A P Uβ ˆSOpnq ` ˘ @ y, A P Uαβ ˆSOpnq, α, β P A. Thus, every collection as in (3.5) satisfying (3.6) determines a double cover ‰ ` ˘ ` ˘ “ r @ y, A r P Uα ˆSpinpnq, α P A, r “ y, qn pAq q : Pr ÝÑ P, q ry, As (3.9) satisfying the conditions of Definition 1.1(b).

Proof of Theorem 1.4(1): Classical Perspective

47

If q 1 : Pr ÝÑ P is another double cover satisfying the conditions of Definition 1.1(b), there exists a collection ( tfα : Uα ÝÑ SOpnq αPA s.t. ` ˘ “ ‰ ` ˘ r “ y, fα pyqqn pAq r r P Uα ˆSpinpnq, α P A. (3.10) @ y, A q 1 ry, As This collection satisfies ˇ ˇ gαβ “ fα ˇU gαβ fβ´1 ˇU αβ

αβ

@ α, β P A.

(3.11)

The Spin-structures q and q 1 are equivalent if there exists a collection (  s.t. frα : Uα ÝÑ Spinpnq αPA ˇ ˇ @ α, β P A, qn ˝ frα “ fα @ α P A; grαβ “ frα ˇU grαβ frβ´1 ˇU αβ

αβ

such a collection tfrα u induces a Spinpnq-equivariant automorphism ‰ ` ˘ “ r : Pr ÝÑ Pr, r ry, As r “ x, frα pyq Ψ Ψ ` ˘ r P Uα ˆSpinpnq, α P A, @ y, A r satisfying q 1 “ q˝ Ψ. Combining the last three paragraphs with Section A.4, we obtain a map ` ˘ ` ˘ q 1 Y ; SOpnq q 1 Y ; Spinpnq ÝÑ H H from the set of equivalence classes of principal Spinpnq-bundles to the set of equivalence classes of principal SOpnq-bundles so that for every principal Spinpnq-bundle Pr in the preimage of a principal SOpnqbundle P , there exists a double cover q : Pr ÝÑ P satisfying the conditions of Definition 1.1(b). However, two such double covers need not be equivalent; see the last part of Section 3.2. The analogous considerations apply in the setting of Definition 1.1(a) with the map ˘ ` ` ˘ q 1 Y ; Øpnq q 1 Y ; Pin˘ pnq ÝÑ H H from the set of equivalence classes of principal Pin˘ pnq-bundles to the set of equivalence classes of principal Øpnq-bundles induced by the Lie group homomorphism (2.26).

48

3.2

Spin/Pin-Structures & Real Enumerative Geometry

The Sets P˘ pV q and SppV, oq

ˇ We now apply the Cech cohomology perspective of the previous section to describe the structure of the sets P˘ pV q of Pin˘ -structures on V and SppV, oq of Spin-structures on pV, oq up to equivalence. By Proposition 3.3, the statements (a) and (b) of the SpinPin 2 property hold if Y is paracompact and satisfies the k “ 1 case of the mild topological condition in the following definition: Definition 3.2. Let k P Zě0 . A topological space Y is locally H k p¨; Z2 q-simple if it is locally path-connected and p lim ÝÑ H pU ; Z2 q “ 0

@ y P Y, p “ 1, . . . , k,

(3.12)

U Qy

where the direct limit is taken with respect to the cohomology homomorphisms induced by the inclusions of neighborhoods U of y. The direct limit in (3.12) vanishes if and only if for every neighborhood U Ă Y of y and every η P H p pU ; Z2 q there exists a neighborhood U 1 Ă U of y such that η|U 1 “ 0. If Y is locally H 1 -simple, then the sheaf homomorphisms ˘ ` ˘ ` ˘ ` ` ˘ SY Spinpnq ÝÑ SY SOpnq and SY Pin˘ pnq ÝÑ SY Øpnq between the sheaves of germs of continuous functions on Y with values in Spinpnq, SOpnq, Pin˘ pnq, and Øpnq induced by the Lie group homomorphisms (2.5) and (2.26) are surjective; see the paragraph after Definition A.4. All CW complexes are locally H k -simple for every k P Z` and paracompact. Proposition 3.3. Let V be a vector bundle over a paracompact locally H 1 -simple space Y: (a) If V admits a Pin˘ -structure, then the group H 1 pY ; Z2 q acts naturally, freely, and transitively on the set P˘ pV q. (b) If o P OpV q and pV, oq admits a Spin-structure, then the group H 1 pY ; Z2 q acts naturally, freely, and transitively on the set SppV, oq. Proof. By Proposition A.1, it is sufficient to establish the two q 1 pY ; Z2 q. Let n “ rk V , o P OpV q, claims with H 1 pY ; Z2 q replaced by H and s ” pSpinpV, oq, qV q be a Spin-structure on pV, oq.

Proof of Theorem 1.4(1): Classical Perspective

49

q 1 pY ; Z2 q. Choose a collection tr Suppose η P H gαβ u of transition data for the principal Spinpnq-bundle Pr ” SpinpV, oq as in (3.5) and (3.6) so that the collection (3.7) consists of transition data for the principal SOpnq-bundle P ” SOpV, oq and qV is given by (3.9). Refining the open cover tUα u if necessary, we can assume that η is represented by a collection ˇ ˇ ˇ  ( ηαβ : Uαβ ÝÑ Z2 α,βPA s.t. ηβγ ˇU ´ηαγ ˇU `ηαβ ˇU “0 αβγ

@ α, β, γ P A.

αβγ

αβγ

(3.13)

We take ` ˘ η¨s ” s1 ” Spin1 pV, oq, qV1

(3.14)

to be the Spin-structure on pV, oq determined by the transition data (  1 η (3.15) grαβ ” grαβpIn αβ : Uαβ ÝÑ Spinpnq α,βPA. 1 gαβ because pIn is This collection satisfies (3.6) with r gαβ replaced by r in the center of Spinpnq. For the same reason, the equivalence class of s1 depends only on the equivalence classes of s and on η. It is q 1 pY ; Z2 q on SppV, oq. immediate that this defines a group action of H 1 Suppose the Spin-structure s in (3.14) is equivalent to s. Refining the open cover tUα u if necessary, we can assume that there exists a collection (  (3.16) frα : Uα ÝÑ Spinpnq αPA

of continuous maps such that ˇ ˇ η gαβpIn αβ “ frα ˇU grαβ frβ´1 ˇU r αβ αβ ˘ ` ˘ ` r “ qn frα pxqAr qn A

@ α, β P A, ` ˘ r P Uα ˆSpinpnq, α P A. @ x, A (3.17)

The first condition above means that the collection (3.16) determines an equivalence between the principal Spinpnq-bundles determined by the collections (3.5) and (3.15). The second condition means that

50

Spin/Pin-Structures & Real Enumerative Geometry

this equivalence commutes with the corresponding projections qV and qV1 to the principal SOpnq-bundle SOpV, oq. It implies that for every α P A, there exists a continuous function μα : Uα ÝÑ Z2

s.t.

frα “ pInμα .

Since pIn is in the center of Spinpnq, this and the first condition in (3.17) imply that ˇ ˇ @ α, β P A, ηαβ “ μα ˇU ´ μβ ˇU αβ

αβ

q 1 pY ; Z2 q q 1 pY ; Z2 q. Thus, H i.e. η represents the trivial element of H acts freely on SppV, oq. Suppose s2 ” pSpin2 pV, oq, qV2 q is another Spin-structure on pV, oq. Refining the open cover tUα u if necessary, we can assume that this pair is determined by the collections (  (  2 and fα : Uα ÝÑ SOpnq αPA gαβ : Uαβ ÝÑ Spinpnq r satisfying (3.6) with r g replaced by gr2 , (3.10) with q 1 replaced by qV2 , (3.11), and 2 gαβ q “ gαβ qn pr

@ α, β P A.

(3.18)

If n ě 2, let ηn P H 1 pSOpnq; Z2 q denote the generator. Since Y is locally H 1 -simple, we can assume that ` ˘ @ αP A fα˚ ηn “ 0 P H 1 Uα ; Z2 and thus there exists a collection (  s.t. frα : Uα ÝÑ Spinpnq αPA

qn ˝ frα “ fα @ α P A.

(3.19)

If n “ 1, the existence of such a collection is immediate. By (3.11) and (3.18), the existence of the collection (3.19) implies that there exists a collection as in (3.13) such that ˇ ` ˇ η ˘ grαβpIn αβ fr´1 ˇ @ α, β P A. (3.20) g 2 “ frα ˇ r αβ

Uαβ

β

Uαβ

2 u gαβ r replaced Since tr gαβ uαβ satisfies (3.6), tr αβ satisfies (3.6) with g 2 p by gr , and In is in the center of Spinpnq, the collection tηαβ u satisfies the (cocycle) condition in (3.13) and thus determines an element η

Proof of Theorem 1.4(1): Classical Perspective

51

q 1 pY ; Z2 q. By (3.20) and the condition in (3.19), the collection of H in (3.19) determines an equivalence between the Spin-structures η¨s q 1 pY ; Z2 q acts transitively on SppV, oq. and s2 . Thus, H This concludes the proof of (b). The same reasoning, with Spin and SOpnq replaced by Pin˘ and Øpnq everywhere, applies  to 3.3. Example 3.4. Suppose V is a vector bundle over a paracompact locally H 1 -simple space Y , n ” rk V , Ψ is as in (3.1), and p ” pPin˘ pV q, qV q is a Pin˘ -structure on V . Let (  and grαβ : Uαβ ÝÑ Pin˘ pnq α,βPA (  gαβ q : Uαβ ÝÑ Øpnq α,βPA gαβ “ qn˘ pr be a collection of transition data for the principal Pin˘ pnq-bundle Pin˘ pV q and the associated collection of transition data for the principal Øpnq-bundle ØpV q. Define ` ( ˘n´1  by p´1qηαβ “ det gαβ . ηαβ : Uαβ ÝÑ Z2 α,βPA ˇ By Corollary A.7, the Cech cohomology class rηs represented by tηαβ u is pn´1qw1 pV q. The Pin˘ -structure ˘ ` η¨p ” p1 ” Pin1˘ pV q, qV1 on V is determined by the transition data (  1 η gαβ ” grαβpIn αβ : Uαβ ÝÑ Pin˘ pnq α,βPA. r 1 r In;n . Thus, gαβ “ grαβ By (2.20) and the second equation in (2.30), rIn;n r the map  ( r : Ψ˚ Pin˘ pV q ” pp, prq : p “ ´qV pr pq ÝÑ Pin1˘ pV q, Ψ ˘ “ ‰ ` ˘ ` r rx, As r “ x, rIn;n A r @ x, A r P Uα ˆPin˘ pnq, α P A, r rx, ´qn pAqs, Ψ

r We conis well defined and Pin˘ pnq-equivariant and Ψ˚ qV “ qV1 ˝ Ψ. clude that ` ˘ rΨ˚ ps “ pn´1qw1 pV q ¨rps, establishing the rk V 2 “ 0 case of (1.13).

52

Spin/Pin-Structures & Real Enumerative Geometry

Proof of SpinPin 3 property. Let V be a rank n vector bundle over a paracompact space Y , o P OpV q, and s ” pSpinpV, oq, qV q be a Spin-structure on pV, oq. Let (  srα : Uα ÝÑ SpinpV, oq αPA be a collection of local sections so that tUα uαPA is an open cover of Y , (3.5) be the transition data for the principal Spinpnq-bundle SpinpV, oq determined by the associated collection (3.4) of trivializations, and (3.7) be the transition data for the principal SOpnq-bundle SOpV, oq Ă ØpV q determined by the associated collection (3.8) of trivializations. Let sα . sα “ qV ˝r The sections sα ” sα ¨In;1 determine trivializations I´1 n;1 ¨Φα of the oriented orthonormal frame bundle SOpV, oq Ă ØpV q for the opposite orientation o on V with the transition data (  ´1 In;1 gαβ In;1 : Uαβ ÝÑ SOpnq α,βPA. The collection 

( rαβrIn;1 : Uαβ ÝÑ Spinpnq α,βPA gpαβ ” rI´1 n;1 g

(3.21)

then determines a Spin-structure s on pV, oq, ˆğ ˙M qV : SpinpV, oq ” Uα ˆSpinpnq „ ÝÑ SOpV, oq, `

αPA

˘ r r “ sα pyq¨In;1 qn pAq, q V ry, As ` ˘ ` ˘ r „ y, A r P Uβ ˆSpinpnq Uα ˆSpinpnq Q y, gpαβ pyqA ` ˘ r P Uαβ ˆSpinpnq, α, β P A. @ y, A The equivalence class of s depends only on the equivalence class of s. The above construction induces a bijection from SppV, oq to SppV, oq. This bijection is natural with respect to continuous maps and isomorphisms of vector bundles. Since pIn lies in the center of Spinpnq, this bijection is equivariant with respect to the H 1 pY ; Z2 qactions of Proposition 3.3. It is immediate that this bijection satisfies  the last condition of the SpinPin 3 property.

Proof of Theorem 1.4(1): Classical Perspective

53

Remark 3.5. Since the space Øpnq´SOpnq is path-connected, the equivalence class of s above would be the same if it were conr of Pin˘ pnq´Spinpnq instead of In;1 . An structed using any element B explicit equivalence between the two resulting Spin-structures would be induced by the maps ` ˘ ` ˘ r ÝÑ y, B r ´1rIn;1 A r . y, A Uα ˆSpinpnq ÝÑ Uα ˆSpinpnq, The bijections (1.14) constructed in the proofs of the SpinPin 3 property in the perspectives of Definitions 1.2 and 1.3 in Sections 4.3 and 4.3, respectively, likewise would not change if they were constructed using any element of Øpnq´SOpnq instead of In;1 . Example 3.6. Suppose V is a vector bundle over a paracompact locally H 1 -simple space Y with n ” rk V odd, Ψ is as in (3.1), o P OpV q, and s ” pSpinpV, oq, qV q is a Spin-structure on pV, oq. Let (  and s ” pSpinpV, oq, q V q srα : Uα ÝÑ SpinpV, oq αPA be as in the proof of the SpinPin 3 property above. By (2.20), rI´1 rIn;n grαβ “ gpαβrI´1 rIn;n . n;1 n;1 Thus, the map  ( r : Ψ˚ SpinpV, oq ” pp, prq : p “ ´qV pr pq ÝÑ SpinpV, oq, Ψ ˘ “ ‰ ` r srα pyq¨ A r “ y, rI´1 rIn;n A r r sα pyq¨p´qn pAqq, Ψ n;1 ` ˘ r P Uα ˆSpinpnq, α P A, @ y, A r We conis well defined and Spinpnq-equivariant and Ψ˚ qV “ qV ˝ Ψ. ˚ clude that rΨ ss “ rss in SppV, oq. Proof of SpinPin 4 property. Let V be a rank n vector bundle over a paracompact space Y and o P OpV q. Thus, ØpV q « SOpV, oqˆZ2

(3.22)

as topological spaces. By the SpinPin 1 property, V admits a Pin˘ structure if and only if pV, oq admits a Spin-structure. We can thus

54

Spin/Pin-Structures & Real Enumerative Geometry

assume that there exists a Pin˘ -structure ˘ ` p ” Pin˘ pV q, qV on V as in Definition 1.1(a). The restriction ˇ ` ˘ qV ˇq´1 pSOpV,oqˆt0uq : SpinpV, oq ” qV´1 SOpV, oqˆt0u ÝÑ Y V

is then a Spin-structure on pV, oq. This induces a well-defined natural ˘ H 1 pY ; Z2 q-equivariant map R˘ o from P pV q to SppV, oq. Since the 1 H pY ; Z2 q-actions of Proposition 3.3 are free and transitive, this map is bijective. We now establish the last claim of the SpinPin 4 property. With p and SpinpV, oq as above, let srα , grαβ , and q V be as in the proof of the SpinPin 3 property. In particular, srα grαβ “ srβ on Uαβ . The map ˘ ` r : SpinpV, oq ÝÑ q ´1 SOpV, oqˆt0u Ă Pin˘ pV q, Ψ V ` ˘ ` ˘ r @ y, A r P Uα ˆSpinpnq, α P A, r ry, As r “ srα pyq¨rIn;1 A Ψ r Thus, is then well defined and Spinpnq-equivariant and q V “ qV ˝ Ψ. the Spin-structure s on pV, oq constructed in the proof of the SpinPin 3 property is equivalent to the Spin-structure obtained by  restricting qV to qV´1 pSOpV, oqˆt0uq. For a topological space Y and a Lie group G, let CpY ; Gq denote the group of continuous maps from Y to a Lie group G. If Y is paracompact locally H 1 -simple, the short exact sequence qn

t1u ÝÑ Z2 ÝÑ Spinpnq ÝÑ SOpnq ÝÑ t1u

(3.23)

of Lie groups induces an exact sequence t1u

/

CpY ; Z2 q

H 1 pY ; Z2 q

/

qq0 / C `Y ; SOpnq˘ g g g g q δ gggg 0 ggggg g g g g g s gggg/ |1 ` qq1 ˘ ` ˘ /H |1 Y ; SOpnq H Y ; Spinpnq

` ˘ C Y ; Spinpnq

q δ 1

/

H 2 pY ; Z2 q

(3.24) of based sets; see Proposition A.6. The first three non-trivial maps in this sequence are group homomorphisms.

Proof of Theorem 1.4(1): Classical Perspective

55

Suppose n ě 2. By the naturality of (3.24), there is a commutative diagram ` ˘ C SOpnq; Spinpnq 

qq0

`

˘

/ C SOpnq; SOpnq

f˚

` ˘ C Y ; Spinpnq

qq0

`



δq0

/ H 1 pSOpnq; Z2 q

f˚

˘

/ C Y ; SOpnq

δq0



f˚

/ H 1 pY ; Z2 q

of group homomorphisms for every continuous map f : Y ÝÑ SOpnq. Such a map lies in the image of qq0 in (3.24) if and only if f ˚ ηn “ 0 P H 1 pY ; Z2 q, where ηn P H 1 pSOpnq; Z2 q is the generator as before. Thus, ` ˘ ` ˘ δq0 idSOpnq “ ηn P H 1 SOpnq; Z2 , ` ˘ δq0 pf q “ δq0 f ˚ idSOpnq “ f ˚ ηn P H 1 pY ; Z2 q,  ` ˘( Im δq0 “ f ˚ ηn : f P C 0 Y ; SOpnq Ă H 1 pY ; Z2 q . By Proposition 3.3, H 1 pY ; Z2 q acts freely and transitively on the set SppV, oq of Spin-structures on an oriented vector bundle pV, oq over Y that admits a Spin-structure. By the exactness of (3.24), H 1 pY ; Z2 q{Im δq0 acts freely and transitively on the set of principal Spinpnq-bundles that doubly cover the trivial SOpnq-bundle over Y . The latter action is induced by the former. Thus, Im δq0 in (3.24) acts freely and transitively on the set of equivalence classes of double covers qV : SpinpV, oY q ÝÑ SOpV, oY q “ Y ˆSOpnq as in Definition 1.1 with fixed domain and target. The same considerations apply to Pin˘ -structures on a real vector bundle V . If n “ 2 and Y is a CW complex, then all groups in (3.23) are abelian and Im δq0 in (3.24) is the entire space H 1 pY ; Z2 q. Thus, for every rank 2 oriented vector bundle pV, oq over Y , there is at most one principal Spinp2q-bundle SpinpV, oq that doubly covers the principal SOp2q-bundle SOpV, oq and H 1 pY ; Z2 q acts freely and transitively on the set of equivalence classes of associated projections qV with the domain and target fixed.

56

3.3

Spin/Pin-Structures & Real Enumerative Geometry

Correspondences and Obstructions to Existence

We next establish the SpinPin 5 property for the Spin- and Pin˘ structures of Definition 1.1. Combining it with Examples 1.9–1.12, we then obtain the SpinPin 1 property; it holds as long as Y is a paracompact locally H 2 -simple space in the sense of Definition 3.2. We conclude this section by establishing the SpinPin 6 property. Proof of SpinPin 5 property. Pin˘ pnq ˘ qn



Øpnq

Let n “ rk V , W “ τY ‘V , and

r ι 2n`1;n

ι2n`1;n

/ Pin˘ pn`1q ˘ qn`1

 / Øpn`1q

(3.25)

be as in Section 2.2. We identify Øpnq with a subspace of Øpn`1q via the embedding ι2n`1;n and Pin˘ pnq with the subspace ˇ Pin˘ pn`1qˇØpnq Ă Pin˘ pn`1q via the embedding r ι 2n`1;n . The natural embedding of V into W identifies ØpV q with a subspace of ØpW q preserved by the action of Øpnq Ă Øpn`1q. If qW : Pin˘ pW q ÝÑ ØpW q is a Pin˘ -structure on W , then the subspace ˇ ` ´1 ØpV qq Pin˘ pV q ” Pin˘ pW qˇØpV q ” qW is preserved by the action of Pin˘ pnq Ă Pin˘ pn ` 1q. Thus, the restriction qV : Pin˘ pV q ÝÑ ØpV q of qW determines a Pin˘ -structure on V . Let p ” pPin˘ pV q, qV q be a Pin˘ -structure on V . Transition data 

( gαβ : Uαβ ÝÑ Pin˘ pnq α,βPA r

Proof of Theorem 1.4(1): Classical Perspective

57

for Pin˘ pV q determine the transition data ( gαβ q : Uαβ ÝÑ Øpnq α,βPA, gαβ ” qn˘ pr 2 ( ιn`1;n pgαβ q : Uαβ ÝÑ Øpn`1q α,βPA, (  2 gαβ q : Uαβ ÝÑ Pin˘ pn`1q α,βPA and r ι n`1;n pr 

for ØpV q, ØpW q, and a principal Pin˘ pn ` 1q-bundle Pin˘ pW q. By the commutativity of (3.25), there is a commutative diagram Pin˘ pV q qV



ØpV q

/ Pin˘ pW q 

qW

/ ØpW q

so that the top and bottom arrows are equivariant with respect to the actions of Pin˘ pnq and Øpnq, respectively. The right vertical arrow above determines a Pin˘ -structure StV ppq on W . The two constructions above are mutual inverses and descend to bijections between the collections P˘ pV q and P˘ pW q. This establishes the existence of the second map in (1.16). The construction of the first map in (1.16) is analogous, with Ø and Pin˘ replaced by SO and Spin throughout. It is immediate that it satisfies the first condition after (1.16). The equivariance of the two bijections in (1.16) follows from (2.10). It remains to establish (1.17). Suppose o P OpV q, s ” pSpinpV, oq, qV q is a Spin-structure on pV, oq, and ` ˘ StV psq ” SpinpW, StV poqq, qW is the Spin-structure on pW, StV poqq constructed from a collection 

( gαβ : Uαβ ÝÑ Spinpnq α,βPA r

(3.26)

of transition data for SpinpV, oq as above. If p is a Pin˘ -structure as before and s is determined by p as in the proof of the SpinPin 4

58

Spin/Pin-Structures & Real Enumerative Geometry

property, then the Pin˘ -structure StV ppq on W can be constructed using the same collection tr gαβ uα,βPA. The restriction of the projection qW : PinpW q ÝÑ ØpW q to the preimage of SOpW, StV poqq Ă ØpW q is then the projection for StV psq. Thus, StV psq is the Spin-structure on pW, StV poqq determined by the Pin˘ -structure StV ppq on W as in the proof of the SpinPin 4 property. This establishes the last equality in (1.17). Let tr sα uαPA be a collection of local sections of SpinpV, oq Ă SpinpW, StV poqq determining a collection (3.26) of transition data for SpinpV, oq. The collection (  2 gαβ q : Uαβ ÝÑ Spinpn`1q α,βPA (3.27) r ι n`1;n pr then consists of the transition data for SpinpW, StV poqq determined by tr sα uαPA. Let q V : SpinpV, oq ÝÑ SOpV, oq and ` ˘ ` ˘ ` ˘ q W : Spin W, StV o ÝÑ SO W, StV o “ SO W, StV o be the Spin-structures determined by the collections (3.26) and (3.27) as in the proof of the SpinPin 3 property and ` ˘ ` ˘ q 1W : Spin W, StV o ÝÑ SO W, StV o be the Spin-structure on pW, StV oq determined by the collection (3.21) as above. The last two Spin-structures are determined by the collections (  ´1 rI ι 2n`1;n pr gαβ qrIn`1;1 : Uαβ ÝÑ Spinpn`1q α,βPA and n`1;1r (  2 gαβrIn;1 q : Uαβ ÝÑ Spinpn`1q α,βPA r ι n`1;n prI´1 n;1 r of transition data. By the first equation in (2.25) with pn, mq replaced by pn`1, nq, these two collections are the same. This establishes the  first equality in (1.17). Proof of SpinPin 1 property. γn ÝÑ Gpnq

For n P Z` , let

and

r γ rn ÝÑ Gpnq

be the real tautological n-plane bundle over the infinite Grassmannian of real n-planes and the oriented tautological n-plane bundle over the infinite Grassmannian of oriented n-planes:

Proof of Theorem 1.4(1): Classical Perspective

59

(a) By Proposition A.6, the short exact sequence t1u ÝÑ Z2 ÝÑ Pin˘ pnq ÝÑ Øpnq ÝÑ t1u of Lie groups induces a commutative diagram ` ˘ q 1 Gpnq; Pin˘ pnq H

`

˘

/H q 1 Gpnq; Øpnq

f˚



˘ ` q 1 Y ; Pin˘ pnq H

`



`

δq1

/ H 2 Gpnq; Z2

f˚

˘

δq1

/H q 1 Y ; Øpnq



˘

f˚

/ H 2 pY ; Z2 q

(3.28) of pointed sets for every continuous map f : Y ÝÑ Gpnq. By the last part of Section 3.1 and the exactness of the bottom row in this diagram, a rank n real vector bundle V over Y admits a Pin˘ -structure if and only if the image under δq1 of the equivalence class rØpV qs of q 1 pY ; Øpnqq vanishes. ØpV q in H By [34, Theorem 7.1], H 2 pGpnq; Z2 q is generated by w2 pγn q and w12 pγn q. By [34, Theorem 5.6], for every rank n real vector bundle V over Y , there exists a continuous map f : Y ÝÑ Gpnq such that V « f ˚ γn . Along with the commutativity of (3.28), these statements ˘ imply that there exist a˘ n , bn P Z2 such that ` ˘ ˘ 2 δq1 rØpV qs “ a˘ n w2 pV q ` bn w1 pV q for every rank n real vector bundle V over every paracompact locally ˘ H 2 -simple space Y . By the SpinPin 5 property, a˘ n , bn do not depend ˘ ˘ on n; we thus denote them by a , b . By Examples 1.9 and 1.10, b` “ 0 and b´ “ 1. By Example 1.12 and the already established SpinPin 4 property, a` “ 1 and a´ “ 1. (b) By Proposition A.6, the short exact sequence (3.23) induces a commutative diagram ` ˘ r q 1 Gpnq; Spinpnq H 

`

˘

q 1 Gpnq; r /H SOpnq

f˚

` ˘ q 1 Y ; Spinpnq H

`



`

δq1

r / H 2 Gpnq; Z2

f˚

˘

f˚

˘

/H q 1 Y ; SOpnq

δq1

 / H 2 pY ; Z2 q

(3.29)

60

Spin/Pin-Structures & Real Enumerative Geometry

r of pointed sets for every continuous map f : Y ÝÑ Gpnq. By the last part of Section 3.1 and the exactness of the bottom row in this diagram, a rank n-oriented vector bundle V over Y admits a Spin-structure if and only if the image under δq1 of the equivalence q 1 pY ; SOpnqq vanishes. class rSOpV qs of SOpV q in H r Z2 q is generated by w2 pr γn q. By By [34, Theorem 12.4], H 2 pGpnq; [34, Theorem 5.6], for every rank n-oriented vector bundle V over Y , r there exists a continuous map f : Y ÝÑ Gpnq such that V « f ˚ γ rn . Along with the commutativity of (3.29), these statements imply that there exists an P Z2 such that ` ˘ δq1 rSOpV qs “ an w2 pV q for every rank n-oriented vector bundle V over every paracompact locally H 2 -simple space Y . By the SpinPin 5 property, an does not  depend on n; we thus denote it by a. By Example 1.12, a “ 1. Proof of SpinPin 6 property. Let V˘ be as in (1.6), o˘ V be the canonical orientations on these vector bundles, n “ rk V , n˘ “ rk V˘ , and Pin˘ pnq ˘ qn



Øpnq

r ι˘ n

ι˘ n

/ Spinpn˘ q 

qn˘

/ SOpn˘ q

(3.30)

be as in Corollary 2.7. By the SpinPin 1 property, pV˘ , o˘ V q admits a Spin-structure if and only if V admits a Pin˘ -structure. We can thus assume that there exists a Pin˘ -structure p ” pPin˘ pV q, qV q on V . Transition data (  gαβ : Uαβ ÝÑ Pin˘ pnq α,βPA r for Pin˘ pV q determines the transition data (  gαβ q : Uαβ ÝÑ Øpnq α,βPA, gαβ ” qn˘ pr ( ˘ ιn pgαβ q : Uαβ ÝÑ SOpn˘ q α,βPA, (  ˘ gαβ q : Uαβ ÝÑ Spinpn˘ q α,βPA and r ι n pr

Proof of Theorem 1.4(1): Classical Perspective

61

for ØpV q, SOpV˘ q, and a principal Spinpn˘ q-bundle ` ˘ Co˘ V ppq ” SpinpV˘ , o˘ q, qV˘ . By the commutativity of (3.30), there is a commutative diagram Pin˘ pV q qV



ØpV q

/ SpinpV˘ , o˘ q 

qV˘

/ SOpV˘ , o˘ q

so that the top and bottom arrows are equivariant with respect to the actions of Pin˘ pnq and Øpnq, respectively. The equivalence class of Spin-structures on V˘ represented by the right vertical arrow in the above diagram is determined by the equivalence class of Pin˘ -structures on V represented by the left vertical arrow. By the first column in (2.40), the resulting maps (1.20) are equivariant with respect to the actions of H 1 pY ; Z2 q provided by Proposition 3.3. Since these actions are free and transitive, these maps are bijective. By the above construction and the proof of ˘ the SpinPin 5 property, the Spin-structures Co˘ τY ‘V pStV ppqq and ˘ St˘ V˘ pCoV ppqq on (1.19) are determined by the collections ˘ ( ˘ ` 2 gαβ q : Uαβ ÝÑ Spinpn˘ `1q α,βPA, ι n`1;n pr r ιn`1 r ˘ (  ` ˘ gαβ q : Uαβ ÝÑ Spinpn˘ `1q α,βPA r ιn˘ `1;n˘ r ι n pr of transition data. By (2.46) with pn, mq replaced by pn`1, nq, these two collections are the same. This establishes the last claim of the  SpinPin 6 property. Example 3.7. Let V be the infinite Mobius band line bundle of Examples 1.23 and 1.24. Thus, ` ˘ ` ˘ 1 0, I2;2 A „ p1, Aq, SOpV´ , o´ V q “ r0, 1sˆSOp2q {„ÝÑ RP , “ ‰ rt, As ÝÑ eπit , ` ˘ ` ˘ 1 0, I4;4 A „ p1, Aq, SOpV` , o` V q “ r0, 1sˆSOp4q {„ÝÑ RP , “ ‰ rt, As ÝÑ eπit ,

62

Spin/Pin-Structures & Real Enumerative Geometry

with I2;2 ” ´I2 and I4;4 ” ´I4 as in (2.11). Under the conventions specified by (1.33) and (2.43), the Pin˘ -structure p˘ 0 pV q on V constructed in Examples 1.23 and 1.24 corresponds to the OSpinstructure os0 pV˘ , o˘ V q given by ` ˘ ` ˘ ` ˘ ˘ Spin0 V˘ , o˘ V “ r0, 1sˆSpinp3˘1q {„ ÝÑ SO V˘ , oV , ˘ ` ‰ ` ˘ “ r „ p1, Aq, r r , r “ t, q3˘1 pAq qp3˘1qV rt, As 0, rI3˘1;3˘1 A on pV˘ , o˘ V q, with the homomorphism q3˘1 given by (2.5). 3.4

Short Exact Sequences

We conclude the verification of the properties of Spin- and Pinstructures in the perspective of Definition 1.1 by establishing the SpinPin 7 property and using it in combination with Examples 3.4 and 3.6 to obtain (1.13). Proof of SpinPin 7 property. A short exact sequence e of vector bundles over a paracompact space Y as in (1.21) determines a homotopy class of isomorphisms V « V 1 ‘V 2 so that ι is the inclusion as the first component on the right-hand side above and j is the projection to the second component. Thus, it is sufficient to establish the SpinPin 7 property for the direct sum exact sequences as in (1.22). From the definition of the induced orientation before the statement of the SpinPin 7 property, it is immediate that ` ˘ o1 o2 “ o1 o2 “ o1 o2 P O V 1 ‘V 2 and ` ˘ ` ˘ o11 o12 o2 “ po11 o12 qo2 P O V11 ‘V21 ‘V

(3.31)

for all o1 P OpV 1 q, o11 P OpV11 q, o12 P OpV21 q, o2 P OpV 2 q, and vector bundles V 1 , V11 , V21 , V over Y . For vector bundles V 1 and V 2 over Y ,

Proof of Theorem 1.4(1): Classical Perspective

63

we define ι1V 1 ,V 2 : V 1 ÝÑ V 1 ‘V 2

and

ι2V 1 ,V 2 : V 2 ÝÑ V 1 ‘V 2

to be the canonical inclusions and ` ˘ ιV 1 ,V 2 : ØpV 1 qˆY ØpV 2 q ÝÑ Ø V 1 ‘V 2 to be the induced map. Let V 1 and V 2 be vector bundles over Y of ranks m and n´m, respectively, and V “ V 1 ‘V 2 . Suppose ` ` ˘ ˘ o1 P OpV 1 q, os1 ” SpinpV 1 , o1 q, qV 1 , and p2 ” Pin˘ pV 2 q, qV 2 are an orientation on V 1 , an OSpin-structure on V 1 , and a Pin˘ structure on V 2 , respectively. Choose collections (  2 (  1 srα : Uα ÝÑ Pin˘ pV 2 q αPA srα : Uα ÝÑ SpinpV 1 , o1 q αPA and (3.32) of local sections so that tUα uαPA is an open cover of Y . Let (  1 gαβ : Uαβ ÝÑ Spinpmq α,βPA and r (  2 gαβ : Uαβ ÝÑ Pin˘ pn´mq α,βPA r

(3.33)

be the associated transition data for the principal Spinpmq-bundle SpinpV 1 , o1 q and for the principal Pin˘ pmq-bundle Pin˘ pV 2 q, respectively. The collections (  1 1 gαβ q : Uαβ ÝÑ SOpmq α,βPA and gαβ “ qm pr (  2 ˘ 2 pr gαβ q : Uαβ ÝÑ Øpn´mq α,βPA gαβ “ qn´m then consist of the transition data for the principal SOpmq-bundle SOpV 1 , o1 q and for the principal Øpm´nq-bundle ØpV 1 q induced by the collections (  1 s1α : Uα ÝÑ SOpV 1 , o1 q αPA and sα ” qV 1 ˝r (  2 s2α : Uα ÝÑ ØpV 2 q αPA sα ” qV 2 ˝r of local trivializations of these bundles.

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Spin/Pin-Structures & Real Enumerative Geometry

The transition data for the principal Øpnq-bundle determined by the collection (  (3.34) sα ” ιV 1 ,V 2 ps1α , s2α q : Uα ÝÑ ØpV q αPA of local sections is given by (  1 2 , gαβ q : Uαβ ÝÑ Øpnq α,βPA. gαβ ” ιn;mpgαβ The collection  ( 1 2 gαβ ”r r ιn;m pr gαβ , grαβ q : Uαβ ÝÑ Pin˘ pnq α,βPA

(3.35)

then determines a Pin˘ -structure p on V , ˆğ ˙M ˘ ˘ Uα ˆPin pnq „ ÝÑ ØpV q, qV : Pin pV q ” `

αPA

˘ r r “ sα pyq¨qn pAq, qV ry, As ` ˘ ` ˘ r „ y, A r P Uβ ˆPin˘ pnq gαβ pyqA Uα ˆPin˘ pnq Q y, r ` ˘ r P Uαβ ˆPin˘ pnq, α, β P A. @ y, A By the reasoning in Appendix A.4, the equivalence class of p ” xxos1 , p2 yy‘ is independent of the choices of the collections (3.32) and depends only on the equivalence classes of os1 and p2 . Along with the definition of o ” o1 o2 P OpV q, the above construction with p2 and Pin˘ replaced by an OSpin-structure os2 ” pSpinpV 2 , o2 q, qV 2 q on V 2 and by Spin, respectively, yields an OSpin-structure ` ˘ os ” xxos1 , os2 yy‘ ” SpinpV, oq, qV on V ; its equivalence class depends only on the equivalence classes of os1 and os2 . By (2.10) and (2.28), ` 1 2 ˘ ` 1 ˘ ` 1 2˘ r pIn´m “ r r2 “ r r pIn r ,A r pIm , A r ,A ιn;m A ιn;m A r ιn;m A r1 , A r2 q P SpinpmqˆPin˘ pn´mq. @ pA Along with the proof of Proposition 3.3, this implies that the two induced maps xx¨, ¨yy e in (1.24) are H 1 pY ; Z2 q-biequivariant.

Proof of Theorem 1.4(1): Classical Perspective

65

Suppose the vector bundles V 1 and V 2 over Y are split as direct sums of oriented line bundles, os1 “ os0 pV 1 q and os2 “ os0 pV 2 q. We can take A to consist of a single element α and s1α and s2α to be sections of SOpV 1 , o1 q and SOpV 2 , o2 q respecting the ordered splittings and the orientations of all line bundle components. The induced section sα of SOpV, o1 o2 q in (3.34) then respects the induced ordered splitting and the orientations of all line bundle components of pV, o1 o2 q. This implies (ses1). The property (1.25) follows immediately from the construction. If p2 is a Pin˘ -structure on V 2 , o2 P OpV 2 q, and os2 is the OSpinSpin on V 2 obtained from p2 and o2 as in the proof of the SpinPin 4 property, then the collection tr s2α uαPA in (3.32) can be chosen as in 2 the construction for os , i.e. with values in ˇ SpinpV 2 , o2 q ” Pin˘ pV 2 qˇSOpV 2 ,o2 q . The Pin˘ - and OSpin-structures on V obtained in this way then satisfy ˇ OSpinpV q “ Pin˘ pV qˇSOpV,oq . This establishes (1.26). Suppose V11 , V21 , V 2 and os11 , os12 , p2 are as in (ses4) and m1 , m2 , n P Z` are the ranks of V11 , V21 , and V11 ‘V21 ‘V 2 , respectively. Let ts11;α uαPA, ts12;α uαPA, and ts2α uαPA be collections of local sections asso1 1 ciated with os11 , os12 , p2 as in (3.32) and tg1;αβ uα,βPA, tg2;αβ uα,βPA, ˘ 2 and tgαβ uα,βPA be the induced transition data. The Pin -structures xxos11 , xxos12 , p2 yy‘ yy‘ and xxxxos11 , os12 yy‘ , p2 yy‘ are then given by the transitions data ˘ ( ` 1  1 2 ,r ιn´m1 ;m2 pg2;αβ , gαβ q : Uαβ ÝÑ Pin˘ pnq α,βPA and r ιn;m1 g1;αβ ( ` ˘  1 1 2 , g2;αβ q, gαβ ιm1 `m2 ;m1 pg1;αβ : Uαβ ÝÑ Pin˘ pnq α,βPA, r ιn;m1 `m2 r respectively. By (2.39), these two collections are the same. This implies (ses4). With os1 “ os0 pτY q, the local sections sr1α in (3.32) can be chosen 1 “r I1 for all α, β P A. If tr gαβ uα,β is the transition data so that r gαβ ˘ for a Pin -structure p determined by a collection tsα uαPA of local

66

Spin/Pin-Structures & Real Enumerative Geometry

sections, then the Pin˘ -structure xxos0 pτY q, pyy‘ is determined by the transition data (  r ιn`1;1 prI1 , grαβ q : Uαβ ÝÑ Pin˘ pn`1q α,βPA. This collection is the same as the collection specifying the Pin˘ structure St˘ V ppq in the proof of the SpinPin 5 property. This establishes (ses5). Suppose V , o, and p are as in (ses6), n “ rk V , and os is the OSpinstructure on V obtained from p and o as in the proof of the SpinPin 4 property. We can choose sections sr1α for pV, oq and sr2α for p2˘1qλpV q 2 are constant in (3.32) so that the associated transition functions r gαβ and equal rI2˘1 . Since ` ˘ ` ˘ r and r rI1 “ r r “r r ιn`1;n A, ι1n`1;n A ι n´ pAq ` ˘ ` ˘ r r rI3 “ r r “r ι1n`3;n A ι n` pAq r ιn`3;n A, r P Spinpnq, the transition data for xxos, os0 pp2˘1qλpV, oqqyy‘ for all A in the above construction is the same as for Co˘ V ppq in the proof of the SpinPin 6 property. This establishes (ses6). It remains to establish (ses3). We continue with the setup and notation in the paragraph containing (3.32) and in the paragraph immediately after it. Define (  2 by det gαβ “ p´1qηαβ . ηαβ : Uαβ ÝÑ Z2 α,βPA ˇ By Corollary A.7, the Cech cohomology class rηs represented by tηαβ u 2 is w1 pV q. Let ` ˘ s1 ” SpinpV 1 , o1 q, q 1V (3.36) be the Spin-structure on pV 1 , o1 q constructed from the Spinstructure s on pV 1 , o1 q in the proof of the SpinPin 3 property. The analogs of the first collection in (3.33) and of the collection (3.35) for s1 are (  1 1 r gαβ and Im;1 : Uαβ ÝÑ Spinpmq α,βPA gpαβ ” rI´1 m;1 r (3.37) (  1 2 ιn;m pp gαβ , grαβ q : Uαβ ÝÑ Pin˘ pnq α,βPA, gpαβ ”r

Proof of Theorem 1.4(1): Classical Perspective

respectively. By (2.34), ` η ˘ gαβpInαβ rIn;pmq gαβ “ rI´1 p n;pmq r

67

@ α, β P A.

(3.38)

The OSpin-structure (3.36) is described by ˆğ ˙M 1 1 q V 1 : SpinpV , o q ” Uα ˆSpinpmq „ ÝÑ SOpV 1 , o1 q, αPA

` ˘ r r “ s1α pyq¨Im;1 qm pAq, q V 1 ry, As ` 1 ˘ ` ˘ r „ y, A r P Uβ ˆSpinpmq Uα ˆSpinpmq Q y, gpαβ pyqA ` ˘ r P Uαβ ˆSpinpmq, α, β P A. @ y, A The collections (3.37) arise from the local sections of SpinpV 1 , o1 q given by sp1α : Uα ÝÑ Uα ˆSpinpmq Ă SpinpV 1 , o1 q,

sp1α pyq “ ry, rIm s .

The analog of the collection in (3.34) determined by these local sections is (  sα ” ιV 1 ,V 2 ps1α ¨Im;1 , s2α q “ sα ¨In;pmq : Uα ÝÑ ØpV q αPA . The Pin˘ -structure p ” xxos1 , p2 yy‘ on V is then given by ˆğ ˙M ˘ ˘ y Uα ˆPin pnq „ ÝÑ ØpV q, qpV : Pin pV q ” `

αPA

˘ r r “ sα pyq¨In;pmq qn pAq, qpV ry, As ` ˘ ` ˘ r „ y, A r P Uβ ˆPin˘ pnq Uα ˆPin˘ pnq Q y, gpαβ pyqA ` ˘ r P Uαβ ˆPin˘ pnq, α, β P A. @ y, A By the proof of Proposition 3.3, the Pin˘ -structure ` ˘ η¨p ” Pin˘ η pV q, qη on V is described by the displayed equation following (3.35) with grαβ η replaced by grαβpInαβ . By (3.38), the map ‰ ` ˘ “ r y˘ r ry, As r “ y, rI´1 A r : Pin˘ Ψ Ψ η pV q ÝÑ Pin pV q, `

˘ r P Uα ˆPin˘ pmq, α P A, @ y, A

n;pmq

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Spin/Pin-Structures & Real Enumerative Geometry

r is thus well defined. It is Pin˘ pnq-equivariant and satisfies qη “ qpV ˝ Ψ. ˘ We conclude that rη¨ps “ rps in P pV q. This establishes the first property in (ses3). ` 1 2 Let n “ n`2 ˘1 as before. The Spin-structures Co˘ V xxos , p yy‘ q 2 and xxos1 , Co˘ V 2 pp qyy‘ are specified by the transition data ˘ (  ˘` 1 2 , gαβ q : Uαβ ÝÑ Spinpn˘ q α,βPA and ιn;m pgαβ r ιn r ` 1  ˘ ( 2 r ιn˘ ;m gαβ ,r ι˘ n´m pgαβ q : Uαβ ÝÑ Spinpn˘ q α,βPA, respectively. By (2.46), these two collections are the same. This estab lishes the second property in (ses3). Completion of proof of SpinPin 2 property. In light of Proposition 3.3, it remains to establish (1.13). It is immediate that this identity holds if the rank of V 1 is 0. By Example 3.4, it also holds if the rank of V 2 is 0. We thus assume that the ranks of V 1 and V 2 are positive. We define automorphisms of vector bundles V 1 , V 2 , V 1 ‘V 2 by Ψ1 : V 1 ÝÑ V 1 ,

Ψ2 : V 2 ÝÑ V 2 ,

Ψ1 pv 1 q “ ´v 1 ,

Ψ2 pv 2 q “ ´v 2 ,

p : V 1 ‘V 2 ÝÑ V 1 ‘V 2 , Ψ p 1 , v 2 q “ pv 1 , ´v 2 q. Ψpv

By Example 3.4 and the SpinPin 4 property, # os1 if rk V 1 P 2Z; Ψ1˚ os1 “ os1 if rk V 1 R 2Z;

(3.39)

these two statements were also established in Examples 3.1 and 3.6 directly. By the naturality of the xx¨, ¨yy‘ -operation of the SpinPin 7 property, Ψ˚ xxos1 , p2 yy‘ “ xxΨ1˚ os1 , p2 yy‘ , p ˚ xxos1 , p2 yy‘ “ xxos1 , Ψ2˚ p2 yy‘ Ψ

(3.40)

for all os1 P OSpinpV 1 q and p2 P OSpinpV 2 q. Suppose V 1 is orientable and V 2 admits a Pin˘ -structure p2 . By the SpinPin 1 property, V 1 then admits an OSpin-structure if and

Proof of Theorem 1.4(1): Classical Perspective

69

only if V 1 ‘V 2 admits a Pin˘ -structure. Along with the H 1 pY ; Z2 qequivariance of xx¨, ¨yy‘ in the first input and Proposition 3.3, this implies that the map

OSppV 1 q ÝÑ P˘ pV 1 ‘V 2 q,

os1 ÝÑ xxos1 , p2 yy‘ ,

is a bijection. It is thus sufficient to assume that p “ xxos1 , p2 yy‘ in this case. By the first statement in (3.40), (3.39), and the first statement in the SpinPin 7(ses3) property, ` ˘ Ψ˚ xxos1 , p2 yy “ prk V 1 qw1 pV 2 q ¨xxos1 , p2 yy. Thus, (1.13) holds if V 1 is orientable and V 2 admits a Pin˘ -structure. Suppose V 1 is orientable and admits an OSpin-structure os1 . By the SpinPin 1 property, V 2 then admits a Pin˘ -structure if and only if V 1 ‘V 2 does. Along with the H 1 pY ; Z2 q-equivariance of xx¨, ¨yy‘ in the second input and Proposition 3.3, this implies that the map

P˘ pV 2 q ÝÑ P˘ pV 1 ‘V 2 q,

p2 ÝÑ xxos1 , p2 yy‘ ,

is an H 1 pY ; Z2 q-equivariant bijection. Combining this with the second statement in (3.40) and Example 3.4, we obtain @@ ` ˘ DD p ˚ xxos1 , p2 yy‘ “ xxos1 , Ψ2˚ p2 yy‘ “ os1 , prk V 2 ´1qw1 pV 2 q ¨p Ψ ‘ ` ˘ @@ 1 2 DD 2 2 “ prk V ´1qw1 pV q ¨ os , p ‘ for all p2 P P˘ pV 2 q. Thus, ` ˘ p ˚ p “ prk V 2 ´1qw1 pV 2 q ¨p Ψ

@ p P P˘ pV 1 ‘V 2 q .

(3.41)

By symmetry, this implies (1.13) if V 2 is orientable and admits an OSpin-structure. By Proposition 3.3, for every p P P˘ pV 1‘V 2 q, there exists a unique ηppq P H 1 pY ; Z2 q such that Ψ˚ p “ ηppq¨p. ˘ For os P OSppV 1 ‘V 2 q, let ηposq ” ηppq if os “ R˘ o ppq with p P P pV q; by the SpinPin 4 property, this class is the same for the two possibilities for p. Since Ψ is orientation-preserving on a fiber if and only if

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Spin/Pin-Structures & Real Enumerative Geometry

rk V 1 is even, # Ψ˚ os “ ηposq¨

os,

if rk V 1 P 2Z;

os,

if rk V 1 R 2Z.

Along with the H 1 pY ; Z2 q-equivariance of xx¨, ¨yy‘ in the first input and the first statement in the SpinPin 7(ses3) property, this implies that @@ ˚ ` ˘ DD Ψ os, p3 ‘ “ ηposq`prk V 1 qw1 pV 3 q ¨xxos, p3 yy‘

(3.42)

for every vector bundle V 3 over Y and every p3 P P˘ pV 3 q. Suppose V 1 admits a Pin˘ -structure p1 and V 1 ‘V 2 is orientable; in particular, w1 pV 1 q “ w1 pV 2 q. By the SpinPin 1 property and the first assumption, V 1 ‘ V 2 admits an OSpin-structure if and only if V 1 ‘ V 2 ‘ V 1 admits a Pin˘ -structure. Along with the H 1 pY ; Z2 qequivariance of xx¨, ¨yy‘ in the first input and Proposition 3.3, this implies that the map

OSppV 1 ‘V 2 q ÝÑ P˘ pV 1 ‘V 2 ‘V 1 q,

os ÝÑ xxos, p1 yy‘ ,

is an H 1 pY ; Z2 q-equivariant bijection. By the orientability of V 2 ‘V 1 and the conclusion following (3.41) with V 2 replaced by V 2 ‘V 1 , ` (˚  Ψ‘idV 1 xxos, p1 yy‘ “ prk V 1 ´1qw1 pV 1 q

˘ `prk V 1 qw1 pV 2 ‘V 1 q ¨xxos, p1 yy‘ ` ˘ @@ DD “ prk V 1 ´1qw1 pV 1 q ¨ os, p1 ‘

for all os P OSppV 1 ‘V 2 q. By the naturality of the xx¨, ¨yy‘ -operation and (3.42), @@ (˚ DD  Ψ‘idV 1 xxos, p1 yy‘ “ Ψ˚ os, p1 ‘ ` ˘ “ ηposq`prk V 1 qw1 pV 1 q ¨xxos, p1 yy‘ . Combining the last two equations, we obtain ηposq “ w1 pV 1 q. Thus, (1.13) holds if V 1 admits a Pin˘ -structure and V 1 ‘V 2 is orientable.

Proof of Theorem 1.4(1): Classical Perspective

71

We now consider the general case and show that ηppq “ prk V 1 ´1qw1 pV 1 q`prk V 1 qw1 pV 2 q

@ p P P˘ pV 1 ‘V 2 q. (3.43) By the surjectivity of the Hurewicz homomorphism on π1 [42, Proposition 7.5.2] and the Universal Coefficient Theorem for Cohomology [35, Theorem 53.5], the homomorphism ` ˘ κ : H 1 pY ; Z2 q ÝÑ Hom π1 pY q, H 1 pS 1 ; Z2 q , (3.44) ˘  (` κpηq α : S 1 ÝÑ Y “ α˚ η, is injective. It is thus sufficient to show that α˚ ηppq “ prk V 1 ´1qα˚ w1 pV 1 q`prk V 1 qα˚ w1 pV 2 q P H 1 pS 1 ; Z2 q for every α P LpY q. By the naturality of ηp¨q and w1 p¨q, this is equivalent to ηpα˚ pq “ prk V 1 ´1qw1 pα˚ V 1 q`prk V 1 qw1 pα˚ V 2 q P H 1 pS 1 ; Z2 q. (3.45) By the SpinPin 1 property, the bundles α˚ V 1 , α˚ V 2 , and α˚ pV 1‘V 2 q over S 1 admit a Pin˘ -structure; each of them admits an OSpinstructure if orientable. At least one of the three vector bundles is orientable. Thus, (3.45) holds because the three cases under  which (1.13) is shown to hold above cover all cases if Y “ S 1 .

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Chapter 4

Proof of Theorem 1.4(1): Trivializations Perspectives

We establish the statements of Section 1.2 for the notions of Spinstructure and Pin˘ -structure arising from Definitions 1.2 and 1.3 in Sections 4.3 and 4.2, respectively. The proofs of these statements in the perspective of Definition 1.3 rely heavily on the simple topological observations of Section 4.1. 4.1

Topological Preliminaries

Let V be a rank n vector bundle over a topological space Y . For o P OpV q, we denote by TrivpV, oq the set of homotopy classes of trivializations of the oriented vector bundle pV, oq. The following lemma is used to establish the last equality in (1.17) for vector bundles V of ranks 1, 2. Lemma 4.1. Let V be an even-rank vector bundle over a CW complex Y of dimension at most 2. For every o P OpV q, the maps ` ˘ TrivpτY ‘V, oY oq ÝÑ Triv τY ‘V ‘τY , oY ooY , rΦs ÝÑ rΦ1 s, rΦ2 s, ` ` ˘ ˘ Φ1 pr1 , v, r2 q “ Φpr1 , vq, r2 , Φ2 pr1 , v, r2 q “ r1 , Φpr2 , vq , (4.1) are the same.

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Spin/Pin-Structures & Real Enumerative Geometry

74

Proof.

We can assume that n ” rk V ě 2. Let ι1 , ι2 : SOpn`1q ÝÑ SOpn`2q

be the homomorphisms induced by the inclusions Rn`1 ÝÑ Rn`2 ,

pr, vq ÝÑ pr, v, 0q, p0, v, rq @ pr, vq P RˆRn .

Since the Lie group homomorphisms ι1 and ι2 differ by an automorphism of SOpn`2q, ` ˘ ` ˘ ι1˚ “ ι2˚ : π1 SOpn`1q ÝÑ π1 SOpn`2q “ Z2 . (4.2) Given a trivialization Φ of pτY ‘V, oY oq, let Φ1 , Φ2 be the trivializations of pτY ‘V ‘τY , oY ooY q as in (4.1) and h : Y ÝÑ SOpn`2q be the continuous map so that Φ2 “ hΦ1 . Since V is orientable, we can assume that its restriction to the 1-skeleton Y1 of Y is trivial. The restriction of Φ to Y1 is then given by ˇ Φ|Y1 : pτY ‘V qˇY1 ÝÑ Y1 ˆRn`1 , ` ˘ Φ|Y1 py, r, vq “ y, φpyqpr, vq @ py, r, vq P Y1 ˆRˆRn , for some continuous map φ : Y1 ÝÑ SOpn`1q. Thus, ` ˘ Φ1 |Y1 py, wq “ y, pι1 pφpyqqqw , ˇ ` ˘ @ py, wq P pτY ‘V ‘τY qˇY1 , Φ2 |Y1 py, wq “ y, pι2 pφpyqqqw with ι2 ˝ φ “ h ¨ pι1 ˝ φq. By (4.2), h|Y1 is homotopic to a constant map. Since π2 pSOpn`2qq is trivial, it follows that h itself is homotopic to a constant map. Thus, the trivializations Φ1 and Φ2 are  homotopic. For o, o1 P OpV q, we define a bijection Ro1 ,o : TrivpV, oq ÝÑ TrivpV, o1 q by sending the homotopy class of a trivialization Φ of pV, oq to the homotopy class Ro1 ,o prΦsq containing the trivialization hΦ for some locally constant map h : Y ÝÑ Opnq. The following statement is the key input used to establish (1.13) and the first statement of the SpinPin 7(ses3) property in the perspectives of Definitions 1.2 and 1.3. Lemma 4.2. Let V 1 and V 2 be vector bundles over a topological space Y , Ψ be as in (1.12), o P OpV 1 ‘V 2 q, and  P TrivpV 1 ‘V 2 , oq.

Proof of Theorem 1.4(1): Trivializations Perspectives

75

(a) If Y is a CW complex of dimension at most 2, V 1 is orientable, and rkpV 1 ‘V 2 q ě 3, then Ψ˚  “ RΨ˚ o,o pq. (b) If Y is paracompact and locally path connected and V 1 is not orientable, then Ψ˚  ‰ RΨ˚ o,o pq. Proof. The assumptions in both cases imply that V 1 ‘ V 2 is orientable. We denote by m and n the ranks of V 1 and V 1 ‘V 2 , respectively. Let Φ : V 1 ‘V 2 ÝÑ Y ˆRn

and h : Y ÝÑ Opnq

be a trivialization in  and the continuous map so that Ψ˚ Φ “ hΦ, respectively. (a) Since V 1 and V 2 are orientable, we can assume that their restrictions to the 1-skeleton Y1 of Y are trivial. The restriction of Φ to Y1 is then given by ˇ ` ˘ Φ|Y1 : pV 1 ‘V 2 qˇY1 ÝÑ Y1 , Φ|Y1 py, vq “ y, φpyqv , for some continuous map φ : Y1 ÝÑ SOpnq. Thus, ` ` ˘˘ Ψ˚ Φ|Y1 py, vq “ y, A A´1 φpyqA v , where A P Opnq is the diagonal matrix with the first m diagonal entries equal ´1 and the remainder equal 1. The map A´1 φA : Y1 ÝÑ SOpnq is homotopic to the identity if and only if φ is. Since π1 pSOpnqq “ Z2 , it follows that the maps φ and A´1 φA are homotopic. Thus, h|Y1 is homotopic to a constant map. Since π2 pSOpnqq is trivial, it follows that h itself is homotopic to a constant map. r rm ÝÑ Gpmq be the tautological vec(b) Let γm ÝÑ Gpmq and γ tor bundles over the Grassmannians of real m-planes and oriented rm for some m-planes, respectively. By [34, Theorem 5.6], V 1 « f ˚ γ continuous map f : Y ÝÑ Gpmq. Since the map “ ‰ ` ˘ α : S 1 ÝÑ Gpmq ÝÑ α˚ w1 pγm q, π1 Gpmq ÝÑ H 1 pS 1 ; Z2 q, is an isomorphism and V 1 is not orientable, [36, Lemma 79.1] implies that there exists α P LpY q such that α˚ w1 pV 1 q ‰ 0 (otherwise, f would r lift over the projection Gpmq ÝÑ Gpmq and V 1 would be a pullback of the oriented vector bundle γ rm ). Thus, it is sufficient to establish the claim for Y “ RP1 .

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Let γR;1 ÝÑ RP1 be the tautological line bundle. By the assumptions on V 1 and V 2 , V 1 « γR;1 ‘pm´1qτRP1 ,

V 2 « γR;1 ‘pn´m´1qτRP1 .

By (a), the negation of each τRP1 component sends  to Ro,o pq. Thus, we can assume that m “ 1. Under the standard identification of R2 with C, two trivializations of 2γR;1 over RP1 are given by ˘ ` ˘ ` Φ` reiθ s, aeiθ , beiθ “ reiθ s, eiθ pa`ibq @ a, b P R, ` iθ ˘ ` ˘ Φ´ re s, aeiθ , beiθ “ reiθ s, e´iθ pa`ibq @ a, b P R. They extend to trivializations of V 1 ‘V 2 by the identity on the τRP1 components and thus determine elements ´ , ` of TrivpV 1 ‘V 2 , oq, for one of the two orientations o on V 1 ‘V 2 . Since the conjugation on C identifies the pullback of Φ` by the negation of the second copy of γR;1 with Φ´ , ` ˘ (4.3) Ψ˚ ˘ “ Ro,o ¯ . On the other hand, the trivializations Φ` and Φ´ differ by the multiplication by the map RP1 ÝÑ S 1 “ SOp2q,

reiθ s ÝÑ e2iθ .

Since this map generates π1 pSOp2qq and the homomorphism π1 pSOp2qq ÝÑ π1 pSOpnqq induced by the inclusion R2 ÝÑ Rn is surjective, it follows that ` ‰ ´ . Combining this with (4.3), we obtain  the claim. For a bordered surface Σ, we denote by rΣsZ2 P H2 pΣ, BΣ; Z2 q the Z2 -fundamental homology class of pΣ, BΣq. The first statement and a special case of the second statement of Lemma 4.3 are used to establish the SpinPin 1 and 2 properties in the perspective of Definition 1.3. The general case of the second statement and the third statement are used for similar purposes in Chapter 6. Lemma 4.3. Let X be a topological space and Y Ă X. (a) For every collection α1 , . . . , αk P LpY q such that α1 ` ¨ ¨ ¨ ` αk is a singular Z2 -boundary in Y , there exists a continuous map

Proof of Theorem 1.4(1): Trivializations Perspectives

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F : Σ ÝÑ Y from a bordered surface so that F |BΣ is the disjoint union of the loops α1 , . . . , αk . If Y is path-connected, then Σ can be chosen to be connected. (b) For every b P H2 pX, Y ; Z2 q, there exists a continuous map F : pΣ, BΣq ÝÑ pX, Y q from a bordered surface so that ` ˘ b “ F˚ rΣsZ2 P H2 pX, Y ; Z2 q. (4.4) (c) For every continuous map u : pΣ, BΣq ÝÑ pX, Y q from a bordered surface such that u˚ rΣsZ2 vanishes in H2 pX, Y ; Z2 q, there exists a continuous map u1 : Σ1 ÝÑ Y from a bordered surface so that BΣ “ BΣ1 , u|BΣ “ u1 |BΣ1 , and the continuous map uYu1 : ΣYΣ1 ÝÑ X obtained by gluing u and u1 along the boundaries of their domains vanishes in H2 pX; Z2 q. Proof. As in [51, Section 2.1], we denote by Δk Ă Rk the standard k-simplex and by Δkj with j “ 0, 1, . . . , k its facets. A singular Z2 chain b in X is a sum of finitely singular 2-simplices, i.e. continuous maps Fi : Δ2 ÝÑ X. (a) Suppose α1 , . . . , αk P LpY q and b is a singular Z2 -chain in Y as above so that Bb “ α1 `¨ ¨ ¨`αk P S1 pY ; Z2 q.

(4.5)

We can then identify k of the restrictions Fi |Δ2 with the maps αi and j pair up the remaining restrictions as in [51, Lemma 2.11]. By [36, Theorem 77.5], the topological space Σ obtained by identifying the domains of Fi along the paired up edges is a closed surface with some open disks removed. The closures of these disks may intersect but only at the vertices of the glued 2-simplices. The maps Fi induce a continuous map F : Σ ÝÑ Y so that its restriction to the boundary of Σ is the sum in (4.5). The vertices of the restrictions of Fi |Δ2 corresponding to some of j the maps αi may be identified with the vertices of other such restrictions, instead of or in addition to being identified with just each other; see the left side of Figure 4.1. If this happens with one or both of the vertices of the restriction Fi |Δ2 corresponding to one of the j loops αi , we subdivide Fi into three singular simplices and insert two additional singular 2-simplices each of which takes constant value

Spin/Pin-Structures & Real Enumerative Geometry

78

x1 α1 x1

c a

x1

x1

a

x1

α1

α2

c

x1

x1

x1

x1 c

x1

x1

a

α2

c a

x1

Figure 4.1. Modification of a bounding chain for α1 and α2 to produce a bounding surface.

αi p0q “ αi p1q along one of the edges; see the right side of Figure 4.1. We then pair up the two constant edges to separate the edge corresponding to αi into its own boundary component, adding an extra handle in the process. After separating off the edges corresponding to all loops αi into their own boundary components in this way, we obtain a bordered surface and a continuous map as in (a). If Y is path-connected, we can connect the topological components of Σ by thin cylinders and map them to appropriate paths in Y . (b) If b above is a cycle in pX, Y q, then the restrictions Fi |Δ2 j that are not mapped into Y come in pairs as in [51, Lemma 2.10]. The topological space Σ obtained by identifying the domains of Fi along the paired up edges is a closed surface with some open disks removed; the edges forming its boundary BΣ are mapped into Y . After a modification as in (a), Σ can be assumed to be a bordered surface. The maps Fi induce a continuous map F : pΣ, BΣq ÝÑ pX, Y q so that ` ˘ rbs “ F˚ rΣsZ2 P H2 pX, Y ; Z2 q. This establishes (b). (c) Choose a triangulation of Σ and a collection Fi : Δ3 ÝÑ X of singular 3-simplices that bound the 2-chain determined by the triangulation of Σ in the relative singular group S2 pX, Y ; Z2 q. It can be assumed that no more than two singular 1-simplices obtained by restricting the maps Fi to the edges of Δ3 are the same (same 1-simplices come precisely in pairs). The restrictions Fi |Δ3j that do not come in pairs are either identified with the 2-simplices of the triangulation of Σ or are mapped to Y . The restrictions of such singular 2-simplices Fi |Δ3 to the edges come in pairs. The topological j space Σ1 obtained by identifying the domains of the 2-simplices Fi |Δ3 j

Proof of Theorem 1.4(1): Trivializations Perspectives

79

that are neither paired nor associated with the 2-simplices of Σ along the paired up edges is a closed surface with some open disks removed; the edges forming its boundary BΣ1 correspond to the edges forming the boundary BΣ of Σ. After a modification as in (a), Σ1 can be assumed to be a bordered surface with BΣ1 “ BΣ and the induced map u : Σ1 ÝÑ Y satisfying u|BΣ “ u1 |BΣ1 . The glue map u Y u1 bounds the collection of the singular 3-simplices Fi and thus vanishes in H2 pX; Z2 q.  4.2

The Spin- and Pin-Structures of Definition 1.3

Throughout this section, the terms Spin-structure and Pin˘ structure refer to the notions arising from Definition 1.3. It is sufficient to establish the statements of Section 1.2 under the assumption that the base topological space Y is path-connected; this will be assumed to be the case. Proof of SpinPin 2 property. By the definition of Spin- and Pin˘ -structures in this perspective, it is sufficient to establish the claims of this property other than (1.13) for Spin-structures s ” psα qα on oriented vector bundles pV, oq with n ” rk V ě 3. By (1.7), the oriented vector bundle α˚ pV, oq over S 1 has two homotopy classes of trivializations for every α P LpY q. Let s be a Spin-structure on pV, oq and η P H 1 pY ; Z2 q. We define the Spinstructure η ¨ s ” pη ¨ sα qα on pV, oq by # “ sα , if α˚ η “ 0 P H 1 pS 1 ; Z2 q; @ α P LpY q. η ¨ sα ‰ sα , if α˚ η “ 0 P H 1 pS 1 ; Z2 q; Suppose F : Σ ÝÑ Y is a continuous map from a connected bordered surface and α1 , . . . , αm : S 1 ÝÑ Y

(4.6)

are the restrictions of F to the boundary components of Σ. Since m ÿ @

D @ D α˚i η, rS 1 s “ F ˚ η, BΣ “ 0 P Z2 ,

i“1

the number of boundary components αi of F such that pη ¨ sqα ‰ sα is even. Since the trivializations in s extend to trivializations of

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F ˚ pV, oq, Corollary A.14 then implies that this is also the case for the trivializations in η ¨ s. Thus, η ¨ s is indeed a Spin-structure on pV, oq. It is immediate that this construction defines a group action of H 1 pY ; Z2 q on the set of such structures. By the injectivity of the homomorphism (3.44), η ¨ sα ‰ sα for some α P LpY q if η ‰ 0 and so this action of H 1 pY ; Z2 q is free. Suppose s1 ” ps1α qα is another Spin-structure on pV, oq. Define # 0, if s1α “ sα ; η : LpY q ÝÑ Z2 , ηpαq “ 1, if s1α ‰ sα . This determines a linear map from the Z2 -vector space generated by LpY q to Z2 . Suppose F : Σ ÝÑ Y is a continuous map from a bordered surface with boundary components (4.6). By Definition 1.3, the homotopy classes of trivializations of F ˚ pV, oq|BΣ determined by s and s1 extend to a trivialization of F ˚ pV, oq. Corollary A.14 then implies that the number of boundary components αi of F such that s1α ‰ sα is even and so m ÿ ηpαi q “ 0 P Z2 . i“1

Along with the surjectivity of the Hurewicz homomorphism for π1 and Lemma 4.3(a), this implies that η descends to a homomorphism η : H1 pY ; Z2 q ÝÑ Z2 . By the Universal Coefficient Theorem for Cohomology, such a homomorphism corresponds to an element of H 1 pY ; Z2 q, which we still denote by η. By the definition of η and the construction above, η ¨ s “ s1 . Thus, the action of H 1 pY ; Z2 q described above is transitive. By the definition of the above action of H 1 pY ; Z2 q on SppV, oq, ˚

α pη ¨ sq

# “ α˚ s,

if α˚ η “ 0 P H 1pS 1 ; Z2 q;

‰ α˚ s,

if α˚ η ‰ 0 P H 1pS 1 ; Z2 q;

@ s P SppV, oq, η P H 1 pY ; Z2 q, α P LpY q.

(4.7) Along with Lemma 4.2, this implies that # s, if rk V 1 P 2Z; Ψ˚ s “ w1 pV 1 q ¨ Ro,o psq, if rk V 1 R 2Z; with Ψ as in (1.13).

@ o P OpV 1 ‘V 2 q, s P SppV 1 ‘V 2 , oq, (4.8)

Proof of Theorem 1.4(1): Trivializations Perspectives

81

Let V 1 and V 2 be as in (1.13) and # pV 1 , V 2 ‘p2˘1qλpV 1 qbλpV 2 qq if rk V 1 P 2Z; 1 2 pW , W q “ pV 1 ‘p2˘1qλpV 1 qbλpV 2 q, V 2 q if rk V 1 R 2Z. The automorphism Ψ in (1.12) induces the analogous automorr on W 1 ‘ W 2 . Since a Pin˘ -structure p on V 1 ‘ V 2 in this phism Ψ perspective is a Spin-structure s˘ on pW 1‘W 2 , o˘ V 1 ‘V 2 q, the first case r in (4.8) with Ψ replaced by Ψ gives ` ˘ r ˚ s˘ “ w1 pW 1 q ¨ s˘ ” w1 pV 1 q`prk V 1 qpw1 pV 1 q`w1 pV 2 qq ¨ p . Ψ˚ p ” Ψ This establishes (1.13).



Proof of SpinPin 1 property. By the definition of Spin- and Pin˘ -structures in this perspective, it is sufficient to establish this claim for Spin-structures on oriented vector bundles pV, oq with n ” rk V ě 3. Suppose pV, oq admits a Spin-structure s ” psα qα . For every continuous map F : Σ ÝÑ Y from a closed surface, the bundle F ˚ V is then trivializable and so D @ D @ w2 pV q, F˚ prΣsZ2 q “ w2 pF ˚ V q, rΣsZ2 “ 0. Along with Lemma 4.3(b) with pX, Y q replaced by pY, Hq, this implies that xw2 pV q, by “ 0 for every b P H2 pY ; Z2 q. By the Universal Coefficient Theorem for Cohomology [35, Theorems 53.5], the homomorphism ` ˘  ( κpq pbq “ x, by, κ : H 2 pY ; Z2 q ÝÑ HomZ2 H2 pY ; Z2 q, Z2 , is an isomorphism. Combining the last two statements, we conclude that w2 pV q “ 0. Suppose w2 pV q “ 0. Choose a collection C ” tαi ui of loops in Y that form a basis for H1 pY ; Z2 q and a trivialization φi of α˚i pV, oq for each loop in C. Given α P LpY q, let α1 , . . . , αk P C be so that α`α1 `¨ ¨ ¨`αk P BS2 pY ; Z2 q. By Lemma 4.3(a), there exists a continuous map F : Σ ÝÑ Y from a connected bordered surface which restricts to α, α1 , . . . , αk : S 1 ÝÑ Y

(4.9)

for some parametrization of BΣ by k ` 1 copies of S 1 . By Corollary A.13, there exists a trivialization Φ of F ˚ V ÝÑ Σ so that its

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Spin/Pin-Structures & Real Enumerative Geometry

restriction to each αi agrees with φi . We take the homotopy class sα of the trivializations for α˚ V ÝÑ S 1 to be the homotopy class of the restriction of Φ. Suppose F 1 : Σ1 ÝÑ Y is another continuous map satisfying the conditions of the previous paragraph and Φ1 is a trivialization of F 1˚ V ÝÑ Σ so that its restriction to each αi agrees with φi . p (resp. Σ) r the closed (resp. bordered) surface obtained Denote by Σ 1 from Σ and Σ by identifying them along the boundary components r has two corresponding to α, α1 , . . . , αk (resp. α1 , . . . , αk ). Thus, Σ p is a boundary components, each of which corresponds to α, and Σ r connected surface obtained from Σ by identifying these two boundary components. The maps F and F 1 induce continuous maps p ÝÑ Y and Fr : Σ r ÝÑ Y, Fp : Σ p Σ. r The trivializations Φ which restrict to F and F 1 over Σ, Σ1 Ă Σ, 1 ˚ r of Fr pV, oq over Σ. r The bunand Φ induce a trivialization Φ ˚ n p r p dle F pV, oq over Σ is obtained from ΣˆR by identifying the copies of α˚ V via the clutching map ϕ : S 1 ÝÑ SOpnq determined by the difference between the trivializations of α˚ V induced by Φ and Φ1 . Since ˘ ` p Z2 q, w2 Fp˚ V “ Fp˚ w2 pV q “ 0 P H 2 pΣ; Corollary A.12 implies that ϕ is homotopically trivial. Thus, Φ and Φ1 determine the same homotopy class sα of trivializations of α˚ pV, oq. It remains to verify that s ” psα qα satisfies the condition of Definition 1.3. Suppose F : Σ ÝÑ Y is a continuous map from a connected bordered surface and α11 , . . . , α1m : S 1 ÝÑ Y are the restrictions of F to the boundary components of Σ. If m “ 0, the vector bundle F ˚ W is trivializable by Corollary A.11. In the following, we consider the case m P Z` . For each i “ 1, . . . , m, let αij : S 1 ÝÑ Y with j “ 1, . . . , ki , and

Φi : Fi˚ V

Fi : Σi ÝÑ Y,

ÝÑ Σi ˆRn

be the loops in C, a continuous map from a connected bordered surface, and a trivialization of Fi˚ pV, oq, respectively, as in

Proof of Theorem 1.4(1): Trivializations Perspectives

83

the construction of the homotopy class sα1i of trivializations of α1˚ i pV, oq below (4.9). Thus, Fi |BΣi is the disjoint union of the loops α1i , αi1 , . . . , αiki and the restriction of Φi to each αij agrees with the initially chosen trivialization φij . Let φ1i be the restriction of Φi to ˚ ˚ α1˚ i V Ă Fi V, F V.

By Corollary A.13, the trivializations φ11 , . . . , φ1m´1 extend to a trivialization Φ of F ˚ pV, oq. We show in the following that the restriction 1 of Φ to α1˚ m V is homotopic to φm , thus confirming the condition of Definition 1.3. Since the loops in C are linearly independent in H1 pY ; Z2 q and ki m ÿ ÿ i“1 j“1

αij “

m ÿ

α1i “ BF P S1 pY ; Z2 q,

i“1

p (resp. Σ) r the closed the loops αij come in pairs. Denote by Σ (resp. bordered) surface obtained from Σ1 , . . . , Σm , Σ by identifying them along the paired up boundary components corresponding to αij and along the boundary components corresponding to α11 , . . . , α1m r has two boundary components, each (resp. α11 , . . . , α1m´1 ). Thus, Σ 1 p is a connected surface obtained of which corresponds to αm , and Σ r from Σ by identifying these two boundary components. The maps F1 , . . . , Fm and F induce continuous maps p ÝÑ Y Fp : Σ

r ÝÑ Y, and Fr : Σ

p Σ. r The trivializawhich restrict to Fi and F over Σi , Σ Ă Σ, r r tions Φ1 , . . . , Φm and Φ induce a trivialization Φ of Fr˚ V over Σ. ˚ n p r p The bundle F V over Σ is obtained from ΣˆR by identifying the 1 copies of α1˚ m V via the clutching map ϕ : S ÝÑ SOpnq determined by the difference between the trivializations of α1˚ m V induced by Φm and Φ. Since ˘ ` p Z2 q, w2 Fp˚ V “ Fp˚ w2 pV q “ 0 P H 2 pΣ; Corollary A.12 implies that ϕ is homotopically trivial. Thus, Φm and Φ determine the same homotopy class of trivializations  of α1˚ mV .

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Proof of SpinPin 3, 4 property. For the purposes of establishing the first of these properties, we can assume that n ” rk V ě 3. For o P OpV q and s P SppV, oq, we define the Spin-structure s P SppV, oq by  ( sα “ φ ” In;1 φ : φ P sα (4.10) @ α P LpY q, with In;1 P Opnq as below (2.21). If F : Σ ÝÑ Y is a continuous map from a bordered surface with boundary components (4.6) and Φ is a trivialization of F ˚ pV, oq so that its restriction to the boundary component pBΣqi of Σ corresponding to αi lies in sαi for every i, then the restriction of the trivialization In;1 Φ of F ˚ pV, oq to pBΣqi lies in sαi . Thus, s is indeed a Spin-structure on pV, oq. By (4.7), the resulting bijection (1.14) is H 1 pY ; Z2 q-equivariant. It also satisfies the last condition of the SpinPin 3 property. We next describe a bijection between SppV, oq and ˘ ˘ P˘ pV q ” SppV ‘p2˘1qλpV q, o˘ V “ SppV ‘p2˘1qτY , op2˘1qoY ; the last equality holds because the orientation o on V determines a canonical homotopy class of trivializations of α˚ λpV, oq for every loop α in Y . By the SpinPin 1 property, pV, oq admits a Spin-structure if and only if V admits a Pin˘ -structure. We can thus assume that pV, oq admits a Spin-structure. If rk V ě 3, we associate a Spin-structure s ” psα qα on pV, oq with ˘ the Spin-structure s˘ ” ps˘ α qα on pV˘ , oV q given by  ( (4.11) @ α P LpY q. s˘ α “ φ˘ ” φ‘p2˘1qpdet φq : φ P sα If F and Φ are as below (4.10), then the restriction of the trivialization n`2˘1 Φ˘ ” Φ‘p2˘1qpdet Φq : F ˚ pV˘ , o˘ V q ÝÑ ΣˆR

to the boundary component of Σ corresponding to αi lies in s˘ αi . ˘ ˘ Thus, s is indeed a Spin-structure on pV˘ , oV q. By (4.7), the map

SppV, oq ÝÑ P˘ pV q,

s ÝÑ s˘ ,

(4.12)

is H 1 pY ; Z2 q-equivariant. Along with the SpinPin 2 property, this implies that this map is a bijection. We take the map R˘ o in (1.15)

Proof of Theorem 1.4(1): Trivializations Perspectives

85

to be its inverse. If φ is a trivialization of α˚ pV, oq as in (4.10), then ` ˘ ` ˘ φ ˘ “ In`2˘1;3˘1 φ˘ : α˚ V ‘p2˘1qλpV q ÝÑ S 1 ˆRn`2˘1 . Since this trivialization is homotopic to φ˘ , the last claim in the SpinPin 4 property holds. If rk V “ 2, we first identify the trivializations of α˚ pτY ‘V q with the trivializations of α˚ pV ‘τY q in the obvious way and then extend them to trivializations of α˚ pV ‘ 3τY q by the identity on the last two τY -summands. If rk V “ 1, we first identify the trivializations of ˘ ` α˚ p2τY ‘V q ” α˚ τY ‘pτY ‘V q with the trivializations of α˚ ppτY ‘V q‘τY q and of α˚ pV ‘ 2τY q in the obvious orientation-preserving ways and then extend the latter to the trivializations of α˚ pV ‘3τY q by the identity on the last τY -summand. These identifications of sections extend to pullbacks by continuous maps F as below (4.10). Thus, the collection s˘ ” ps˘ α q induced by s via these identifications is indeed a Spin-structure on pV˘ , o˘ V q. We note that ` ˘ ´ ´ ´ R´ o “ RStV poq : P pV q “ P pτY ‘V q ÝÑ SppV, oq “ Sp StpV, oq (4.13) under the identifications (1.19) if rk V “ 1 and o P OpV q. In all cases, the resulting bijections (1.15) are H 1 pY ; Z2 q-equivariant and satisfy  the last requirement of the SpinPin 4 property. Proof of SpinPin 5, 6 property. The two sides of (1.20) are the same by definition; we take this map to be the identity. The requirement after (1.20) is forced by the definition of the second map of the SpinPin 5 property in (4.15). Let n “ rk V . By the SpinPin 1 property, an oriented vector bundle pV, oq admits a Spin-structure if and only if pτY ‘ V, Stpoqq does. In order to establish the SpinPin 5 property, we can thus assume that there exists a Spin-structure on pV, oq. If n ě 3, we associate a Spin-structure s ” psα qα on pV, oq with the Spin-structure StV s ” pStV sα qα on pτY ‘V, Stpoqq given by  ( (4.14) @ α P LpY q. StV sα “ StV φ ” idτS 1 ‘φ : φ P sα If F : Σ ÝÑ Y is a continuous map from a bordered surface with boundary components (4.6) and Φ is a trivialization of F ˚ pV, oq

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so that its restriction to the boundary component pBΣqi of Σ corresponding to αi lies in sαi for every i, then the restriction of the trivialization ˘ ` StV Φ ” idτΣ ‘Φ : F ˚ τY ‘V, Stpoq ÝÑ ΣˆRn`1 to pBΣqi lies in StV sαi . Thus, StV s is indeed a Spin-structure on pτY ‘ V, Stpoqq. It is immediate that it satisfies the first condition after (1.16). With φ as in (4.10), ` StV φ “ StV φ : α˚ τY ‘V q ÝÑ S 1 ˆRn`1 . Thus, the resulting first map in (1.16) satisfies the first equality in (1.17). We take the second map in (1.16) to be ˇ ˇ : P˘ pV q ” SppV˘ , o˘ St˘ V ” StV˘ SppV˘ ,o˘ Vq Vq ` ˘ ÝÑ Sp StpV˘ , o˘ Vq ˘ ` ˘ ` ˘ “ Sp pτY ‘V q˘ , o˘ τY ‘V ” P τY ‘V ; (4.15) the identifications (1.2) imply the equality above. By (4.7), both maps (1.16) are then H 1 pY ; Z2 q-equivariant. In light of the SpinPin 2 property, this implies that they are bijections. With φ˘ as in (4.11), ˘ ` ˘ ` ˚ ˚ 1 n`3˘1 . StV φ ˘ “ St˘ V φ˘ : α pτY ‘V q˘ “ α τY ‘V˘ ÝÑ S ˆR Thus, the second equality in (1.17) also holds. For n “ 1, 2, the two sides of the first map in (1.16) are the same by definition. We take this map to be the identity then and define the second map in (1.16) by (4.15). The second equality in (1.17) is satisfied by (4.13) in the Pin´ -case if n “ 1 and by Lemma 4.1 in the three remaining cases. The first map in (1.16) satisfies the first condition after (1.16) and the first equality in (1.17) in both cases by definition. The two maps are H 1 pY ; Z2 q-equivariant in all six cases, by definition in three of the cases and by (4.7) in the remaining three  cases.

Proof of Theorem 1.4(1): Trivializations Perspectives

87

Proof of SpinPin 7 property. For every α P LpY q, a short exact sequence e of vector bundles over Y as in (1.21) determines a homotopy class of isomorphisms α˚ V « α˚ V 1 ‘α˚ V 2 so that α˚ ι is the inclusion as the first component on the right-hand side above and α˚ j is the projection to the second component. Thus, it is sufficient to establish the SpinPin 7 property for the direct sum exact sequences as in (1.22). Furthermore, an orientation o1 on V 1 determines a homotopy class of trivializations of α˚ λpV 1 q and thus of isomorphisms ˘ ` ˘ ` α˚ V 1 ‘V 2 ˘ ” α˚ V 1 ‘V 2 ‘p2˘1qλpV 1 ‘V 2 q ˘ ` ` ˘ (4.16) « α˚ V 1 ‘V 2 ‘p2˘1qλpV 2 q ” α˚ V 1 ‘V˘2 for every α P LpY q. Let V 1 and V 2 be vector bundles over Y of rank n1 and n2 , respectively. Suppose n1 , n2 ě 3, o1 P OpV 1 q, and o2 P OpV 2 q. For a Spin-structure s1 ” ps1α qα on pV 1 , o1 q and a Spin-structure s2 ” ps2α qα on pV 2 , o2 q, we define a Spin-structure xxs1 , s2 yy‘ ” ps1α ‘s2α qα on pV 1 ‘V 2 , o1 o2 q by  ( s1α ‘s2α “ φ1 ‘φ2 : φ1 P s1α , φ2 P s2α (4.17) @ α P LpY q. If F : Σ ÝÑ Y is a continuous map from a bordered surface with boundary components (4.6) and Φ1 , Φ2 are trivializations of F ˚ pV 1 , o1 q and F ˚ pV 2 , o2 q so that their restrictions to the boundary component pBΣqi of Σ corresponding to αi lie in s1αi and s2αi , respectively, for every i, then the restriction of the trivialization 1

Φ1 ‘Φ2 : F ˚ pV 1 ‘V 2 , o1 o2 q ÝÑ ΣˆRn `n

2

to pBΣqi lies in s1αi ‘s2αi . Thus, xxs1 , s2 yy‘ is indeed a Spin-structure on pV 1 ‘V 2 , o1 o2 q. The above construction determines the first map xx¨, ¨yy e in (1.24) if n1 , n2 ě 3. Along with (4.16), it also determines the second map xx¨, ¨yy e if n1 ě 3 and rk V˘2 ě 3 (i.e. not V´2 if n2 “ 1). By (4.7), both maps are H 1 pY ; Z2 q-biequivariant. These two maps satisfy (ses1),

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(1.25), and (ses4) for vector bundles of ranks at least 3 by definition. With φ˘ as in (4.11), ˘ ` 1 φ ‘φ2 ˘ “ φ1 ‘φ2˘ : α˚ pV 1 ‘V 2 q˘ ` ˘ ` ˘ 1 2 “ α˚ V 1 ‘ α˚ V˘2 ÝÑ S 1 ˆRn `n `2˘1 . Thus, (1.26) is also satisfied in these cases. For a vector bundle V over Y and a P Zě0 , let ` ˘ StaV : OSppV q ÝÑ OSp aτY ‘V ˘ ` and StaV : P˘ pV q ÝÑ P˘ aτY ‘V

(4.18)

denote a iterations of the maps (1.16). By the proof of the SpinPin 5 property and the construction above, StaV ppq “ xxos0 paτY q, pyy‘

(4.19)

for all a ě 3 and p P P˘ pV q if rk V˘ ě 3. For vector bundles V 1 , V 2 over Y and a, b P Zě0 , let ΨV 1 ,V 2 : aτY ‘bτY ‘V 1 ‘V 2 ÝÑ aτY ‘V 1 ‘bτY ‘V 2 be the obvious bundle isomorphism. If b P 2Zě0 , then ˘ ` a 1 b 1 2 ˚ 2 @ o1 P OpV 1 q, o2 P OpV 2 q. Sta`b V 1 ‘V 2 po o q “ ΨV 1 ,V 2 StV 1 po qStV 2 po q (4.20) If b P 2Zě0 and V 1 , V 2 split as direct sums of oriented line bundles, then ˘ ˘ ` ` a 1 2 ˚ 1 b 2 pV ‘V q “ Ψ os Sta`b 1 ,V 2 os0 StV 1 pos0 pV qq‘StV 2 pos0 pV qq . 1 2 0 V V ‘V (4.21) If b P 2Zě0 and rk V 1 , rk V 2 ě 3, then ` ˘ @@ a DD 1 2 ˚ 1 b 2 Sta`b V 1 ‘V 2 xxos , os yy‘ “ ΨV 1 ,V 2 StV pos q, StV pos q ‘ , DD ` ˘ @@ a 1 2 ˚ 1 b 2 (4.22) Sta`b V 1 ‘V 2 xxos , p yy‘ “ ΨV 1 ,V 2 StV pos q, StV pp q ‘ for all os1 P OSppV 1 q, os2 P OSppV 2 q, and p2 P P˘ pV 2 q; this follows immediately from the construction of xx¨, ¨yy‘ above. By the

Proof of Theorem 1.4(1): Trivializations Perspectives

89

SpinPin 6 property, the proofs of the SpinPin 5 and 6 properties, and the construction above, ` b ˘ ` ˘ ˘ b Co˘ bτY ‘V StV p “ StV CoV ppq ˘DD @@ ` b “ Ψ˚V,p2˘1qλpV,oq R˘ o ppq, StV os0 pp2˘1qλpV, oqq ‘ (4.23) for all b P 2Zě0 , o P OpV q, and p P P˘ pV q if rk V ě 3. For vector bundles V 1 , V 2 over Y not both of ranks at least 3, we define the two maps in (1.24) by (4.22) with b P 2Zě0 . By (4.22) for vector bundles V 1 , V 2 of ranks at least 3 and the SpinPin 5 property, the resulting maps are well defined and H 1 pY ; Z2 q-biequivariant. Furthermore, the definition of the second map agrees with the definition above if the rank of V˘2 is at least 3. By (ses1) for V 1 , V 2 of ranks at least 3, (4.21), and the first property after (1.16), the first map satisfies (ses1) for all V 1 , V 2 . By (4.20), this map satisfies (1.25). By (1.26) for V 1 , V 2 of ranks at least 3 and the second equality in (1.17), the two maps satisfy (1.26) for all V 1 , V 2 . Since the map (1.20) is identity under the identification (4.16), they satisfy the second property in (ses3). By (ses4) for V11 , V21 , V 2 of ranks at least 3, they satisfy (ses4) for all vector bundles V11 , V21 , V 2 . By the SpinPin 5 property and (4.19), the two maps satisfy (ses5) for all V . By the SpinPin 6 property and (4.23), they also satisfy (ses6) for all V . By the SpinPin 3 and 5 properties and (4.22), it is sufficient to establish the first property in (ses3) under the assumption that the ranks of V 1 and V 2 are at least 3 and the former is odd (after possibly to both sides for some a, b P Zě0 with b even). It is applying Sta`b V immediate from the definition of the map (1.9) and the construction of the map (1.14) in the perspective of Definition 1.3 that (3.39) holds in this perspective. Combining this with the first equation in (3.40) and (1.13), we obtain xxos1 , p2 yy‘ “ xxΨ1˚ os1 , p2 yy‘ “ Ψ˚ xxos1 , p2 yy‘ “ w1 pV 2 q ¨ xxos1 , p2 yy‘ . This establishes the SpinPin 7 property for all vector bundles V 1 and V 2 over Y . 

90

4.3

Spin/Pin-Structures & Real Enumerative Geometry

The Spin- and Pin-Structures of Definition 1.2

Throughout this section, the terms Spin-structure and Pin˘ structure refer to the notions arising from Definition 1.2. As these notions are restricted to CW complexes, the properties of Section 1.2 are fairly easy to establish in this setting. For n P Zě0 , a topological space Z is called n-connected if πk pZq is trivial for every k P Zě0 with k ď n. For topological spaces X and Z, we denote by rX, Zs the set of homotopy classes of continuous maps from X to Z. If in addition Y Ă X and z0 P Z, let rpX, Y q, pZ, z0 qs be the set of homotopy classes of continuous maps from X to Z sending Y to z0 . If Z is a group, then so are rX, Zs and rpX, Y q, pZ0 , idqs. If Y is a CW complex and k P Zě0 , we denote by Yk the k-skeleton of Y . Lemma 4.4 is used in the proof of the SpinPin 2 property and in Section 6.4. Lemma 4.4. Suppose n P Z` , Z is an pn´1q-connected topological space with |πn pZq| “ 2 and πn`1 pZq trivial, z0 P Z, and ` ˘ ` ˘ ηZ P H n Z, tz0 u; Z2 “ H n Z; Z2 is the generator. Let X be a CW complex and Y Ă X be a subcomplex. The map ‰ ` ˘ “ pXn`1 , Yn q, pZ, z0 q ÝÑ H n Xn`1 , Yn ; Z2 , rhs ÝÑ h˚ ηZ , (4.24) is then bijective, while the map ‰ ` ˘ “ pXn`2 , Yn q, pZ, z0 q ÝÑ H n Xn`2 , Yn ; Z2 , rhs ÝÑ h˚ ηZ , (4.25) is surjective. Proof. By the Hurewicz theorem [42, Proposition 7.5.2] and the Universal Coefficient Theorem for Cohomology [35, Theorem 53.5], the homomorphism @ D πn pXq ÝÑ Hn pX; Zq ÝÑ Z2 , rf : S n ÝÑ Xs ÝÑ ηZ , f˚ rS n sZ2 , is an isomorphism. Along with [42, Theorem 8.1.15], this implies that the map ‰ ` ˘ “ (4.26) pXn , Yn q, pZ, z0 q ÝÑ H n Xn , Yn ; Z2 , rhs ÝÑ h˚ ηZ , is a bijection.

Proof of Theorem 1.4(1): Trivializations Perspectives

91

Suppose δ “ 1, 2 and η P H n pXn`δ , Yn ; Z2 q. By the bijectivity of (4.26), there exists a continuous map f from pXn , Yn q to pZ, z0 q so that ˇ ` ˘ (4.27) η ˇpXn ,Yn q “ f ˚ ηZ P H n Xn , Yn ; Z2 . By [42, Theorem 8.1.17], f extends to a continuous map h from Xn`1 to Z. Since πn`1 pZq is trivial, h then extends to a continuous map from Xn`2 to Z. Thus, there exists a continuous map h from pXn , Yn q to pZ, z0 q so that ˇ ˇ ` ˘ (4.28) η ˇpXn ,Yn q “ ph˚ ηZ qˇpXn ,Yn q P H n Xn , Yn ; Z2 . By the cohomology exact sequence for the triples Yn Ă Xn Ă Xn`1 and Yn Ă Xn Ă Xn`2 , the restriction homomorphisms ` ˘ ` ˘ H n Xn`1 , Yn ; Z2 ÝÑ H n Xn , Yn ; Z2 ` ˘ ` ˘ and H n Xn`2 , Yn ; Z2 ÝÑ H n Xn , Yn ; Z2 are injective. Along with (4.28), this establishes the surjectivity of (4.24) and (4.25). Since πn`1 pZq is trivial, any two extensions of a continuous map f : Xn ÝÑ Z to a continuous map h : Xn`1 ÝÑ Z are homotopic with h|Xn fixed. By the bijectivity of (4.26), any two maps f from pXn , Yn q to pZ, z0 q satisfying (4.27) are homotopic. Combining the last two statements, we conclude that the map (4.24) is  injective. Corollary 4.5. Suppose n ě 3 and Y is a CW complex. Let ηn P H 1 pSOpnq; Z2 q be the generator. The homomorphism ‰ “ (4.29) Y2 , SOpnq ÝÑ H 1 pY2 ; Z2 q, rhs ÝÑ h˚ ηn , is an isomorphism. The homomorphism ‰ “ Y3 , SOpnq ÝÑ H 1 pY3 ; Z2 q,

rhs ÝÑ h˚ ηn ,

is surjective. Proof. This follows immediately from (1.7) and Lemma 4.4 with  n “ 1, Z “ SOpnq, and pX, Y q replaced by pY, Hq.

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Proof of SpinPin 1 property. By the definition of Spin- and Pin˘ -structures in this perspective, it is sufficient to establish this claim for Spin-structures on oriented vector bundles pV, oq with rk V ě 3. If the vector bundle V |Y2 is trivializable, then w2 pV q|Y2 “ 0. Since the restriction homomorphism H 2 pY ; Z2 q ÝÑ H 2 pY2 ; Z2 q

(4.30)

is injective by the cohomology exact sequence for the pair Y2 Ă Y , it follows that w2 pV q “ 0. Suppose w2 pV q “ 0 and o P OpV q. By the SpinPin 1 property for perspective of Definition 1.1, pV, oq then admits a Spin-structure qV : SpinpV, oq ÝÑ SOpV, oq in the sense of Definition 1.1. By (1.8), the principal Spinpnq-bundle SpinpV, oq|Y2 is trivializable. For any section sr of SpinpV, oq|Y2 , the s of SOpV, oq|Y2 determines a trivialization of pV, oq|Y2 and section qV ˝r  thus a Spin-structure on pV, oq in the sense of Definition 1.2. Proof of SpinPin 2 property. By the definition of Spin- and Pin˘ -structures in this perspective, it is sufficient to establish the claims of this property other than (1.13) for Spin-structures on oriented vector bundles pV, oq with n ” rk V ě 3. By the cohomology exact sequence for the pair Y2 Ă Y , the restriction homomorphism H 1 pY ; Z2 q ÝÑ H 1 pY2 ; Z2 q

(4.31)

is an isomorphism. If o P OpV q and pV, oq admits a trivialization over Y2 , then the natural action of rY2 , SOpnqs on the set of homotopy equivalence classes of such trivializations given by ` ˘ “ rhs, rΦs ÝÑ hΦs, (4.32) rY2 , SOpnqsˆ SppV, oq ÝÑ SppV, oq, is free and transitive. Combining this action with the isomorphisms (4.31) and (4.29), we obtain the claims of the SpinPin 2 property other than (1.13). By the naturality of the above action of H 1 pY ; Z2 q on SppV, oq with respect to continuous maps, it satisfies (4.7) and thus (4.8). The last paragraph in the proof of the SpinPin 2 property in the perspective of Definition 1.3 now applies verbatim and implies that  (1.13) holds in the perspective of Definition 1.3 as well.

Proof of Theorem 1.4(1): Trivializations Perspectives

93

Proof of SpinPin 3, 4 property. For the purposes of establishing the first of these properties, we can assume that n ” rk V ě 3. For each o P OpV q and a homotopy class s P SppV, oq of trivializations Φ of pV, oq over the 2-skeleton Y2 of Y , we take s P SppV, oq to be the homotopy class of trivializations of pV, oq over the 2-skeleton Y2 of Y given by  ( s “ Φ ” In;1 Φ : Φ P s , (4.33) with In;1 as below (2.21). By (4.7), the resulting bijection (1.14) is H 1 pY ; Z2 q-equivariant. It also satisfies the last condition of the SpinPin 3 property. We next describe a bijection between SppV, oq and ˘ ˘ P˘ pV q ” SppV ‘p2˘1qλpV q, o˘ V “ SppV ‘p2˘1qτY , op2˘1qoY ; the last equality holds because the orientation o on V determines a canonical homotopy class of trivializations of λpV q. By the SpinPin 1 property, pV, oq admits a Spin-structure if and only if V admits a Pin˘ -structure. We can thus assume that pV, oq admits a Spinstructure. If rk V ě 3, we identify a homotopy class s of trivializations of V |Y2 with the homotopy class of trivializations of pV˘ , o˘ V q|Y2 given by  ( (4.34) s˘ “ Φ˘ ” Φ‘p2˘1qpdet Φq : Φ P s . By (4.7), the resulting map (4.12) is again H 1 pY ; Z2 q-equivariant and thus a bijection. We take the map R˘ o in (1.15) to be its inverse. If Φ is a trivialization of pV, oq|Y2 as in (4.33), then ` ˘ˇ ` ˘ Φ ˘ “ In`2˘1;3˘1 Φ˘ : V ‘p2˘1qλpV q ˇY2 ÝÑ Y2 ˆRn`2˘1 . Since this trivialization is homotopic to Φ˘ , the last claim in the SpinPin 4 property holds. If rk V “ 2, we first identify the trivializations of pτY ‘V q|Y2 with the trivializations of pV ‘ τY q|Y2 in the obvious way and then extend them to trivializations of pV ‘ 3τY q|Y2 by the identity on the last two τY -summands. If rk V “ 1, we first identify the trivializations of ˇ ` ˘ˇ p2τY ‘V qˇ ” τY ‘pτY ‘V q ˇ Y2

Y2

with the trivializations of ppτY ‘ V q ‘ τY q|Y2 and of pV ‘ 2τY q|Y2 in the obvious orientation-preserving ways and then extend the

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latter to the trivializations of pV ‘ 3τY q|Y2 by the identity on the last τY -summand. In all cases, the resulting bijections (1.15) are H 1 pY ; Z2 q-equivariant and satisfy (4.13) and the last requirement  of the SpinPin 4 property. Proof of SpinPin 5, 6 property. Let n “ rk V . By the SpinPin 1 property, an oriented vector bundle pV, oq admits a Spin-structure if and only if pτY ‘ V, StV poqq does. In order to establish the SpinPin 5 property, we can thus assume that pV, oq admits a Spin-structure. If n ě 3, we identify a homotopy class s of trivializations of V |Y2 with the homotopy class of trivializations of pτY ‘ V, StV poqq|Y2 given by  ( (4.35) StV s “ StV Φ ” idτY2 ‘ Φ : Φ P s . It is immediate that StV s satisfies the first condition after (1.16). With Φ as in (4.33), ˇ ` StV Φ “ StV Φ : τY ‘V qˇY2 ÝÑ Y2 ˆRn`1. Thus, the resulting first map in (1.16) satisfies the first equality in (1.17). We define the second map in (1.16) by (4.15). By (4.7), both maps (1.16) are then H 1 pY ; Z2 q-equivariant bijections. With Φ˘ as in (4.34), ˇ ˘ ` ` ˘ n`3˘1 ˇ . StV Φ ˘ “ St˘ V Φ˘ : pτY ‘V q˘ Y2 “ τY ‘V˘ |Y2 ÝÑ Y2 ˆR Thus, the second equality in (1.17) also holds. For n “ 1, 2, the last paragraph of the proof of the SpinPin 5 property in the perspective of Definition 1.3 now applies verbatim. The two sides of (1.20) are again the same by definition; we take this map to be the identity. The requirement after (1.20) is then again forced by the definition of the second map of the SpinPin 1  in (4.15). Proof of SpinPin 7 property. As in the perspective of Definition 1.1, it is sufficient to establish this property for the direct sum exact sequences as in (1.22). Furthermore, an orientation o1 on V 1 determines a homotopy class of trivializations of λpV 1 q and thus of isomorphisms ˘ ` 1 V ‘V 2 ˘ ” V 1 ‘V 2 ‘p2˘1qλpV 1 ‘V 2 q « V 1 ‘V 2 ‘p2˘1qλpV 2 q ” V 1 ‘V˘2 .

(4.36)

Proof of Theorem 1.4(1): Trivializations Perspectives

95

Let V 1 and V 2 be vector bundles over Y of rank n1 and n2 , respectively, o1 P OpV 1 q, and o2 P OpV 2 q. If s1 and s2 are homotopy classes of trivializations of pV 1 , o1 q|Y2 and pV 2 , o2 qY2 , we take xxs1 , s2 yy to be the homotopy class of trivializations of pV 1 ‘V 2 , o1 o2 q|Y2 given by  ( (4.37) xxs1 , s2 yy “ Φ1 ‘Φ2 : Φ1 P s1 , Φ2 P s2 . This determines the first map in (1.24) if n1 , n2 ě 3. Along with (4.36), this also determines the second map of this property if n1 ě 3 and rk V˘2 ě 3 (i.e. not V´2 if n2 “ 1). By (4.7), both maps are then H 1 pY ; Z2 q-biequivariant. These two maps satisfy (ses1), (1.25), and (ses4) for vector bundles of ranks at least 3 by definition. With Φ˘ as in (4.34), ` 1 ` ˘ ˘ ˘ 1 2 Φ ‘Φ2 ˘ “ Φ1 ‘Φ2˘ : V 1 |Y2 ‘pV˘2 |Y2 ÝÑ Y2 ˆRn `n `2˘1. Thus, (1.26) is also satisfied in these cases. For vector bundles V 1 , V 2 over Y not both of ranks at least 3, we define the two maps in (1.24) as in the proof of this property in the perspective of Definition 1.3, i.e. by (4.22), but with all relevant bijections taken between the sets of Spin- and Pin˘ -structures in the perspective of Definition 1.2. The entire part of that proof starting with (4.18) applies verbatim in this case, even though the domains and targets of the relevant bijections now refer to different sets.  Example 4.6. Let γR;1 ÝÑ RP1 , oT RP1 , Bθ , f , and g be as in Example 1.14, Φ0 be as in (1.37), and os1 p2γR;1 , o´ γR;1 q be the OSpin´ structure on p2γR;1 , oγR;1 q as in Example 1.15. The composition of a right inverse for the vector bundle homomorphism g with the trivialization of T pRP1 q determined by the vector field Bθ is given by ˆ ˙ ` ˘ p´yqpx, yq xpx, yq h : τRP1 ÝÑ 2γR;1 , h rx, ys, b “ rx, ys, b 2 2 , b 2 2 . x `y x `y Since Φ0 ˝pf ‘hq : 2τRP1 “ RP1 ˆC ÝÑ RP1 ˆC,

`

˘ ` ˘ reiθ s, c “ reiθ s, e2iθ c ,

the automorphism id ‘ pΦ0 ˝ pf ‘ hqq of 3τRP1 is homotopically nontrivial. Thus, ˘ ` ˘DD ` ˘ @@ ` os0 τRP1 , oRP1 , os0 T pRP1 q, oT RP1 p1.35q “ os1 2γR;1 , o´ γR;1 . (4.38)

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For n P Z` , we denote by ˚ NRPn RP1 « pn´1qγR;1 « pn´1qγR;1

the normal bundle for the standard embedding of RP1 into RPn . Let 0 ÝÑ T pRP1 q ÝÑ T pRPn q|RP1 ÝÑ NRPn RP1 ÝÑ 0

(4.39)

be the associated short exact sequence of vector bundles over RP1 . Example 4.7. Let os0 pRP1 , oT RP1 q, os0 p2γR;1 , o´ γR;1 q, os1 p2γR;1 , ´ ` oγR;1 q, and os1 p4γR;2 , oγR;2 q be the OSpin-structures on RP1 , 2γR;1 , ´ 2 and 4γR;2 provided by Examples 1.14–1.16. Let p´ 1 pRP q be the Pin ´ 2 2 structure on RP provided by Example 1.16 and os1 pRP q be the 2 OSpin-structure on pT RP2 q´ corresponding to p´ 1 pRP q under the ´ ´ 1 bijection (1.20). We denote by p0 pNRP2 RP q the Pin -structure on NRP2 RP1 « γR;1 corresponding to os0 p2γR;1 , o´ γR;1 q under the bijection (1.20). Euler’s exact sequences (1.35) for RP1 and RP2 and an ˚ induce the commutative square of identification of NRP2 RP1 with γR;1 exact rows and columns of vector bundles over RP1 in Figure 4.2. By 2 ` the definitions of os´ 1 pRP q and os1 p4γR;2 , oγR;1 q and the SpinPin 6 property, @@

ˇ DD ` ˘ ` ˘ˇ 2 ˇ ` ˇ os0 τRP1 , oRP1 , os´ 1 pRP q RP1 p1.35q “ os1 4γR;1 , oγR;1 RP1 @@ DD ´ “ os1 p2γR;1 , o´ γR;1 q, os0 p2γR;1 , oγR;1 q ‘ .

Figure 4.2. Commutative square of exact rows and columns of vector bundles ˚ ˚ ‘γR;1 is the projection over RP1 used in Example 4.7; the vertical map from 3γR;1 to the last two components.

Proof of Theorem 1.4(1): Trivializations Perspectives

97

By (4.38) and (1.28), @@

DD ´ os1 p2γR;1 , o´ γR;1 q, os0 p2γR;1 , oγR;1 q ‘ @@@@ ` DD ˘ ` ˘DD “ os0 τRP1 , oRP1 , os0 T pRP1 q, oT RP1 p1.35q , os0 p2γR;1 , o´ γR;1 q ‘ @@ ` DD ˘ @@ ` ˘ DD “ os0 τRP1 , oRP1 , os0 RP1 , oT RP1 , os0 p2γR;1 , o´ γR;1 q p4.39q ‘ .

Along with the second statement in the SpinPin 7(ses3) property, the last two equations give ˇ @@ ` 1 ˘ ´` ˘DD 2 ˇ 1 . (4.40) p´ 1 pRP q RP1 “ os0 RP , oT RP1 , p0 NRP2 RP p4.39q Example 4.8. Let os0 pRP1 , oT RP1 q, os0 p2γR;1 , o´ γR;1 q, os1 p2γR;1 , ` q be the OSpin-structures on RP1 , 2γ q, and os p4γ , o o´ 1 R;3 γR;3 R;1 , γR;1 and 4γR;3 provided by Examples 1.14–1.17 and os0 pNRP3 RP1 q the OSpin-structure on NRP3 RP1 « 2γR;1 corresponding to 1 3 os0 p2γR;1 , o´ γR;1 q. Euler’s exact sequences (1.35) for RP and RP and ˚ induce the commua standard identification of NRP3 RP1 with 2γR;1 tative square of exact rows and columns of vector bundles over RP1 in Figure 4.2 with the middle and bottom entries in the right colˇ 3 ˇ umn replaced by T pRP q RP1 and NRP3 RP1 , respectively. The same computation as in Example 4.7 yields ˇ DD ˘ @@ ` os0 τRP1 , oRP1 , os1 pRP3 qˇRP1 p1.35q ˘ @@ ` ˘ DD @@ ` DD “ os0 τRP1 , oRP1 , os0 RP1 , oT RP1 , os0 p2γR;1 , o´ γR;1 q p4.39q ‘ . Along with the SpinPin 7(ses5) and 5 properties, this implies that ˇ @@ ` ˘ ` ˘DD (4.41) os1 pRP3 qˇRP1 “ os0 RP1 , oT RP1 , os0 NRP3 RP1 p4.39q .

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Chapter 5

Equivalence of Definitions 1.1–1.3

It remains to establish the second and third statements of Theorem 1.4. This is done in Sections 5.1 and 5.2, respectively, implementing the approach indicated after the statement of this theorem. We first confirm a basic observation that underpins this approach. Proof of Lemma 1.5. Let n “ rk V . As in Section 2.1, we denote by rIn P Spinpnq the identity element and by pIn P Spinpnq the other preimage of the identity In P SOpnq under the projection (2.5). For r “ 0, 1, let αn;r : r0, 1s ÝÑ SOpnq and

α rn;r : r0, 1s ÝÑ Spinpnq

be a loop representing r times the generator of π1 pSOpnqq with αn;r p0q, αn;r p1q “ In and its lift to a path with α rn;r p0q “ rIn and α rn;r p1q “ pIrn . Since every orientable vector bundle over S 1 is trivializable, we can assume that pV, oq “ npτS 1 , oS 1 q. By the SpinPin 1 and 2 properties in Section 1.2, there are two equivalence classes of Spin-structures in the sense of Definition 1.1 on this vector bundle. They are represented by ` ˘L qV ;r : Spinr pV, oq ” r0, 1sˆSpinpnq „ÝÑ S 1 ˆSOpnq, ` ˘ ` ˘ r „ 1, A r @A r P Spinpnq, qV ;r rt, As r p0, pIrn Aq ˘ ` r @ pt, Aq r P r0, 1sˆSpinpnq, “ e2πit , qn pAq

99

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Spin/Pin-Structures & Real Enumerative Geometry

with r “ 0, 1. Sections of these Spinpnq-principal bundles and the induced sections of S 1 ˆSOpnq are given by ‰ ` ˘ “ srV ;r : S 1 ÝÑ Spinr pV, oq, srV ;r e2πit “ t, α rn;r ptq , (5.1) ` ˘ ` ˘ sV ;r : S 1 ÝÑ S 1 ˆSOpnq, sV ;r e2πit “ e2πit , αn;r ptq . Since the sections sV ;0 and sV ;1 are not homotopic, the homotopy classes of the associated trivializations of S 1 ˆRn are different.  5.1

Proof of Theorem 1.4(2)

Let pV, oq be a rank n ě 3 oriented vector bundle over a paracompact locally contractible space Y . By the SpinPin 1 property, pV, oq admits a Spin-structure in the sense of Definition 1.1(b) if and only if it admits a Spin-structure in the sense of Definition 1.3. We can thus assume that pV, oq admits a Spin-structure qV : SpinpV, oq ÝÑ SOpV, oq

(5.2)

in the sense of Definition 1.1(b). For every α P LpY q,

` ˘ α˚ qV : α˚ SpinpV, oq ÝÑ α˚ SOpV, oq “ SO α˚ pV, oq

(5.3)

is a Spin-structure over S 1 in the sense of Definition 1.1(b). Since Spinpnq is path-connected, the principal bundle α˚ SpinpV, oq admits a section srα . Since in addition Spinpnq is simply connected, any two such sections are homotopic. We take the homotopy class sα of trivializations of α˚ pV, oq associated with the Spin-structure (5.2) to be the homotopy class represented by the trivialization induced by the sα of SOpα˚ pV, oqq. section pα˚ qV q˝r For every continuous map F : Σ ÝÑ Y from a compact bordered surface, ˘ F ˚ qV : F ˚ SpinpV, oq ÝÑ F ˚ SOpV, oq “ SOpF ˚ pV, oq is a Spin-structure over Σ in the sense of Definition 1.1(b). Since Spinpnq is path-connected and simply connected, the principal bundle F ˚ SpinpV, oq admits a section srF . The restrictions of the trivialsF of SOpF ˚ pV, oqq ization of F ˚ pV, oq induced by the section pF ˚ qV q˝r

Equivalence of Definitions 1.1–1.3

101

to the boundary components α1 ” F |C1 , . . . , αm ” F |Cm : S 1 ÝÑ Y of F are induced by the sections srαi ” srF |Ci of F ˚ SpinpV, oq|Ci . Thus, the homotopy classes of trivializations constructed in the previous paragraph satisfy the condition of Definition 1.3. We conclude that a Spin-structure (5.2) on pV, oq in the sense of Definition 1.1(b) determines a Spin-structure ` ˘ psα qαPLpY q ” Θ3 SpinpV, oq, qV on pV, oq in the sense of Definition 1.3. It is immediate that the latter depends only on the equivalence class of the Spin-structure (5.2). By the construction, the resulting map Θ3 : OSp1 pV q ÝÑ OSp3 pV q

(5.4)

from the set of equivalence classes of OSpin-structures on V in the sense of Definition 1.1(b) to the set of OSpin-structures on V in the sense of Definition 1.3 is natural with respect to the pullbacks induced by continuous maps. We show in the following that Θ3 is compatible with the SpinPin 2(b) property, i.e. ˘ ` ˘ ` Θ3 η¨pSpinpV, oq, qV q “ η¨ Θ3 pSpinpV, oq, qV q @ η P H 1 pY ; Z2 q. (5.5) Along with the SpinPin 2(b) property, this implies that (5.4) is a bijection. Suppose pV, oq ” npτY , oY q is the trivial rank n-oriented vector bundle over Y and qV is the Spin-structure on pV, oq given by (1.5). For each α P LpY q, we can then take srα : S 1 ÝÑ α˚ SpinpV, oq “ S 1 ˆSpinpnq,

srα pxq “ px, rIn q.

The induced homotopy class sα of trivializations of pV, oq is then the canonical homotopy class of trivializations of S 1ˆRn . For any oriented vector bundle pV, oq split into oriented line bundles, the map (5.4) thus takes the distinguished element s0 pV, oq in the perspective of Definition 1.1 to the the distinguished element s0 pV, oq in the perspective of Definition 1.3.

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Compatibility with SpinPin 2(b) property. Let η P H 1 pY ; Z2 q and α P LpY q. By the naturality of the H 1 pY ; Z2 q-action of Propositions 3.3(b), ˘  (` ` ˘ (5.6) α˚ η¨pSpinpV, oq, qV q “ α˚ η ¨ α˚ SpinpV, oq, α˚ qV . By the SpinPin 2 property with Y “ S 1 , the Spin-structure (5.6) is equivalent to the Spin-structure (5.3) if and only if α˚ η “ 0. Along with Lemma 1.5, this implies that the homotopy class of trivializations of α˚ V determined by the Spin-structure η ¨pSpinpV, oq, qV q is the same as sα if and only if α˚ η “ 0. Combining this with (4.7), we obtain (5.5).  Compatibility with SpinPin 3 property. Suppose Y “ S 1 and pV, oq is the trivial rank n-oriented vector bundle over Y . For r “ 0, 1, rr , qV ;r , Spinr pV, oq, srV ;r , and sV ;r be as in the proof of let αr , α Lemma 1.5. In particular, the homotopy class sid of trivializations of pV, oq associated with the OSpin-structure po, qV ;r q is represented by the trivialization φn;r induced by sV ;r . By the paragraph containing (3.21), (1.14) takes the Spinstructure qV ;r to the Spin-structure on the oriented vector bundle pV, oq given by ` ˘ q V ;r : Spinr pV, oq ” S 1 ˆSpinpnq ÝÑ SOpV, oq ” S 1 ˆ Opnq´SOpnq , ˘ ` ˘ ` r “ e2πit , αn;r ptqIn;1 qn pAq r q V ;r e2πit , A r P r0, 1sˆSpinpnq. ˆ @ pt, Aq A section of this Spinpnq-principal bundle and the induced section of S 1 ˆSOpnq are given by ` ` ˘ ˘ @ t P r0, 1s. e2πit ÝÑ e2πit , rIn , e2πit ÝÑ e2πit , αn;r ptqIn;1 The trivialization of pV, oq induced by the latter section is In;1 φn;r . Along with (4.10), this implies that the homotopy class s1id of trivializations of pV, oq associated with the OSpin-structure po, q V ;r q is the homotopy class sid of trivializations of pV, oq determined by sid . By the naturality of (1.14), the conclusion of the previous paragraph implies that ` ˘ @ os P OSp1 pV q (5.7) Θ3 os “ Θ3 posq

Equivalence of Definitions 1.1–1.3

103

for every vector bundle V of rank at least 3 over a paracompact locally contractible space Y .  Compatibility with SpinPin 5 property. We continue with the setup at the beginning of the proof of the compatibility with the SpinPin 3 property. By the proof of the SpinPin 5 property in Section 3.3 and (2.10), the first map in (1.16) takes the Spin-structure qV ;r to the Spin-structure ` qStpV q;r : Spinr StpV, oqq ÝÑ S 1 ˆSOpn`1q. A section srStpV q;r of this principal Spinpn ` 1q-bundle over S 1 is αn;r q in (5.1). The triviobtained by replacing α rn;r with ι2n`1;n pr alization of StpV, oq induced by the trivialization qStpV q;r ˝ α rn;r of SOpStpV, oqq is the trivialization StV φn;r in (4.14). Thus, the homotopy class pStV sq1id of trivializations of StV pV, oq associated with the OSpin-structure StV po, qV ;r q is the homotopy class StV sid determined by sid . By the naturality of the first map in (1.16), this implies that ˘ ˘ ` ` (5.8) Θ3 StV posq “ StV Θ3 posq @ os P OSp1 pV q for every vector bundle V of rank at least 3 over a paracompact  locally contractible space Y . We define the map Θ3 in (5.4) for rank 2 vector bundles V over Y and then for rank 1 vector bundles V over Y by (5.8). By the SpinPin 5 property and by the already established properties of (5.4) for vector bundles of ranks at least 3, the resulting maps Θ3 for vector bundles of ranks 1 and 2 are natural H 1 pY ; Z2 q-equivariant bijections which associate the distinguished elements s0 pV, oq in the perspectives of Definitions 1.1 and 1.3 with each other for all oriented vector bundles pV, oq split into oriented line bundles. These maps are compatible with the SpinPin 5 property, i.e. satisfy (5.8), by definition. Along with the first equality in (1.17) and (5.7) for vector bundles of rank at least 3, this implies that the maps Θ3 satisfy (5.7) for all vector bundles V . Compatibility with SpinPin 7 property. By the naturality of the first map in (1.24), it is sufficient to show that DD ` ˘ @@ Θ3 xxos1 , os2 yy e “ Θ3 pos1 q, Θ3 pos2 q e ˆ @ os1 P OSp1 pV 1 q, os2 P OSp1 pV 2 q,

(5.9)

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for all short exact sequences e of trivial oriented vector bundles over Y “ S 1 as in (1.21). By the SpinPin 2(b) property and the H 1 pY ; Z2 q-equivariance of Θ3 and of the first map in (1.24) in the first input, it is sufficient to check the Y “ S 1 case of (5.9) with os1 being the distinguished OSpin-structure os0 pmpτS 1 , oS 1 qq. By (ses1), (ses4), and (ses5) in the SpinPin 7 property, DD @@ 2 os0 pmpτY , oY qq, os2 ‘ “ Stm V pos q. Along with (5.8), this implies that (5.9) holds with os1 “ os0 pmpτS 1 , oS 1 qq and thus for all short exact sequences (1.21) of vector bundles over a paracompact locally contractible space Y .  With P1˘ pV q and P3˘ pV q denoting the sets P˘ pV q of the Pin˘ structures on a vector bundle V over Y in the perspectives of Definitions 1.1 and 1.3, respectively, define ˘ ` ` ˘ ˘ Θ3 : P1˘ pV q ÝÑ P3˘ pV q by Co˘ V Θ3 ppq “ Θ3 CoV ppq . (5.10) By the SpinPin 6 property and the already established properties of (5.4), (5.10) is a natural H 1 pY ; Z2 q-equivariant bijection which is compatible with the SpinPin 5 property, i.e. (5.8) holds for Pin˘ structures p in place of the OSpin-structures os. The map Θ3 is compatible with the SpinPin 6 property by definition. By the second statement in the SpinPin 7(ses3) property and (5.9), (5.10) is compatible with the second map in (1.24), i.e. (5.9) holds for Pin˘ structures p2 in place of the OSpin-structures os2 . By the SpinPin 7(ses6) property, (5.10) is compatible with the SpinPin 4 property, i.e. ˘ ˘ ` ` ˘ @ p P P1˘ pV q, o P OpV q, Θ3 R˘ o ppq “ Ro Θ3 ppq for every vector bundle V over a paracompact locally contractible space Y . This concludes the proof of Theorem 1.4(2). 5.2

Proof of Theorem 1.4(3)

Let pV, oq be a rank n ě 3-oriented vector bundle over a CW complex Y . By the SpinPin 1 property, pV, oq admits a Spin-structure in the sense of Definition 1.1(b) if and only if it admits a Spin-structure

Equivalence of Definitions 1.1–1.3

105

in the sense of Definition 1.2. We can thus assume that pV, oq admits a Spin-structure in the sense of Definition 1.1(b) as in (5.2). Since Spinpnq is path-connected and simply connected, the principal bundle SpinpV, oq|Y2 admits a section sr2 . Since in addition π2 pSpinpnqq is trivial, any two such sections are homotopic. We take the Spin-structure s ” Θ2 pSpinpV, oq, qV q

(5.11)

on pV, oq in the sense of Definition 1.2 to be the homotopy class of trivializations of pV, oq|Y2 represented by the trivialization Φ induced by the section qV ˝r s2 of SOppV, oq|Y2 q. It is immediate that s depends only on the equivalence class of the Spin-structure (5.2). By the construction, the resulting map Θ2 : OSp1 pV q ÝÑ OSp2 pV q

(5.12)

from the set of equivalence classes of OSpin-structures on V in the sense of Definition 1.1(b) to the set of OSpin-structures on V in the sense of Definition 1.2 is natural with respect to the pullbacks induced by continuous maps. We show in the following that Θ2 is compatible with the SpinPin 2(b) property, i.e. ˘ ` ˘ ` Θ2 η¨pSpinpV, oq, qV q “ η¨ Θ2 pSpinpV, oq, qV q ˆ @ η P H 1 pY ; Z2 q.

(5.13)

Along with the SpinPin 2(b) property, this implies that (5.12) is a bijection. Suppose pV, oq ” npτY , oY q is the trivial rank n-oriented vector bundle over Y and qV is the Spin-structure on pV, oq given by (1.5). We can then take ˇ sr2 : Y2 ÝÑ SpinpV, oqˇY2 “ Y2 ˆSpinpnq, sr2 pxq “ px, rIn q. The induced trivialization of pV, oq is then the canonical trivialization of Y2 ˆRn . For any oriented vector bundle pV, oq split into oriented line bundles, the map (5.12) thus takes the distinguished element s0 pV, oq in the perspective of Definition 1.1 to the the distinguished element s0 pV, oq in the perspective of Definition 1.2.

106

Spin/Pin-Structures & Real Enumerative Geometry

Compatibility with SpinPin 2(b) property. Suppose η P H 1 pY ; Z2 q, ` ˘ ` ˘ Spin1 pV, oq, qV1 ” η¨ SpinpV, oq, qV is as in (3.14), and h : Y2 ÝÑ SOpnq is a continuous map such that η|Y2 “ h˚ ηn ; see the proof of the SpinPin 2(b) property in the perspective of Definition 1.2 in Section 4.3. Let sr 12 be a section of Spin1 pV, oq|Y2 . Since π2 pSpinpnqq is trivial, (5.13) holds if and only if the trivializations of SOppV, oq|Y1 q induced by the restrictions of qV1 ˝r s 12 and h¨pqV ˝r s2 q are homotopic. Let α P LpY q. By (5.5) and (4.7), the trivializations of SOpα˚ pV, oqq induced by  ( ` ˘ ` ˘  ( ` ˘ ` ˘ s2 “ α˚ qV ˝ α˚ sr2 s 12 “ α˚ qV1 ˝ α˚ sr 12 and α˚ qV1 ˝r α˚ qV ˝r are homotopic if and only if  (˚ (5.14) h˝α ηn “ α˚ η “ 0 P H 1 pS 1 ; Z2 q.  ( s2 and The trivializations of SOpα˚ pV, oqq induced by α˚ qV ˝r (   ( s2 q “ ph˝αq¨α˚ qV ˝r s2 α˚ h¨pqV ˝r are homotopic if and only if the loop h˝α : S 1 ÝÑ SOpnq is homotopically trivial. The latter is also equivalent to (5.14). Thus, the trivializations of SOppV, oq|Y1 q induced by the restrictions of qV1 ˝ sr 12 and h¨pqV ˝r s2 q are homotopic. This establishes (5.13) for every vector bundle V of rank at least 3 over Y .  Compatibility with SpinPin 3 property. sr2 , and Φ are as above and

Suppose Y , pV, oq,

qV : SpinpV, oq ÝÑ SOpV, oq is as below (3.21). Let sr 12 be a section of SpinpV, oq|Y2 . Since π2 pSpinpnqq is trivial, ` ˘ (5.15) Θ2 SpinpV, oq, q V “ Θ2 pSpinpV, oq, qV q if and only if the trivialization of SOppV, oq|Y1 q induced by the restriction of q V ˝ sr 12 is homotopic to In;1 Φ|Y1 ; see (4.33). The latter is the

Equivalence of Definitions 1.1–1.3

107

case if and only if the trivialization of SOpα˚ pV, oqq induced by  ( ` ˘ ` ˘ s 12 “ α˚ qV ˝ α˚ sr 12 α˚ qV ˝r is homotopic to α˚ pIn;1 Φq “ In;1 pα˚ Φq for every α P LpY q. The last statement is implied by (5.7). This establishes (5.15) for every vector  bundle V of rank at least 3 over Y . Compatibility with SpinPin 5 property. Suppose Y , pV, oq, sr2 , and Φ are as above and ` ˘ ` ˘ qStpV q : Spin StpV, oq ÝÑ S 1 ˆSO StpV, oq is as in the proof of the SpinPin 5 property in Section 3.3. Let sr 12 be a section of SpinpStpV, oqq|Y2 . Since π2 pSpinpnqq is trivial, ˘ ˘ ` ` Θ2 StV pSpinpV, oq, qV q “ StV Θ2 pSpinpV, oq, qV q (5.16) if and only if the trivialization of SOppV, oq|Y1 q induced by the restrics 12 is homotopic to idτY1 ‘Φ|Y1 ; see (4.35). The latter is tion of qStpV q˝r the case if and only if the trivialization of SOpα˚ StpV, oqq induced by  ( ` ˘ ` ˘ s 12 “ α˚ qStpV q ˝ α˚ sr 12 α˚ qStpV q ˝r is homotopic to α˚ pidτY1 ‘Φ|Y1 q “ idτS1 ‘pα˚ Φq for every α P LpY q. The last statement is implied by (5.8). This establishes (5.16) for  every vector bundle V of rank at least 3 over Y . We define the map Θ2 in (5.12) for rank 2 vector bundles V over Y and then for rank 1 vector bundles V over Y by (5.16). By the same reasoning as in Section 5.1, the resulting maps Θ2 are natural H 1 pY ; Z2 q-equivariant bijections which associate the distinguished elements s0 pV, oq in the perspectives of Definitions 1.1 and 1.2 with each other for all oriented vector bundles pV, oq split into oriented line bundles and are compatible with the SpinPin 3 and 5 properties. Compatibility with SpinPin 7 property. By Definition 1.2, the SpinPin 2(b) property, and the H 1 pY ; Z2 q-equivariance of Θ2 and of the first map in (1.24) in the first input, it is sufficient to show that DD ` ˘ @@ Θ2 xxos1 , os2 yy e “ Θ2 pos1 q, Θ2 pos2 q e ˆ@ os1 P OSp1 pV 1 q, os2 P OSp1 pV 2 q, (5.17)

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Spin/Pin-Structures & Real Enumerative Geometry

for all short exact sequences e of trivial oriented vector bundles as in (1.21) with os1 being the distinguished OSpin-structure os0 pmpτY , oY qq. By (ses1), (ses4), and (ses5) in the SpinPin 7 property, DD @@ 2 os0 pmpτY , oY qq, os2 ‘ “ Stm V pos q. Along with (5.16), this implies that (5.17) holds with os1 “ os0 pmpτS 1 , oS 1 qq and thus for all short exact sequences (1.21) of vector bundles over Y .  With P1˘ pV q and P2˘ pV q denoting the sets P˘ pV q of the Pin˘ structures on a vector bundle V over Y in the perspectives of Definitions 1.1 and 1.2, respectively, define ˘ ` ` ˘ ˘ Θ2 : P1˘ pV q ÝÑ P2˘ pV q by Co˘ V Θ2 ppq “ Θ2 CoV ppq . (5.18) By the same reasoning as in Section 5.1, (5.18) is a natural H 1 pY ; Z2 q-equivariant bijection which is compatible with the SpinPin 4–7 properties. This concludes the proof of Theorem 1.4(3). Example 5.1. With V ÝÑ RP1 denoting the infinite Mobius band line bundle of Examples 1.23, 1.24, and 3.7, ` ˘L πV´ : V´ “ r0, 1sˆC „ÝÑ RP1 “ S 1 {Z2 , ` ˘ “ ‰ p0, ´cq „ p1, cq, πV rt, cs “ eπit , ` ˘L πV` : V` “ r0, 1sˆC2 „ÝÑ RP1 “ S 1 {Z2 , ` ˘ “ ‰ p0, ´cq „ p1, cq, πV rt, cs “ eπit . ` The orientations o´ V of V´ and oV of V` described below (1.18) are the complex orientations of these bundles. Let

γ˘ : r0, 1s ÝÑ SOp3˘1q

and

γ˘ : r0, 1s ÝÑ Spinp3˘1q r

be the paths as in the n “ 2 case of the proof of Lemma 2.2 and their r´ p1q “ rI2;2 and lifts with γ r˘ p0q “ rI3˘1 . By the conventions in (2.15), γ γ r` p1q “ I4;4 . Thus, ˘ “ ` ‰ sr3˘1 reπit s “ t, γ r˘ ptq´1 , sr3˘1 : RP1 ” S 1 {Z2 ÝÑ Spin0 pV˘ , o˘ V q,

Equivalence of Definitions 1.1–1.3

109

is a well-defined continuous section of the Spinp3 ˘ 1q-bundle Spin0 pV˘ , o˘ V q of Example 3.7. The induced trivializations Φ2 ` of pV´ , o´ q V and Φ4 of pV` , oV q are given by ` πit ˘ “ ´πit ‰ 1 Φ´1 c , re s, c “ t, e Φ´1 2 : RP ˆC ÝÑ V´ , 2 ` πit ˘ “ ´πit ‰ ´1 1 2 ´1 Φ4 re s, c “ t, e c . Φ4 : RP ˆC ÝÑ V` , Thus, the homotopy class of Φ4 is the canonical OSpin-structure os0 pV` , o` V q on 4V of Example 1.13. The trivialization Φ2 is the composition of the trivialization (1.37) of 2γR;1 with the direct sum of two copies of the identification (1.40). Remark 5.2. The map γ´ of Example 5.1 is denoted by rot in the second case of [46, Section 1.2.2]. By Example 5.1, the OSpinstructure os0 p2γR;1 , o´ γR;1 q of Example 1.15 is not the base OSpinstructure on 2γR;1 in the second case of [46, Section 1.2.2] and in the corresponding case in [45, Section 2.2]. Thus, the base OSpinstructure on 2γR;1 in [45, 46] is the OSpin-structure os1 p2γR;1 , o´ γR;1 q of Example 1.15. Along with (4.41), this implies that the base OSpin-structure on T pRP3 q|RP1 in [45, 46] is the OSpin-structure os0 pRP3 q|RP1 of Example 1.17. While the choice of neither OSpinstructure on 2γR;1 nor Pin´ -structure on γR;1 appears more canoni´ cal, some of the advantages os0 p2γR;1 , o´ γR;1 q over os1 p2γR;1 , oγR;1 q and ´ of the associated Pin´ -structure p´ 0 pγR;1 q on γR;1 over p1 pγR;1 q are described at the end of Section 7.2, after the CROrient 6 property.

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Chapter 6

Relative Spin- and Pin-Structures

We recall the definitions of relative Spin- and Pin-structures from the CW perspective of [12], introduce a completely intrinsic perspective on these structures, and state the second theorem of Part I, that these two perspectives are essentially equivalent, in Section 6.1. Section 6.2 describes key properties of relative Spin- and Pin-structures; many of the stated properties resemble the properties of Spin- and Pinstructures collected in Section 6.2. We establish these properties for the two perspectives in Sections 6.5 and 6.3, respectively, and show the two perspectives to be equivalent, when restricted to CW complexes, in Section 6.6. The proofs of the properties of Section 6.2 in the perspective of Definition 6.1 rely heavily on the simple topological observations of Section 6.4. 6.1

Definitions and Main Theorem

We begin with the CW perspective of [12, Definition 8.1.2] on relative Spin-structures. Its extension to Pin-structures appearing in [39, Section 1.2] is described after we introduce a completely intrinsic notion of relative Spin-structures. We call pX, Y q a CW pair if X is a CW complex and Y is a CW subcomplex of X. Definition 6.1. Let pX, Y q be a CW pair and V be a real vector bundle over Y . (a) A relative Pin˘ -structure p on V is a tuple pE, oE , pE,V q consisting of an oriented vector bundle pE, oE q over the 3-skeleton X3 111

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of X and a Pin˘ -structure pE,V on E|Y2 ‘V |Y2 in the perspective of Definition 1.2. (b) If o P OpV q, a relative Spin-structure s on pV, oq is a tuple pE, oE , sE,V q consisting of an oriented vector bundle pE, oE q over X3 and a Spin-structure sE,V on pE, oE q|Y2 ‘ pV, oq|Y2 in the perspective of Definition 1.2. For m P Zě0 , we denote by Stm and Stm V the compositions of m-copies of the maps in (1.1) and (1.16). Relative Spin-structures pE, oE , sE,V q and pE 1 , oE 1 , sE 1 ,V q on pV, oq as in Definition 6.1 are called equivalent if there exist m, m1 P Zě0 and an isomorphism ` ˘ m1 1 Ψ : Stm pE, oE q ” mτX3 ‘E, Stm E poE q ÝÑ St pE , oE 1 q ˘ ` 1 ” m1 τX3 ‘E 1 , Stm E poE 1 q of oriented vector bundles over X3 so that the induced isomorphism ˇ ˇ Ψ|mτY2 ‘E|Y2 ‘ idV |Y2 : Stm pE, oE qˇY2 ‘pV, oqˇY2 ˇ ˇ 1 ÝÑ Stm pE 1 , oE 1 qˇ ‘pV, oqˇ Y2

Y2

Stm E|Y2 ‘V |Y2 psE,V q

1

identifies the Spin-structures and Stm E 1 |Y2 ‘V |Y2 psE 1 ,V q. The equivalence classes of relative Spin-structures on pV, oq for different triangulations of the pair Y Ă X can be canonically identified; see [12, Proposition 8.1.6]. Notions of equivalence for relative Pin˘ -structures of Definition 6.1(a) are defined similarly. When there is no ambiguity, we will not distinguish between the Spin- and Pin-structures of Definition 6.1 and their equivalence classes. For a relative Pin- or Spin-structure s on V as in Definition 6.1, let w2 psq “ w2 pEq P H 2 pX3 ; Z2 q “ H 2 pX; Z2 q; we use the same notation for equivalence classes of such structures. Remark 6.2. In [39, Section 1.2], a relative Pin˘ -structure on a real vector bundle V over Y Ă X is defined (with some typos) to be a tuple pE, oE , pE,V q consisting of an oriented vector bundle pE, oE q over X3 of X and a Pin˘ -structure pE,V on pE, oE q|Y2 ‘pV, oq|Y2 . For the primarily qualitative purposes of [39], it is not material whether the latter is taken in the perspective of Definition 1.1, 1.2, or 1.3, but the approach in [39] leads to some difficulties in studying and applying the disk invariants defined there.

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Let pV, oq be an oriented vector bundle over a compact non-empty one-dimensional manifold Y , i.e. a union of circles, with rkR V ě 3. We call two homotopy classes of trivializations of pV, oq equivalent if the number of connected components of Y on which they differ is even. We call such an equivalence class a mod 2 homotopy class of trivializations of pV, oq. By (1.7), there are two such classes. If Y is empty, we define a mod 2 homotopy class of trivializations to be an element of Z2 . If Y and Y 1 are compact one-dimensional manifolds and pV, oq is an oriented vector bundle over Y \Y 1 , a mod 2 homotopy class s of trivializations of pV, oq|Y and a mod 2 homotopy class s1 of trivializations of pV, oq|Y 1 determine a mod 2 homotopy class ss1 of trivializations of pV, oq. If neither Y nor Y 1 is empty, ss1 is obtained by simply combining the trivializations in s and s1 . If Y 1 is empty, we take ss1 “ s if s1 “ 0 and ss1 ‰ s if s1 ‰ 0. We call ss1 the disjoint union of the trivializations s and s1 . Suppose pV, oq is an oriented vector bundle over a bordered surface Σ with rkR V ě 3. Let Σ˝ Ă Σ be the union of the components Σ1 of Σ with BΣ1 ‰ H, Σ‚ Ă Σ be the union of the remaining components of Σ, and D @ D @ sΣ‚ pV, oq “ w2 pV q, rΣ‚ sZ2 “ w2 pV q, rΣsZ2 P Z2 . If Σ˝ ‰ H, let sΣ˝ pV, oq be the mod 2 homotopy class of trivializations of ˇ ˇ pV, oqˇBΣ “ pV, oqˇBΣ˝ containing the restriction of a trivialization Φ of pV, oq|Σ˝ ; by Corollary A.14, sΣ˝ pV, oq is well defined. If Σ˝ “ H, we take sΣ˝ pV, oq “ 0. In either case, sΣ pV, oq ” sΣ‚ pV, oqsΣ˝ pV, oq

(6.1)

is a mod 2 homotopy class of trivializations of pV, oq|BΣ . In particular, sH pV, oq “ 0 P Z2 . Let Y be a subspace of a topological space X. We denote by LX pY q the collection of all continuous maps u : pΣ, BΣq ÝÑ pX, Y q from bordered surfaces. For such a map, let Bu “ u|BΣ . We call two such maps u : pΣ, BΣq ÝÑ pX, Y q and u1 : pΣ1 , BΣ1 q ÝÑ pX, Y q

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relatively equivalent if Bu “ Bu1 and the continuous map uYu1 : ΣYΣ1 ÝÑ X obtained by gluing u and u1 along the boundaries of their domains represents the zero class in H2 pX; Z2 q. Definition 6.3. Let X be a topological space and pV, oq be an oriented vector bundle over Y Ă X with rkR V ě 3. A relative Spinstructure s on pV, oq is a collection psu quPLX pY q of mod 2 homotopy classes su of trivializations of tBuu˚ pV, oq such that for every continuous map F : Σ ÝÑ Y from a bordered surface relatively equivalent to the disjoint union of continuous maps ui : pΣi , BΣi q ÝÑ pX, Y q,

i “ 1, . . . , k,

the mod 2 homotopy class sΣ pF ˚ pV, oqq of trivializations of tBF u˚ pV, oq equals to the disjoint union of the mod 2 homotopy classes sui of trivializations of tBui u˚ pV, oq. By the Σ “ H case of the condition above, the existence of a relative Spin-structure s in the sense of Definition 6.3 on an oriented vector bundle pV, oq with rkR V ě 3 implies that sui “ 0 for every continuous map ui : Σi ÝÑ X from a closed surface such that ui˚ rΣi s represents the zero element of H2 pX; Z2 q. Thus, the number D @ (6.2) w2 psq, u˚ rΣsZ2 ” su P Z2 depends only on the class in H2 pX; Z2 q determined by a continuous map u : Σ ÝÑ X from a closed surface. Along with Lemma 4.3(b) with Y “ H and the Universal Coefficient Theorem for Cohomology, this implies that s determines an element w2 psq P H 2 pX; Z2 q. In the perspectives of Definition 6.3, a relative Spin-structure s on an oriented vector bundle pV, oq over Y Ă X with rkR V P t1, 2u is a relative Spin-structure on the vector bundle 2τY ‘V with the induced orientation in the first case and on τY ‘V in the second. A relative Pin˘ -structure p on a real vector bundle V over Y Ă X is a relative Spin-structure on the canonically oriented vector bundle (1.6). In either of the two perspectives, a relative OSpin-structure on V is a pair os ” po, sq consisting of an orientation o on V and a relative ˘ pV q the Spin-structure s on pV, oq. We denote by OSpX pV q and PX

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sets of relative OSpin-structures and Pin˘ -structures, respectively, on V (up to equivalence in the perspective of Definition 6.1). For o P OpV q, we denote by SpX pV, oq the set of relative Spin-structures on pV, oq in either perspective. We identity SpX pV, oq with a subset of OSpX pV q in the obvious way. In either perspective, there are natural maps ιX : OSppV q ÝÑ OSpX pV q

˘ and ιX : P˘ pV q ÝÑ PX pV q. (6.3)

In the perspective of Definitions 1.2 and 6.1, these maps are obtained by taking the bundle E over X to be of rank 0. If the rank of V is at least 3 and psα qαPLpY q is a Spin-structure on pV, oq in the perspective of Definition 1.3, we define the Spin-structure ` ˘ (6.4) psu quPLX pY q ” ιX psα qαPLpY q in the perspective of Definition 6.3 as follows. Let u : pΣ, BΣ ÝÑ pX, Y q be a continuous map from a bordered surface. If BΣ “ H, we take su “ 0. If BΣ ‰ H, we take su to be the disjoint union of the (mod 2) homotopy classes su|Br Σ of trivializations of tu|Br Σ u˚ pV, oq under any identification of each topological component Br Σ of BΣ with S 1 . By the condition at the end of Definition 1.3, su does not depend on the choice of such identifications. The collection (6.4) then satisfies the condition at the end of Definition 6.3 and is thus is a relative Spin-structure on pV, oq. This construction induces the first map in (6.3) for vector bundles V of ranks 1 and 2 and the second map in (6.3) for vector bundles. Theorem 6.4. Let X be a topological space and pV, oq be an oriented vector bundle over Y Ă X. p1q The relative OSpin- and Pin-structures on V in the perspective of Definition 6.1 if pX, Y q is a CW pair and in the perspective of Definition 6.3 satisfy all properties of Section 6.2. p2q If pX, Y q is a CW pair, there are canonical identifications of the sets OSpX pV q in the perspectives of Definitions 6.1 and 6.3 ˘ pV q in the two perspectives for every vector and of the sets PX bundle V over Y . These identifications are intertwined with the identifications of Theorem 1.4 via the maps p6.3q and respect all structures and correspondences of Section 6.2.

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Properties of Relative Spin- and Pin-Structures

Let X be a topological space and Y Ă X. An isomorphism Ψ : V 1 ÝÑ V of vector bundles over Y induces bijections Ψ˚X : OSpX pV q ÝÑ OSpX pV 1 q and

˘ ˘ Ψ˚ : PX pV q ÝÑ PX pV 1 q (6.5)

between the relative OSpin-structures on V and V 1 and the Pin˘ structures on V and V 1 in the perspective of Definition 6.1 if pX, Y q is a CW pair and in the perspective of Definition 6.3. The first map sends a relative OSpin-structure pE, oE , osE,V q in the perspective of Definition 6.1 to  (˚ ˘ ` E, oE , idE|Y2 ‘Ψ|V |Y2 osE,V P OSpX pV 1 q, with t. . .u˚ on OSppE|Y2 ‘ V |Y2 q as in (1.9). In the perspective of Definition 6.3, the relative OSpin-structures on V 1 are obtained from the relative OSpin-structures on V by pre-composing the relevant trivializations with Ψ. Let X, X 1 be topological spaces, Y Ă X, Y 1 Ă X 1 , and V be a vector bundle over Y . A continuous map f : pX 1 , Y 1 q ÝÑ pX, Y q induces maps ˘ ˘ ˚ pV q ÝÑ PX f ˚ : OSpX pV q ÝÑ OSpX 1 pf ˚ V q and f ˚ : PX 1 pf V q (6.6)

in the perspective of Definition 6.1 if f is a CW map between CW pairs and in the perspective of Definition 6.3. The first map sends a relative OSpin-structure pE, oE , osE,V q in the perspective of Definition 6.1 to  (˚ ˘ ` ˚ f E, f ˚ oE , idE|Y2 ‘Ψ|V |Y2 osE,V P OSpX pV 1 q, with f ˚ on OSppE|Y2 ‘V |Y2 q as in (1.10). In the perspective of Definition 6.3, the relative OSpin-structures on f ˚ V are obtained from the relative OSpin-structures on V by pre-composing the relevant trivializations with (1.11). We denote by δX,Y : H 1 pY ; Z2 q ÝÑ H 2 pX, Y ; Z2 q

(6.7)

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the coboundary homomorphism in the cohomology exact sequence of the pair Y Ă X. The RelSpinPin properties in the following apply in either of the perspectives of Definitions 6.1 and 6.3 on relative Spin-, Pin˘ -, and OSpin-structures, provided pX, Y q is a CW pair in the first perspective. The naturality properties of the group actions and correspondences in the following refer to the commutativity with the pullbacks (6.5) induced by isomorphisms of vector bundles over Y and the pullbacks (6.6) induced by the admissible continuous maps. RelSpinPin 1 (Obstruction to existence). Let V be a vector bundle over Y Ă X and μ P H 2 pX; Z2 q. (a) The vector bundle V admits a relative Pin´ -structure (resp. Pin` -structure) p with w2 ppq “ μ if and only if μ|Y “ w2 pV q´w12 pV q (resp. μ|Y “ w2 pV q). (b) If V is orientable, V admits a relative Spin-structure s with w2 psq “ μ if and only if μ|Y “ w2 pV q. RelSpinPin 2 (Affine structure). Let V be a vector bundle over Y Ă X. (a) If V admits a relative Pin˘ -structure, the group H 2 pX, Y ; Z2 q ˘ pV q. acts freely and transitively on the set PX (b) If o P OpV q and pV, oq admits a relative Spin-structure, the group H 2 pX, Y ; Z2 q acts freely and transitively on the set SpX pV, oq. Both actions are natural with respect to pullbacks by continuous maps and w2 pη¨pq “ w2 ppq`η|X ,

w2 pη¨sq “ w2 psq`η|X

(6.8)

˘ pV q, and s P SpX pV, oq. If V 1 and for all η P H 2 pX, Y ; Z2 q, p P PX V 2 are vector bundles over Y , with at most one of them possibly of ˘ pV 1‘ rank 0, then the action of the automorphism Ψ in (1.12) on PX 2 V q is given by

` ˘ Ψ˚ p “ δX,Y prk V 1 ´1qw1 pV 1 q`prk V 1 qw1 pV 2 q ¨p ˘ pV 1 ‘V 2 q. @ p P PX

(6.9)

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RelSpinPin 3 (Orientation reversal). Let V be a real vector bundle over Y Ă X. There is a natural H 2 pX, Y ; Z2 q-equivariant involution

OSpX pV q ÝÑ OSpX pV q,

os ÝÑ os,

(6.10)

so that w2 posq “ w2 posq and os P SpX pV, oq for all os P SpX pV, oq and o P OpV q. RelSpinPin 4 (Reduction). Let V be a real vector bundle over X Ă Y . For every o P OpV q, there are natural H 2 pX, Y ; Z2 qequivariant bijections ˘ R˘ o : PX pV q ÝÑ SpX pV, oq

(6.11)

˘ ˘ so that w2 pR˘ o p¨qq “ w2 p¨q and Ro p¨q “ Ro p¨q.

RelSpinPin 5 (Stability). Let V be a vector bundle over Y Ă X. There are natural H 2 pX, Y ; Z2 q-equivariant bijections ˘ ` StV : OSpX pV q ÝÑ OSpX τY ‘V ,

˘ ` ˘ ˘ St˘ V : PX pV q ÝÑ PX τY ‘V (6.12)

so that w2 pStV p¨qq “ w2 p¨q, StV posq “ StV posq @ os P OSpX pV q, ˘ ˘ StV ˝R˘ o “ RStV poq ˝StV @ o P OpV q.

(6.13)

RelSpinPin 6 (Correspondences). Let V be a real vector bundle over Y Ă X. There are natural H 2 pX, Y ; Z2 q-equivariant bijections ` ˘ ˘ ˘ Co˘ V : PX pV q ÝÑ SpX V˘ , oV

(6.14)

˘ ˘ ˘ ˘ so that w2 ˝Co˘ V “ w2 and CoτY ‘V ˝StV “ StV˘ ˝CoV .

RelSpinPin 7 (Exact triples). Every short exact sequence e of vector bundles over Y Ă X as in (1.21) determines natural

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H 2 pX, Y ; Z2 q-biequivariant maps xx¨, ¨yy e : OSpX pV 1 qˆ OSpX pV 2 q ÝÑ OSpX pV q, ˘ ˘ pV 2 q ÝÑ PX pV q xx¨, ¨yy e : OSpX pV 1 qˆ PX

(6.15)

so that the following properties hold: (ses2X ) If e is as in (1.21), o1 P OpV 1 q, and o2 P OpV 2 q, then xxos1 , os2 yy e P OSpX pV, o1eo2 q @ os1 P SpX pV 1 , o1 q, os2 P SpX pV 2 , o2 q, (6.16) D D ` ˘ @ @ 1 2 1 ˘ 2 R˘ o1 o2 xxos , p yy e “ os , Ro2 pp q e e

˘ @ os1 P SpX pV 1 , o1 q, p2 P PX pV 2 q.

(6.17)

(ses3X ) If e is as in (1.21), o1 P OpV 1 q, os1 P SpX pV 1 , o1 q, and p2 P ˘ PX pV 2 q, then ˘ ` xxos 1 , p2 yy e “ δX,Y w1 pV 2 q ¨xxos1 , p2 yy e, DD ` ˘ @@ 1 ˘ 1 2 2 Co˘ V xxos , p yy e “ os , CoV 2 pp q e˘1 , o ` ˘ w2 xxos1 , p2 yy e “ w2 pos1 q`w2 pp2 q.

(6.18)

(ses4X ) If V11 , V21 , V 2 are vector bundles over Y , then @@ DD DD os11 , xxos12 , p2 yy‘ ‘ “ xxos11 , os12 yy‘ , p2 ‘

@@

˘ for all os11 P OSpX pV11 q, os12 P OSpX pV21 q, and p2 P PX pV 2 q. ˘ (ses5X ) If V is a vector bundle over Y and p P PX pV q,

St˘ V ppq “ xxιX pos0 pτY , oY qq, pyy‘ . ˘ pV q, (ses6X ) If V is a vector bundle over Y , o P OpV q, and p P PX then

@@ ˘ ˘DD ` Co˘ V ppq “ Ro ppq, ιX os0 pp2˘1qλpV, oqq ‘ .

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By RelSpinPin 4, (6.17), and the first and third statements in RelSpinPin 7(ses3X ), xxos 1 , os2 yy e “ xxos1 , os 2 yy e “ xxos1 , os2 yy e, ` ˘ w2 xxos1 , os2 yy e “ w2 pos1 q`w2 pos2 q

(6.19)

for every short exact sequence e of vector bundles as in (1.21), os1 P OSpX pV 1 q, and os2 P OSpX pV 2 q. By RelSpinPin 4 and 7(ses4X ) and (6.17), @@ DD DD @@ 1 (6.20) os1 , xxos12 , os2 yy‘ ‘ “ xxos11 , os12 yy‘ , os2 ‘ for all vector bundles V11 , V21 , V 2 over Y , os11 P OSpX pV11 q, os12 P OSpX pV21 q, and os2 P OSpX pV 2 q. By RelSpinPin 4, 5, and 7(ses5X ) and (6.17), StV posq “ xxιX pos0 pτY , oY qq, osyy‘

(6.21)

for every vector bundle V over Y and os P OSpX pV q. Combining the second map in (6.15) with the canonical isomorphism of V 1 ‘ V 2 with V 2 ‘ V 1 , we obtain a natural H 2 pX, Y ; Z2 qbiequivariant map ˘ ˘ pV 1 qˆ OSpX pV 2 q ÝÑ PX pV q. xx¨, ¨yy e : PX

(6.22)

By the RelSpinPin 7 property, this map satisfies the obvious analogs of (6.17), the first and third statements in RelSpinPin (ses3X ), and RelSpinPin (ses4X ) and (ses6X ). RelSpinPin 8 (Compatibility with Spin/Pin-structures). For every real vector bundle V over Y Ă X, the maps (6.3) are equivariant with respect to the homomorphism (6.7), respect all structures and correspondences of SpinPin 3–7 and RelSpinPin 3–7, ˘  ( ` ˘ pV q : w2 ppq “ 0 . (6.23) ιX P˘ pV q “ p P PX By RelSpinPin 4 and 8, ˘  ( ` ιX OSppV q “ os P OSpX pV q : w2 posq “ 0 .

(6.24)

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6.3

121

Proof of Theorem 6.4(1): Definition 6.3 Perspective

We now establish the statements of Section 6.2 for the notions of relative Spin-structure and Pin˘ -structure arising from Definition 6.3. Similar to Section 4.2, the proofs of these statements rely heavily on the topological observations of Section 4.1. Throughout this section, the terms relative to Spin-structure and Pin˘ -structure refer to the notions arising from Definition 6.3. Proof of RelSpinPin 2 property. We construct natural actions ˘ pV q so that of H 2 pX, Y ; Z2 q on OSpX pV q and PX ` ˘ ` ˘ ιX η¨os “ δX,Y pηq¨ιX posq and ιX η¨p “ δX,Y pηq¨ιX ppq (6.25) for all η P H 1 pY ; Z2 q, os P OSppV q, and p P P˘ pV q. By the definition of relative Spin- and Pin˘ -structures in this perspective, it is sufficient to describe the first action and to establish (6.25) and the claims of the RelSpinPin 2 property, other than (6.9), for Spin-structures on oriented vector bundles pV, oq with n ” rk V ě 3. The oriented vector bundle tBuu˚ pV, oq has two mod 2 homotopy classes of trivializations for every u P LX pY q. Let s be a relative Spinstructure on pV, oq and η P H 2 pX, Y ; Z2 q. We define the relative Spinstructure η¨s ” pη¨su qu on pV, oq by # “ su , if xu˚ η, rΣsZ2 y “ 0; η¨su “ ‰ su , if xu˚ η, rΣsZ2 y ‰ 0; (6.26) ` ˘ @ u : pΣ, BΣq ÝÑ pX, Y q P LX pY q. Suppose F : Σ ÝÑ Y is a continuous map from a bordered surface relatively equivalent to the disjoint union of continuous maps ui : pΣi , BΣi q ÝÑ pX, Y q,

i “ 1, . . . , m,

from bordered surfaces. In particular, u1˚ prΣ1 sZ2 q `¨ ¨ ¨` um˚ prΣm sZ2 q “ 0 P H2 pX, Y ; Z2 q, G C m m ÿ ÿ @ ˚ D ui˚ prΣi sZ2 q “ 0 P Z2 . ui η, rΣi sZ2 “ η, i“1

i“1

(6.27)

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Thus, the number of maps ui such that η¨ sui ‰ sui is even. Since the disjoint union of the mod 2 homotopy classes sui of trivializations of tBui u˚ pV, oq equals to the mod 2 homotopy class sΣ pF ˚ pV, oqq of trivializations of tBF u˚ pV, oq, the same is the case for the disjoint union of the mod 2 homotopy classes η ¨ sui . Thus, η ¨s is indeed a relative Spin-structure on pV, oq. It is immediate that the above construction defines a group action of H 2 pX, Y ; Z2 q on the set of such structures. In light of (4.7), this group satisfies the first property in (6.25). If u : Σ ÝÑ X is a continuous map from a closed surface, su “ η¨su if and only if @ D η|X , u˚ rΣsZ2 “ 0 P Z2 . Along with (6.2), Lemma 4.3(b) with Y “ H, and the Universal Coefficient Theorem for Cohomology, this implies the second statement in (6.8). By Lemma 4.3(b), every b P H2 pX, Y ; Z2 q can be represented by a continuous map pΣ, BΣq ÝÑ pX, Y q from a bordered surface. By the Universal Coefficient Theorem for Cohomology, the homomorphism ` ˘  ( κpηq pbq “ xη, by, κ : H 2 pX, Y ; Z2 q ÝÑ Hom H2 pX, Y q, Z2 , is bijective. Thus, η¨su ‰ su for some u P LX pY q if η ‰ 0 and so the action of H 2 pX, Y ; Z2 q above is free. Suppose s1 ” ps1u qu is another relative Spin-structure on pV, oq. Define # 0, if s1u “ su ; η : LX pY q ÝÑ Z2 , ηpuq “ 1, if s1u ‰ su . This determines a linear map from the Z2 -vector space generated by LX pY q to Z2 . Suppose u1 , . . . , um P LX pY q and F are as in and above (6.27). By Definition 6.3, ˘ ` su1 . . . sum “ sΣ F ˚ pV, oq “ s1u1 . . . s1um . Thus, the number of maps ui such that sui ‰ sui is even and so m ÿ i“1

ηpui q “ 0 P Z2 .

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Along with (b) and (c) in Lemma 4.3, this implies that η descends to a homomorphism η : H2 pX, Y ; Z2 q ÝÑ Z2 . By the Universal Coefficient Theorem for Cohomology, such a homomorphism corresponds to an element of H 2 pX, Y ; Z2 q, which we still denote by η. By the definition of η and the construction above, η¨s “ s1 . Thus, the action of H 2 pX, Y ; Z2 q described above is transitive. It remains to establish (6.9). Let η “ prk V 1 ´1qw1 pV 1 q`prk V 1 qw1 pV 2 q P H 1 pY ; Z2 q and ˘ p P PX pV 1 ‘V 2 q.

Let u : pΣ, BΣq ÝÑ pX, Y q be a continuous map from a bordered surface and B1 Σ, . . . , Bk Σ be the components of BΣ. By (1.13) and the SpinPin 2(a) property, pΨ˚ pqu ” Ψ˚ pu equals pu if and only if k ÿ @ D @ D tu|Br Σ u˚ η, rBr ΣsZ2 ” u˚ pδX,Y ηq, rΣsZ2 P Z2 r“1

vanishes. This implies (6.9).



Lemma 6.5 is a reformulation of Corollaries A.12–A.14 for (possibly) disconnected surfaces. It is used in the proof of the RelSpinPin 1 property. If Z´ and Z` are compact one-dimensional manifolds, pVr , r oq is an oriented vector bundle over Z´ \Z` , ` ˘ ` ˘ o |Z` (6.28) ϕ : Vr , r o |Z´ ÝÑ Vr , r is an isomorphism covering an identification of Z´ with Z` , and s´ and s` are mod 2 homotopy classes of trivializations of the two sides in (6.28), we define # 0 P Z2 , if s´ “ ϕ˚ s` ; xs´ , s` yϕ “ 1 P Z2 , if s´ ‰ ϕ˚ s` . If Z´ “ Z` , let xs´ , s` y “ xs´ , s` yid . r is a bordered surface, Z´ and Z` are disLemma 6.5. Suppose Σ r pVr , r joint unions of components of BΣ, oq is an oriented vector bunr r dle over Σ with rkR V ě 3, and (6.28) is an isomorphism covering an

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identification of Z´ with Z` . Let pVp , p oq be the oriented vector bundle p obtained from pVr , r r by gluing via ϕ. over the closed surface Σ oq and Σ If s´ , s` , and s0 are mod 2 homotopy classes of trivializations of pVr , r oq over Z´ , Z` , and BΣ´Z´ YZ` , respectively, then D @ D @ D @ oq “ s´ , s` ϕ ` s0 s´ s` , sΣr pVr , r oq P Z2 . s0 , sΣp pVp , p Proof of RelSpinPin 1 property. By the definition of relative Spin- and Pin˘ -structures in this perspective, it is sufficient to establish this claim for relative Spin-structures on oriented vector bundles pV, oq with rk V ě 3. Suppose pV, oq admits a relative Spin-structure s ” psu qu with w2 psq “ μ. Let u : Σ ÝÑ Y be a continuous map u : Σ ÝÑ Y from a closed surface. By (6.2) and the condition of Definition 6.3 applied with u1 , F “ u, D ` D ˘ @ @ μ|Y , u˚ rΣsZ2 “ su “ sΣ u˚ pV, oq ” w2 pu˚ V q, rΣsZ2 D @ “ w2 pV q, u˚ rΣsZ2 . Along with Lemma 4.3(b) with pX, Y q “ pY, Hq and the Universal Coefficient Theorem for Cohomology, this implies that μ|Y “ w2 pV q. Suppose w2 pV q “ μ|Y . Choose a collection  ( C ” ui : pΣi , BΣi q ÝÑ pX, Y q of maps that form a basis for H2 pX, Y ; Z2 q and a mod 2 homotopy class si of trivializations of tBui u˚ pV, oq for each map in C. Given a continuous map u : pΣ, BΣq ÝÑ pX, Y q from a bordered surface, let u1 , . . . , uk P C be so that ` ˘ ` ˘ ` ˘ u˚ rΣsZ2 `u1˚ rΣ1 sZ2 `¨ ¨ ¨`uk˚ rΣk sZ2 “ 0 P H2 pX, Y ; Z2 q. By Lemma 4.3(c), there exists a continuous map F : Σ0 ÝÑ Y from a bordered surface relatively equivalent to the disjoint union of the maps u, u1 , . . . , uk . In particular, BF “ Bu \ Bu1 \ ¨ ¨ ¨ \ Buk .

Relative Spin- and Pin-Structures

125

We take the mod 2 homotopy class su of trivializations of tBuu˚ pV, oq to be such that ` ˘ su s1 . . . sk “ sΣ0 F ˚ pV, oq .

(6.29)

Suppose F 1 : Σ10 ÝÑ Y is another continuous map satisfying the conditions of the previous paragraph and s1u is the mod 2 homotopy class of trivializations of tBuu˚ pV, oq such that ˘ ` s1u s1 . . . sk “ sΣ10 F 1˚ pV, oq .

(6.30)

p (resp. Σ) r the closed (resp. bordered) surface obtained Denote by Σ 1 from Σ0 and Σ0 by identifying them along the boundary components corresponding to BΣ, BΣ1 , . . . , BΣk (resp. BΣ1 , . . . , BΣk ). Thus, the r split into those forming BΣ Ă Σ0 and boundary components of Σ 1 p r by identifying these those forming BΣ Ă Σ0 ; Σ is obtained from Σ 1 two copies of BΣ. The maps F and F induce continuous maps p ÝÑ Y Fp : Σ

r ÝÑ Y, and Fr : Σ

p Σ. r By (6.29), (6.30), and which restrict to F and F 1 over Σ0 , Σ10Ă Σ, Lemma 6.5, ˘ ` su s1u “ sΣr Fr˚ pV, oq .

(6.31)

Since each of the maps F and F 1 is relatively equivalent to p Z vanishes in H2 pX; Z2 q. Thus, u\u1 . . .\uk , Fp˚ rΣs 2 ˘ @ ` D @ D p Z “ Fp˚ pμ|Y q, rΣs p Z sΣp Fp˚ pV, oq ” w2 pFp˚ V q, rΣs 2 2 D @ p Z “ 0. “ μ|Y , Fp˚ rΣs 2

(6.32)

p is obtained by identifying the two copies The bundle Fp ˚ V over Σ of tBuu˚ V in Fr˚ V . Along with (6.31), (6.32), and Lemma 6.5, this implies that su “ s1u . We next verify that s ” psu qu satisfies the condition of Definition 6.3. Suppose F : Σ ÝÑ Y is a continuous map from a bordered

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surface relatively equivalent to the disjoint union of continuous maps u1i : pΣ1i , BΣ1i q ÝÑ pX, Y q,

i “ 1, . . . , m,

from bordered surfaces. Thus, BF “ Bu11 \ ¨ ¨ ¨ \ Bu1m .

(6.33)

We show in the following that ` ˘ su11 . . .su1m “ sΣ F ˚ pV, oq ,

(6.34)

thus confirming the condition of Definition 6.3. For each i “ 1, . . . , m, let ` ˘ uij : Σij , BΣij ÝÑ pX, Y q with j “ 1, . . . , ki and Fi : Σi0 ÝÑ Y be the maps in C and a continuous map from a bordered surface, respectively, as in the construction of the mod 2 homotopy class su1i of trivializations of tBu1i u˚ pV, oq above. Let sij be the initially chosen mod 2 homotopy class of trivializations of Bu˚ij pV, oq. Thus, BFi “ Bu1i \ Bui1 \ ¨ ¨ ¨ \ Buiki ,

˘ ` su1i si1 . . .siki “ sΣi0 Fi˚ pV, oq . (6.35)

Since the maps in C are linearly independent in H2 pX, Y ; Z2 q and ki m ÿ ÿ

`

˘

uij ˚ rΣij sZ2 “

i“1 j“1

m ÿ

` ˘ ` ˘ u1i˚ rΣ1i sZ2 “ F˚ rΣsZ2

i“1

“ 0 P H2 pX, Y ; Z2 q, the maps uij come in pairs. Along with the first identity in (6.35), this implies that the boundary components of Σ10 , . . . , Σm0 corresponding to the boundaries Buij of these maps come in pairs as well. By (6.33), the remaining boundary components of Σ10 , . . . , Σm0 correspond to the boundary components of Σ. Denote by Σ1 the bordered surface obtained from Σ10 , . . . , Σm0 by identifying them along the paired up p be the closed surboundary components corresponding to Buij . Let Σ 1 face obtained from Σ and Σ by identifying them along the boundary components corresponding to Bu11 , . . . , Bu1m .

Relative Spin- and Pin-Structures

The maps F, F1 , . . . , Fm induce continuous maps p ÝÑ Y, F 1 : Σ1 ÝÑ Y and Fp : Σ

127

(6.36)

which restrict to Fi over Σ0i Ă Σ1 ; the second map also restricts to F p By the second identity in (6.35) and Lemma 6.5, over Σ Ă Σ. ˘ ` su11 . . . su1m “ sΣ1 F 1˚ pV, oq , ` ˘ ` ˘` ˘ sΣ F ˚ pV, oq su11 . . . su1m “ sΣ\Σ1 tF \F 1 u˚ pV, oq . (6.37) Since the disjoint union of F and the maps uij (which come in pairs) p Z vanishes in H2 pX; Z2 q. is relatively equivalent to F1\¨ ¨ ¨\Fm , Fp˚ rΣs 2 Thus, ˘ @ ` D @ D p Z “ Fp˚ pμ|Y q, rΣs p Z sΣp Fp˚ pV, oq ” w2 pFp˚ V q, rΣs 2 2 D @ p Z “ 0. (6.38) “ μ|Y , Fp˚ rΣs 2 p is obtained by identifying the copies of The bundle Fp˚ V over Σ ˚ ˚ tBu1 u V, . . . , tBum u V in tF\F 1 u˚ V . Along with the second identity in (6.37), (6.38), and Lemma 6.5, this implies (6.34). Thus, s is a relative Spin-structure on the oriented vector bundle pV, oq over Y Ă X and so w2 psq|Y “ w2 pV q by the first paragraph of the proof. Along with the cohomology exact sequence for pX, Y q, this implies that μ “ w2 psq`η|X P H 2 pX; Z2 q for some η P H 2 pX, Y ; Z2 q. By the RelSpinPin 2 property, the relative  Spin-structure η¨s on pV, oq satisfies w2 pη¨sq “ μ. Proof of RelSpinPin 3, 4 properties. For the purposes of establishing the first of these properties, we can assume that n ” rk V ě 3. For o P OpV q and s P SpX pV, oq, we define the relative Spin-structure s ” psu qu on pV, oq by # tφ ” In;1 φ : φ P su u, if BΣ ‰ H; su “ if BΣ “ H; su , (6.39) ` ˘ @ u : pΣ, BΣq ÝÑ pX, Y q P LX pY q. Since this orientation-reversal operation commutes with the disjoint union operation on the mod 2 homotopy classes of trivializations over one-dimensional manifolds and sΣ pV 1 , o1 q “ sΣ pV 1 , o1 q

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for any oriented vector bundle pV 1 , o1 q over a bordered surface Σ, the collection psu qu satisfies the condition of Definition 6.3 (because the collection psu qu does) and is thus indeed a relative Spin-structure on pV, oq. By (6.26), the resulting involution (6.10) is H 2 pX, Y ; Z2 qequivariant. It also satisfies the two conditions after (6.10). By (4.10) and (6.39), ` ˘ (6.40) ιX os “ ιX posq @ os P OSppV q. We next describe a bijection between SpX pV, oq and ˘ ˘ ˘ PX pV q ” SpX pV˘ , o˘ V “ SpX pV ‘p2˘1qτY , op2˘1qoY ; the last equality holds because the orientation o on V determines a canonical homotopy class of trivializations of u˚ λpV q for every continuous map u : Σ ÝÑ Y from a bordered surface. By the RelSpinPin 1 property, pV, oq admits a relative Spin-structure if and only if V admits a relative Pin˘ -structure. We can thus assume that pV, oq admits a relative Spin-structure. If rk V ě 3, we associate a relative Spin-structure s ” psu qu ˘ on pV, oq with the relative Spin-structure s˘ ” ps˘ u qu on pV˘ , oV q given by # ( tφ˘ ” φ‘p2˘1qpdet φq : φ P su if BΣ ‰ H; ˘ su “ if BΣ “ H; su , (6.41) ` ˘ @ u : pΣ, BΣq ÝÑ pX, Y q P LX pY q. Since this operation of adding determinant factors commutes with the disjoint union operation on the mod 2 homotopy classes of trivializations over one-dimensional manifolds and ˘˘ ` ˘ ` sΣ pV 1 , o1 q “ sΣ V˘1 , o˘ V1 for any oriented vector bundle pV 1 , o1 q over a bordered surface Σ, the collection ps˘ u qu satisfies the condition of Definition 6.3 (because the collection psu qu does) and is thus indeed a relative Spin-structure on pV˘ , o˘ V q. By (6.26), the map ˘ SpX pV, oq ÝÑ PX pV q,

s ÝÑ s˘ ,

(6.42)

is H 2 pX, Y ; Z2 q-equivariant. Along with the RelSpinPin 2 property, this implies that this map is a bijection. We take the map R˘ o in (6.11)

Relative Spin- and Pin-Structures

129

to be its inverse. By (6.2) and the second case in (6.41), this inverse satisfies the first condition after (6.11). By (4.11) and (6.41), ˘ ˘ ` ` ˘ ιX R˘ o ppq “ Ro ιX ppq

@ p P P˘ pV q.

(6.43)

If φ is a trivialization of tBuu˚ pV, oq as in (6.39) and BΣ ‰ H, then ` ˘ ` ˘ φ ˘ “ In`2˘1;3˘1 φ˘ : tBuu˚ V˘ ÝÑ pBΣqˆRn`2˘1 . Since this trivialization is homotopic to φ˘ , the second condition after (6.11) is also satisfied. We extend (6.42) to vector bundles V with rk V “ 1, 2 as at the end of the proof of the SpinPin 4 property in Section 4.2. Similar to (4.13), ˘ ` ´ ´ ´ R´ o “ RStV poq : PX pV q “ PX τY ‘V ÝÑ SpX pV, oq ` ˘ “ SpX StpV, oq

(6.44)

under the identifications (1.19) if rk V “ 1 and o P OpV q. The resulting extensions are H 2 pX, Y ; Z2 q-equivariant and satisfy (6.43) and the  last requirement of the RelSpinPin 4 property. Proof of RelSpinPin 5, 6 properties. The two sides of (6.14) are the same by definition; we take this map to be the identity. The first condition after (6.14) is then satisfied. Since the map (1.20) in the perspective of Definition 1.3 is also the identity, ˘ ˘ ` ` ˘ ιX Co˘ V ppq “ CoV ιX ppq

@ p P P˘ pV q.

(6.45)

By the RelSpinPin 1 property, an oriented vector bundle pV, oq admits a relative Spin-structure if and only if pτY ‘V, StV poqq does. In order to establish the RelSpinPin 5 property, we can thus assume that pV, oq admits a relative Spin-structure. If rk V ě 3, we associate a relative Spin-structure s ” psu qu on pV, oq with the relative Spin-structure StV s ” pStV su qu on

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pτY ‘V, Stpoqq given by # tStV φ ” idτBΣ ‘φ : φ P su u, if BΣ ‰ H; StV su “ if BΣ “ H; su , ` ˘ @ u : pΣ, BΣq ÝÑ pX, Y q P LX pY q.

(6.46)

Since this stabilization operation commutes with the disjoint union operation on the mod 2 homotopy classes of trivializations over onedimensional manifolds and ˘ ˘ ` ` StV 1 sΣ pV 1 , o1 q “ sΣ StV 1 pV 1 , o1 q for any oriented vector bundle pV 1 , o1 q over a bordered surface Σ, the collection pStV su qu satisfies the condition of Definition 6.3 (because the collection psu qu does) and is thus indeed a relative Spin-structure on pτY ‘V, Stpoqq. By (4.14) and (6.46), ˘ ˘ ` ` ιX StV posq “ StV ιX posq @ os P OSppV q. (6.47) The above construction of the first map in (6.12) with rk V ě 3 induces the first map in (6.12) with rk V “ 1, 2 and the second map in (6.12) as in the proof of the SpinPin 5, 6 properties in Section 4.2. The resulting maps thus satisfy (6.47) for OSpin- and Pin˘ structures on all vector bundles V over Y . By (6.2) and the second case in (6.46), they also satisfy the first condition after (6.12). The verification that the maps in (6.12) and (6.14) satisfy the remaining stated requirements is identical to the argument in Section 4.2.  Proof of RelSpinPin 7 property. For every continuous map u : Σ ÝÑ Y from a bordered surface, a short exact sequence e of vector bundles over Y as in (1.21) determines a homotopy class of isomorphisms u˚ V « u˚ V 1 ‘u˚ V 2 so that u˚ ι is the inclusion as the first component on the right-hand side above and u˚ j is the projection to the second component. Thus, it is sufficient to establish the RelSpinPin 7 property for the direct sum exact sequences as in (1.22). Furthermore, an orientation o1 on V 1 determines a homotopy class of trivializations of u˚ λpV 1 q and thus of isomorphisms ˘ ` ˘ ` u˚ V 1 ‘V 2 ˘ ” u˚ V 1 ‘V 2 ‘p2˘1qλpV 1 ‘V 2 q ˘ ` ` ˘ (6.48) « u˚ V 1 ‘V 2 ‘p2˘1qλpV 2 q ” u˚ V 1 ‘V˘2 for every continuous map u : Σ ÝÑ Y from a bordered surface.

Relative Spin- and Pin-Structures

131

Let V 1 and V 2 be vector bundles over Y of rank n1 and n2 , respectively. Suppose n1 , n2 ě 3, o1 P OpV 1 q, and o2 P OpV 2 q. For a relative Spin-structure s1 ” ps1u qu on pV 1 , o1 q and a relative Spin-structure s2 ” ps2u qu on pV 2 , o2 q, we define a Spin-structure xxs1 , s2 yy‘ ” ps1u ‘s2u qu on pV 1 ‘V 2 , o1 o2 q by # tφ1 ‘ φ2 : φ1 P s1u , φ2 P s2u u, if BΣ ‰ H; s1u ‘ s2u “ 1 su ` s2u P Z2 , if BΣ “ H; (6.49) ` ˘ @ u : pΣ, BΣq ÝÑ pX, Y q P LX pY q. Since this direct sum operation commutes with the disjoint union operation on the mod 2 homotopy classes of trivializations over onedimensional manifolds and ˘ ` sΣ pV 1 , o1 q‘sΣ pV 2 , o2 q “ sΣ pV 1 , o1 q‘pV 2 , o2 q for any oriented vector bundles pV 1 , o1 q and pV 2 , o2 q over a bordered surface Σ, the collection ps1u ‘s2u qu satisfies the condition of Definition 6.3 (because the collections ps1u qu and ps2u qu do) and is thus indeed a relative Spin-structure on pV 1 ‘V 2 , o1eo2 q. By (6.2) and the second case in (6.49), the resulting first map in (6.15) satisfies the second statement in (6.19). By (4.17) and (6.49), ` ˘ ιX os1 ‘os2 “ ιX pos1 q‘ιX pos2 q @ os1 P OSppV 1 q, os2 P OSppV 2 q. (6.50) The above construction determines the first map xx¨, ¨yy e in (6.15) if n1 , n2 ě 3. Along with (6.48), it also determines the second map xx¨, ¨yy e if n1 ě 3 and rk V˘2 ě 3 (i.e. not V´2 if n2 “ 1). By the second statement in (6.19) and the first statement after (6.14), this map satisfies (6.18). By (6.50) and (6.45), it also satisfies (6.50) with OSppV 2 q replaced by P˘ pV 2 q. By (6.26), both maps are H 2 pX, Y ; Z2 q-biequivariant. These two maps satisfy (6.16) and (ses4X ) for vector bundles of ranks at least 3 by definition. With φ˘ as in (6.41), ˘ ˘ ` ˘ ` ` 1 φ ‘φ2 ˘ “ φ1 ‘φ2˘ : tBuu˚ pV 1 ‘V 2 q˘ “ tBuu˚ V 1 ‘ tBuu˚ V˘2 1

ÝÑ pBΣqˆRn `n

2 `2˘1

.

Thus, (6.17) is also satisfied in these cases.

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For vector bundles V 1 , V 2 over Y not both of ranks at least 3, we define the two maps in (6.15) via (4.22) with b P 2Zě0 and with OSpin- and Pin˘ -structures replaced by relative OSpin- and Pin˘ structures. The resulting maps thus satisfy (6.50) for OSpin- and Pin˘ -structures on all vector bundles V 1 , V 2 over Y . By (6.18) for vector bundles of ranks at least 3 and the first statements after (6.11) and (6.12), they also satisfy (6.18) for all vector bundles V 1 , V 2 . The verification that the maps in (6.15) satisfy the remaining stated requirements is identical to the argument in the proof of the Spin Pin 7 property in Section 4.2. Proof of RelSpinPin 8 property. By (6.25), the maps (6.3) are equivariant with respect to the homomorphism (6.7). By (6.40), (6.43), (6.47), (6.45), and (6.50), these maps respect all structures and correspondences of the SpinPin 3–7 and RelSpinPin 3–7 properties. By (6.2) and the definition of the map (6.7), ˘  ( ` ˘ pV q : w2 ppq “ 0 . (6.51) ιX P˘ pV q Ă p P PX ˘ pV q and w2 ppq “ 0. By the RelSpinPin 1 and SpinPin 1 Suppose p P PX properties, this implies that V admits a Pin˘ -structure p1 . By the RelSpinPin 2 property and (6.51), ˘ ` (6.52) η¨ ιX pp1 q “ p and η|X “ 0

for some η P H 2 pX, Y ; Z2 q. By the second condition in (6.52) and the cohomology exact sequence for the pair X Ă Y , η “ δX,Y pη 1 q for some η 1 P H 1 pY ; Z2 q. Since the second map in (6.3) is equivariant with respect to the homomorphism (6.7), the first condition (6.52) implies that ` ˘ ιX η 1 ¨p1 “ p. Along with (6.51), this establishes (6.23). 6.4



Topological Preliminaries

Our proof of Theorem 6.4(1) for the relative Spin- and Pin-structures of Definition 6.1 in Section 6.5 relies on the standard topological statements recalled in this section.

Relative Spin- and Pin-Structures

133

r We denote by γ rm ÝÑ Gpmq the tautological vector bundle over r the the Grassmannian of oriented m-planes in R8 and by Pm P Gpmq 8 canonically oriented subspace of R spanned by the first m coordinate vectors. For m1 , m2 P Zě0 , choose a continuous map ` ˘ ` ˘ r 1 qˆ Gpm r 2 q, pPm1 , Pm2 q ÝÑ Gpm r 1 `m2 q, Pm1 `m2 Fm1 ,m2 : Gpm ˚ s.t. γ rm1 ˆr γm2 “ Fm rm1 `m2 . 1 ,m2 γ If X is a topological space and Y Ă X, then (˚ ` ˘ ` ˘ 2˚ ` ˘  rm1 `m2 “ h1˚ w2 γ rm1 `h w2 γ rm2 Fm1 ,m2 ˝ph1 , h2 q w2 γ P H 2 pX, Y ; Z2 q

(6.53)

r 1 q, Pm1 q and for all continuous maps h1 and h2 from pX, Y q to pGpm 2 r r Pm qs the pGpm q, Pm2 q, respectively. We denote by rpX, Y q, pGpmq, r set of homotopy classes of continuous maps from X to Gpmq which send Y to Pm . Lemma 6.6. Suppose m ě 3 and pX, Y q is a CW pair. The map ‰ “ r Pm q ÝÑ H 2 pX3 , Y2 ; Z2 q, rhs ÝÑ h˚ w2 pr γm q, pX3 , Y2 q, pGpmq, (6.54) is a bijection. r Proof. The space Gpmq is connected and simply connected. By the homotopy exact sequence for the fiber bundle q r r ÝÑ Gpmq, SOpmq ÝÑ Fpmq

(6.55)

r where Fpmq is the space of oriented m-frames on R8 , ` ˘ ` ˘ ` ˘ ` ˘ r r « π1 SOpmq « Z2 and π3 Gpmq « π2 SOpmq “ 0. π2 Gpmq Thus, the claim follows immediately from Lemma 4.4 with n “ 2, r  Z “ Gpmq, and z0 “ Pm . Let X be a topological space and Y Ă X. For each m P Z` , we denote by OVBm pX, Y q the set of isomorphism classes of oriented rank m vector bundles over X with a homotopy class of trivialization r over Y Ă X. For every continuous map h from pX, Y q to pGpmq, Pm q,

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Spin/Pin-Structures & Real Enumerative Geometry

the restriction to Y of the oriented vector bundle h˚ γ rm over X is equipped with a trivialization given by the canonical identification ˇ rm ˇY “ Y ˆr γm |Pm “ Y ˆRm . (6.56) ψh : h˚ γ We denote the homotopy class of ψh by sh . If h is the constant map to Pm , we denote sh by sX;m . By the proof of [34, Theorem 5.6], the map ‰ “ ‰ “ r rm , sh , (6.57) pX, Y q, pGpmq, Pm q ÝÑ OVBm pX, Y q, h ÝÑ h˚ γ is a bijection. We note that ‰ “ rm1 `m2 , sFm1 ,m2 ph1 ,h2 q tFm1 ,m2 ph1 , h2 qu˚ γ ‰ “ rm1 ‘h2˚ γ rm2 , sh1 ‘sh2 “ h1˚ γ

(6.58)

for all m1 , m2 P Z` and continuous maps h1 and h2 from pX, Y q to r 1 q, Pm1 q and pGpm r 2 q, Pm2 q, respectively. pGpm Corollary 6.7. Suppose m1 , m2 P Zě0 , pX, Y q is a CW pair, and “ ‰ “ 1 1 ‰ “ 1 1 ‰ pE1 , s1 q , pE2 , s2 q P OVBm1 pX, Y q, pE 2 , s2 q P OVBm2 pX, Y q s.t. ‰ “ ‰ “ 1 1 pE1 , s1 q‘pE 2 , s2 q “ pE21 , s12 q‘pE 2 , s2 q P OVBm1 `m2 pX, Y q. (6.59) If m1 ě 3, then ‰ “ ‰ “ 1 1 pE1 , s1 q|pX3 ,Y2 q “ pE21 , s12 q|pX3 ,Y2 q P OVBm1 pX3 , Y2 q.

(6.60)

r 2 q, Pm2 q Proof. Let g be a continuous map from pX3 , Y2 q to pGpm 2 2 representing the preimage of rpE , s q|pX3 ,Y2 q s under the bijection (6.57) with m “ m2 . By (6.53) and (6.58), the diagram `

o

H 2 pX3 , Y2 ; Z2 q



˘ 6.54 «

¨`g ˚ w2 pr γm2 q

H 2 pX3 , Y2 ; Z2 q

o

` ˘ 6.54 “ «

“

‰ r 1 q, Pm1 q pX3 , Y2 q, pGpm

Fm1 ,m2 ˝p¨,gq

` ˘ 6.57 «

/ OVBm1 pX3 , Y2 q

¨ ‘rpE 2 ,s2 q|pX ,Y q s 3 2

` ˘  ‰ 6.57 r 1 `m2 q, Pm1 `m2 q / OVBm1 `m2 pX3 , Y2 q pX3 , Y2 q, pGpm «



Relative Spin- and Pin-Structures

135

then commutes. Since the left vertical arrow in this diagram is a bijection, so is the right vertical arrow. This implies the claim.  Proposition 6.8. Let m1 , m2 P Zě0 with m1 ě 3 and pX, Y q be a CW pair. Suppose “ 1 1 ‰ “ 1 1 ‰ pE1 , s1 q , pE2 , s2 q P OVBm1 pX, Y q, E 2 and V are oriented vector bundles over X and Y , respectively, and sE 2 ,V is a homotopy class of trivializations of E 2 |Y ‘V . If there exists an isomorphism Ψ : E11 ‘E 2 ÝÑ E21 ‘E 2 s.t. s11 ‘sE 2 ,V  (˚ ` 1 ˘ “ Ψ|pE11 ‘Eq2 |Y ‘idV s2 ‘sE 2 ,V ,

(6.61)

then (6.60) holds. Proof.

Since the diagram `

˘ ` ˘ 6.54 “ 6.57 ‰ r 1 q, Pm1 q / OVBm1 pX3 , Y2 q H 2 pX3 , Y2 ; Z2 q o « pX3 , Y2 q, pGpm « ¨|pX3 ,Y1 q ˘ ` ˘ ¨|pX3 ,Y1 q   6.54 “ 6.57 ‰ r 1 q, Pm1 q / OVBm1 pX3 , Y1 q H 2 pX3 , Y1 ; Z2 q o « pX3 , Y1 q, pGpm «



¨|pX3 ,Y1 q

`

commutes and the left vertical arrow is injective, so is the right vertical arrow. Thus, it is enough to establish (6.60) with Y2 replaced by Y1 . We can assume that the sum of the ranks of E 2 and V is at least 3. Let s2 and sV be homotopy classes of trivializations of the oriented vector bundles E 2 |Y1 and V |Y1 , respectively, such that s2‘sV “ sE 2 ,V . By (6.61), (6.59) with Y “ Y1 holds. By Corollary 6.7, (6.60) with Y2 replaced by Y1 thus holds.  Suppose pX, Y q is a CW pair. A homotopy class of trivializations of an oriented rank m vector bundle pE, oE q over X3 determines a homotopy class of trivializations of pE, oE q|Y2 . Trivializations of pE, oE q|Y2 obtained from trivializations of pE, oE q differ by the action of h ” f |Y2 as in (4.32) for some continuous function f from X3 to SOpmq. Along with the proof of the SpinPin 2 property

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in Section 4.3, this implies that a trivializable vector bundle pE, oE q over X3 determines a collection ` ˘ ` ˘ H 1 pX3 ; Z2 qs0 pE, oE q|Y2 Ă Sp pE, oE q|Y2 of homotopy classes of trivializations of pE, oE q|Y2 which form an orbit of (  η|Y2 : η P H 1 pX3 ; Z2 q Ă H 1 pY2 ; Z2 q under the action of H 1 pY2 ; Z2 q on SpppE, oE q|Y2 q provided by the SpinPin 2(b) property in Section 1.2. For any η P H 1 pY2 ; Z2 q, let ` ˘ ` ˘ η¨H 1 pX3 ; Z2 qs0 pE, oE q|Y2 Ă Sp pE, oE q|Y2 be the image of this orbit under the action of η. Lemma 6.9. Suppose m ě 3, pX, Y q is a CW pair, η P H 1 pY2 ; Z2 q, r Pm q. If and h is a continuous map from pX3 , Y2 q to pGpmq, ` ˘ γm q P H 2 X3 , Y2 ; Z2 , δX3 ,Y2 pηq “ h˚ w2 pr

(6.62)

rm over X3 is trivializable and then the oriented vector bundle h˚ γ ` ˘ rm |Y2 . sh P η¨H 1 pX3 ; Z2 qs0 h˚ γ

(6.63)

Proof. By (6.62) and the cohomology exact sequence for Y2 Ă X3 , rm q “ 0 in H 2 pX3 ; Z2 q. By the bijectivity of (6.57) and (6.54) w2 ph˚ γ γm is thus trivializable. By the with Y2 “ H, the vector bundle h˚ r bijectivity of (6.57), it is sufficient to establish (6.63) for any map h satisfying (6.62). Let ηm P H 1 pSOpmq; Z2 q be the generator as before. We idenr r tify the fiber of the fiber bundle Fpmq in (6.55) over Pm P Gpmq

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with SOpmq. By [42, Theorem 7.2.8], the homomorphism ` ˘ ` ˘ r r SOpmq ÝÑ πk Gpmq q˚ : πk Fpmq, r is contractible, the is an isomorphism for every k P Z` . Since Fpmq boundary homomorphism ` ˘ ` ˘ r B : π2 Fpmq, SOpmq ÝÑ π1 SOpmq in the homotopy exact sequence for (6.55) is an isomorphism. Let ` ˘ ` ˘ r SOpmq, Im α r2 : D2 , S 1 , 1 ÝÑ Fpmq, be a continuous map generating ` ˘ ` ˘ r SOpmq « π1 SOpmq, Im « Z2 . π2 Fpmq, r Pm q Thus, q˝r α2 induces a continuous map α2 from pP1 , 8q to pGpmq, r and generating π2 pGpmqq ˘ ` r2 |S 1 : pS 1 , 1q ÝÑ SOpmq, Im α1 ” α is a loop generating π1 pSOpmqq « Z2 . Thus, @ D D @ w2 pr γm q, α r2˚ rS 2 sZ2 “ 1 P Z2 . ηm , α1˚ rS 1 sZ2 “ 1 P Z2 and (6.64) Let ηr : C1 pY2 ; Z2 q ÝÑ Z2

and

ηr2 : C2 pX3 , Y2 ; Z2 q ÝÑ Z2

be a cocycle representing η P H 1 pY2 ; Z2 q and the cocycle sending each 2-cell e Ă X2 to ` ˘ ηr2 peq ” ηr pBeqXC1 pY2 ; Z2 q , respectively. Thus, ηr2 represents δX3 ,Y2 pηq P H 2 pX3 , Y2 ; Z2 q. We define a continuous map ˘ ` ˘ ` r r SOpmq h : X3 , Y2 ÝÑ Fpmq, as follows. Let f be a continuous map sending X0 , X1 ´Y1 , and each 1-cell e Ă Y1 with ηrpeq “ 0 to Im and each 1-cell e Ă Y1 with ηrpeq “ 1 to α1 . By the first statement in (6.64), (˚  (6.65) f |Y1 ηm “ η|Y1 P H 1 pY1 ; Z2 q.

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Along with [42, Theorem 8.1.17], this implies that f |Y1 extends to a continuous map from Y2 Y X1 to SOpmq. For the same reason, f extends continuously over the 2-cells e Ă X2 ´Y2 with ηr2 peq “ 0. We r extend the resulting map to a continuous map r h from X2 to Fpmq by sending each 2-cell e Ă X2 ´ Y2 with ηr2 peq “ 1 to α r2 . Since the r r space Fpmq is contractible, the map h extends to X3 . By the cohomology exact sequence for the pair pY2 , Y1 q, the restriction homomorphism H 1 pY2 ; Z2 q ÝÑ H 1 pY1 ; Z2 q is injective. Along with (6.65), this implies that (˚  r h|Y2 ηm “ η P H 1 pY2 ; Z2 q.

(6.66)

The continuous map ` ˘ r Pm h ” q˝r h : pX2 , Y2 q ÝÑ Gpmq, sends every 2-cell e2 Ă X2 with ηr2 peq “ 0 to Pm and every 2-cell e2 Ă X2 with ηr2 peq “ 1 to α2 . Along with the second statement in (6.64), this implies that h satisfies (6.62). r rm over Fpmq has a canonical trivThe oriented vector bundle q ˚ γ r ialization Ψm since each point of Fpmq is an oriented frame for the r differs from the fiber over it. The restriction of Ψm to SOpmq Ă Fpmq canonical identification ˇ ˇ rm ˇSOpmq “ SOpmqˆr γm ˇPm “ SOpmqˆRm q˚γ by the action as in (4.32) of the identity map on SOpmq. Thus, the canonical identification ψh in (6.56) of the restriction of ` γm “ r h˚ q ˚ γ rm q h˚ r γm to Y2 Ă X3 differs from the restriction of the trivialization of h˚ r r induced by Ψm by the multiplication by h|Y2 “ f . Along with (6.66) and the proof of the SpinPin 2 property in Section 4.3, this estab lishes (6.63).

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Corollary 6.10. Let m ě 3, pX, Y q, η, and h be as in Lemma 6.9. γm such that There exists a trivialization Ψh of h˚ r (˚ `  ˘ sX;m ‘pη¨sE 2 ,V q sh ‘sE 2 ,V “ Ψh |ph˚ γrm q|Y2 ‘idE 2 |Y2 ‘V for all oriented vector bundles E 2 and V over X and Y2 , respectively, and homotopy classes sE 2 ,V of trivializations of E 2 |Y2 ‘V . rm |Y2 q be the homotopy class of trivializations of Proof. Let s0 ph˚ γ ˚ rm |Y2 determined by a trivialization Ψ of h˚ γ rm . By Lemma 6.9, h γ there exists η 1 P H 1 pX3 ; Z2 q such that ` ˘ ` ˘ rm |Y2 P Sp h˚ γ rm |Y2 . (6.67) η¨sh “ pη 1 |Y2 q¨s0 h˚ γ By the second statement of Corollary 4.5, η 1 “ f ˚ ηm for some continuous map f from X3 to SOpmq. Let rm ÝÑ mτX . Ψh “ f Ψ : h˚ γ By (6.67) and the construction of the H 1 pY ; Z2 q-action in the proof of the SpinPin 2 property in Section 4.3, (˚  Ψh |ph˚ γrm q|Y2 sX;m “ η¨sh . Along with the SpinPin 7 property in Section 1.2, this implies that ˘ (˚ `  sX;m ‘pη¨sE 2 ,V q Ψh |ph˚ γrm q|Y2 ‘idE 2 |Y2 ‘V ˘ ` ˘ ` “ η¨sh ‘ η¨sE 2 ,V “ sh ‘sE 2 ,V . This establishes the claim. 6.5



Proof of Theorem 6.4(1): Definition 6.1 Perspective

We now establish the statements of Section 6.2 for the notions of relative Spin-structure and Pin˘ -structure arising from Definition 6.1. Throughout this section, the terms relative Spin-structure and Pin˘ structure refer to these notions. Proof of RelSpinPin 1 property. Suppose o P OpV q and pV, oq admits a relative Spin-structure s ” pE, oE , sE,V q with w2 pEq “ μ.

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Since the vector bundle pE, oE q|Y2 ‘ pV, oq|Y2 then admits a Spinstructure, μ|Y2 “ w2 pEq|Y2 by the SpinPin 1(b) property in Section 1.2. Since the restriction homomorphism (4.30) is injective, it follows that μ|Y “ w2 pV q. Suppose μ|Y “ w2 pV q. By Lemma 6.6 with Y “ H, μ|X3 “ w2 pEq for some oriented vector bundle pE, oE q ” h˚ γ r3 over X3 . By the SpinPin 1(b) property, the vector bundle pE, oE q|Y2 ‘pV, oq|Y2 then admits a Spin-structure. The above establishes the RelSpinPin 1(b) property. The RelSpinPin 1(a) property is obtained similarly from the SpinPin 1(a)  property and Lemma 6.6. Proof of RelSpinPin 2 property. By the cohomology exact sequences for the triples Y2 Ă Y Ă X and Y2 Ă X3 Ă X, the restriction homomorphism ` ˘ ` ˘ H 2 X, Y ; Z2 ÝÑ H 2 X3 , Y2 ; Z2

(6.68)

is an isomorphism. It is thus sufficient to establish the claims of this property with H 2 pX, Y ; Z2 q replaced by H 2 pX3 , Y2 ; Z2 q. Suppose s ” pE, oE , sE,V q is a relative Spin-structure on pV, oq and η P H 2 pX3 , Y2 ; Z2 q. Let m ě 3. By Lemma 6.6, ` ˘ γm q P H 2 X3 , Y2 ; Z2 η “ h˚η;m w2 pr

(6.69)

r m , Pm q. We define for some continuous map hη;m from pX3 , Y2 q to pG ‰ “ rm ‘pE, oE q, xshη;m , sE,V y‘ , η¨s “ h˚η;m γ

(6.70)

with shη;m as below (6.56) and x¨, ¨y as in (1.24). By Lemma 6.6, rη ¨ss P SpX pV, oq does not depend on the choice of hη;m . By (6.58), rη ¨ss does not depend on the choice of m or the representative s of rss P SpX pV, oq. By (6.53) and (6.58), the resulting map H 2 pX3 , Y2 ; Z2 qˆ SpX pV, oq ÝÑ SpX pV, oq

(6.71)

is an action of H 2 pX3 , Y2 ; Z2 q on SpX pV, oq. By (6.70) and (6.69), this action satisfies the second statement in (6.8).

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Suppose rη¨ss “ rss. Thus, there exist m1 P Zě0 and an isomorphism ˘ 1` 1 Ψ : Stm h˚η;m γ rm ‘pE, oE q ÝÑ Stm`m pE, oE q of oriented vector bundles over X3 so that `@ D ˘ 1 Stm ph˚ rm ‘Eq|Y2 ‘V |Y2 shη;m , sE,V ‘ η;m γ )˚ ! ˇ ` ˘ 1 “ ΨˇStm1ph˚ γrm ‘Eq| ‘idV |Y2 Stm`m E|Y ‘V |Y sE,V . η;m

Y2

2

2

By Proposition 6.8, this implies that ‰ “ ‰ “ ˚ rm , shη;m “ mτX3 , sX;m P OVBm pX3 , Y2 q. hη;m γ Since the maps (6.54) and (6.57) are bijections, it follows that η “ 0. Thus, the action of H 2 pX3 , Y2 ; Z2 q on SpX pV, oq defined above is free. Suppose s1 ” pE 1 , oE 1 , sE 1 ,V q is another relative Spin-structure on pV, oq. We can assume that the ranks of E and E 1 are the same. By the RelSpinPin 1 property, `

ˇ ˇ ˘ˇ w2 pEq´w2 pE 1 q ˇY2 “ w2 pV qˇY2 ´w2 pV qˇY2 “ 0 P H 2 pY2 ; Z2 q.

Along with the cohomology exact sequence for the pair Y2 Ă X3 and Lemma 6.6, this implies that ` ˘ r3 P H 2 pX3 ; Z2 q w2 pEq´w2 pE 1 q “ h˚2 w2 γ

(6.72)

r P3 q. Let for some continuous map h2 from pX3 , Y2 q to pGp3q, ` ˘ r3 P H 2 pX3 , Y2 ; Z2 q. η2 ” h˚2 w2 γ By (6.72), Lemma 6.6, and the bijectivity of (6.57), there exists an isomorphism r3 ‘pE, oE q ÝÑ 3pτX , oX q‘pE 1 , oE 1 q Ψ2 : h˚2 γ of oriented vector bundles over X3 .

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By the SpinPin 2(b) property in Section 1.2, there exists η1 P H 1 pY2 ; Z2 q such that ` ˘  (˚ ` ˘ η1¨ sh2 ‘sE,V “ Ψ2 |ph˚2 γr3 ‘pE,oE qq|Y ‘idV |Y2 sX3 ;3‘sE 1 ,V . (6.73) 2

r P3 q so that (6.62) Let h1 be a continuous map from pX3 , Y2 q to pGp3q, with pη, hq replaced by pη1 , h1 q holds and Ψ1 be a trivialization provided by Corollary 6.10. By (6.73), ` ˘ Ψ ” Ψ1 ‘Ψ2 : h˚1 γ r3 ‘ h˚2 r γ3 ‘pE, oE q ` ˘ ÝÑ 3pτX , oX q‘ 3pτX , oX q‘pE 1 , oE 1 q is an isomorphisms of oriented vector bundles over X3 such that  (˚ ` ˘ sh1 ‘sh2 ‘sE,V “ Ψ|ph˚1 γr3 ‘h˚2 γr3 ‘pE,oE qq|Y ‘idV |Y2 sX3 ;6 ‘sE,V . 2

By the definition of the H 2 pX, Y ; Z2 q-action above, this implies that `

˘ δX3 ,Y2 pη1 q`η ¨s “ s1 .

Thus, this action is transitive. By Corollary 6.10 with E 2 being the rank 0 bundle, it also satisfies the first identity in (6.25). ˘ pXq by (6.70), replacWe define the action of H 2 pX, Y ; Z2 q on PX ing the Spin structures by Pin˘ -structures. By the same reasoning as in the relative Spin case, this indeed defines a group action. By the SpinPin 6 property in Section 1.2 and the second statement in the SpinPin 7(ses3) property, the H 2 pX, Y ; Z2 q-action on SpX pV˘ , o˘ Vq ˘ pV q being free and transitive implies that H 2 pX, Y ; Z2 q-action on PX is also free and transitive. The SpinPin 6 property and the first identity in (6.25) imply the second identity in (6.25). By (1.13) and Corol lary 6.10, (6.9) holds as well. Proof of RelSpinPin 3, 4 properties. Let o P OpV q. For an oriented vector bundle pE, oE q, we denote by oE |Y2 o|Y2 the orientation on E|Y2 ‘V |Y2 induced by oE and o. With the notation as in (1.14)

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and (1.15), we define the maps (6.10) and (6.11) by ` ˘ OSpX pV q ÝÑ OSpX pV q, os ” E, oE , osE,V ` ˘ ÝÑ os ” E, oE , osE,V , ` ˘ ˘ R˘ p ” E, oE , pE,V o : PX pV q ÝÑ SpX pV, oq, ˘ ` ÝÑ E, oE , Ro˘E |Y o|Y ppE,V q . 2

2

By (1.17), these two maps are well defined on the equivalence classes of relative OSpin- and Pin˘ -structures of Definition 6.1. It is immediate that the two maps preserve w2 and satisfy (6.40) and (6.43). By (1.27) and (1.26), the two maps are H 2 pX, Y ; Z2 q-equivariant. The last requirement in the RelSpinPin 4 property holds by the state ment after (1.15). Proof of RelSpinPin 5, 6 properties. of vector bundles V and W over Y2 , let

For an isomorphism Ψ

Ψ˘ : V˘ ÝÑ W˘ be the induced isomorphism. For vector bundles V 1 , V 2 over Y2 , let ΨV 1 ,V 2 : V 1 ‘V 2 ÝÑ V 2 ‘V 1

(6.74)

be the factor-interchanging isomorphism. If V 3 is another vector bundle over Y2 , then  (  ( (6.75) ΨV 1 ‘V 2 ,V 3 “ ΨV 1 ,V 3 ‘idV 2 ˝ idV 1 ‘ΨV 2 ,V 3 . An orientation o1 on V 1 determines a homotopy class of isomorphisms ` ˘ ` 1 ˘ ˘ 2 ‘V q , o « pV Ψo1 ,V 2 : pV 1 , o1 q‘ V˘2 , o˘ 2 1 2 ˘ V V ‘V of vector bundles over Y2 . If in addition o2 is an orientation of V 2 , then  ( (6.76) Ψo1 o2 ,V 3 „ Ψo1 ,V 2 ‘V 3 ˝ idV 1 ‘Ψo2 ,V 3 ,  (  ( Ψo2 ,V 1 ‘V 3 ˝ idV 2 ‘Ψo1 ,V 3 ˝ ΨV 1 ,V 2 ‘idV˘3  ( (  „ ΨV 1 ,V 2 ‘idV ˘ ˝Ψo1 ,V 2 ‘V 3 ˝ idV 1 ‘Ψo2 ,V 3 . (6.77)

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With the notation as in (1.16), we define the two maps in (6.12) by  ` ˘ ` (˚ StV E, oE , osE,V “ E, oE , ΨE|Y2 ,τY2 ‘idV |Y2 ˘˘ ` StE|Y2 ‘V |Y2 posE,V q ,  ` ˘ ` (˚ St˘ V E, oE , pE,V “ E, oE , ΨE|Y2 ,τY2 ‘idV |Y2 ˘˘ ` StE|Y2 ‘V |Y2 ppE,V q . By (6.75), (1.27), and (6.20), the first map is well defined on the equivalence classes of relative OSpin- and Pin˘ -structures of Definition 6.1 and equivariant with respect to the H 2 pX, Y ; Z2 q-actions. By (6.75) and (ses5) and (ses4) in the SpinPin 7 property in Section 1.2, so is the second map. It is immediate that the two maps preserve w2 and satisfy (6.47) for OSpin- and Pin˘ -structures on all vector bundles V over Y . By (1.17), they also satisfy (6.13). With the notation as in (1.20), we define the map in (6.14) by ` ˘ ` ` ˘ ˘˘ ˚ Co˘ V E, oE , pE,V “ E, oE , ΨoE |Y ,V |Y CoE|Y ‘V |Y ppE,V q . 2

2

2

2

By (6.76) and the second statement in the SpinPin 7(ses3) property, this map is well defined on the equivalence classes of relative Pin˘ - and OSpin-structures of Definition 6.1 and is equivariant with respect to the H 2 pX, Y ; Z2 q-actions. It is immediate that this map preserves w2 and satisfies (6.45). By (6.77) and the equation after (1.20), it also satisfies the last condition of the RelSpinPin 6  property. Proof of RelSpinPin 7 property. Let V 1 , V 2 be vector bundles over Y . With the notation as in (1.24) and (6.74), we define the first map in (6.15) by DD @@ 1 pE , oE 1 , osE 1 ,V 1 q, pE 2 , oE 2 , osE 2 ,V 2 q ‘ `  (˚ “ E 1 ‘E 2 , oE 1 oE 2 , idE 1 |Y2 ‘ΨE 2 |Y2 ,V 1 |Y2 ‘idV 2 |Y2 ` ˘˘ xxosE 1 ,V 1 , osE 2 ,V 2 yy‘ and the second map by the same expression with the OSpinstructure osE 2 ,V 2 replaced by a Pin˘ -structure pE 2 ,V 2 . By (6.75), (6.20), and the SpinPin 7(ses4) property in Section 1.2, these two

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maps are well defined on the equivalence classes of relative OSpinand Pin˘ -structures of Definition 6.1 and are bi-equivariant with respect to the H 2 pX, Y ; Z2 q-actions. It is immediate that the two maps satisfy (6.18) and (6.50) for OSpin- and Pin˘ -structures on all vector bundles V 1 and V 2 over Y . By the first equation in the SpinPin 7(ses3) property and Corollary 6.10, they also satisfy the first equation in the RelSpinPin 7(ses3X ) property. The remaining requirements on the two maps in the RelSpinPin 7 property follow from the corresponding requirements in the SpinPin 7 property.  Proof of RelSpinPin 8 property. The reasoning in the proof of this property for the perspective of Definition 6.3 in Section 6.3  applies verbatim for the perspective of Definition 6.1. 6.6

Equivalence of Definitions 6.1 and 6.3

Let pX, Y q be a CW pair and V be a vector bundle over Y . Since the restriction homomorphism (6.68) is an isomorphism, the RelSpinPin 1 property implies that V admits a relative Pin˘ -structure in the perspective of either Definition 6.1 or 6.3 if and only if the vector bundle V |Y2 over Y2 Ă X3 admits a relative Pin˘ -structure in the perspective of either Definition 6.1 or 6.3. If o P OpV q, then pV, oq admits a relative Spin-structure in the perspective of either Definition 6.1 or 6.3 if and only if the vector bundle pV, oq|Y2 over Y2 Ă X3 admits a relative Spin-structure in the perspective of either Definition 6.1 or 6.3. Furthermore, the natural restriction maps ` ˘ ` ˘ ˘ ˘ PX pV q ÝÑ PX V |Y2 and SpX3pV, oq ÝÑ Sp˘ X3 pV, oq|Y2 3 in either of the two perspectives are equivariant with respect to the isomorphism (6.68). Along with the RelSpinPin 2 property, this implies that the two maps are bijections. These bijections are intertwined with the identifications of Theorem 1.4 via the maps (6.3) and respect all structures and correspondences of Section 6.2. By the previous paragraph, it is thus sufficient to establish Theorem 6.4(2) under the assumption that the dimensions of the CW complexes X and Y are at most 3 and 2, respectively. We can also restrict the consideration to vector bundles V over Y that admit a

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relative Pin˘ -structure in the sense of Definition 6.1 and vector bundles pV, oq over Y that admit a relative Spin-structure in the sense of Definition 6.1, as appropriate. Suppose pV, oq is a rank n ě 3-oriented vector bundle over Y Ă X and s ” pE, oE , sE,V q is a relative Spin-structure on pV, oq in the sense of Definition 6.1. We can assume that the rank of E is at least 3 as well. Let u : pΣ, BΣq ÝÑ pX, Y q be an element of LX pY q and su pE, oE q be the mod 2 homotopy class of trivializations of tBuu˚ pE, oE q as in (6.1). If BΣ ‰ H, we denote by psu the mod 2 homotopy class of trivializations of tBuu˚ pV, oq so that the mod 2 homotopy class su pE, oE q‘psu of trivializations of ` ˘ tBuu˚ pE, oE q|Y ‘pV, oq “ tBuu˚ pE, oE q‘tBuu˚ pV, oq ÝÑ BΣ (6.78) is the mod 2 homotopy class determined by tBuu˚ sE,V . If BΣ “ H, let @ D psu “ w2 pEq, u˚ rΣsZ2 P Z2 . (6.79) Suppose F and u1 , . . . , um are as in (6.27) and just above and p ÝÑ X is the continuous map obtained by gluing these maps Fp : Σ along the boundaries of their domains as in (6.36). Since F is relatively equivalent to u1 \¨ ¨ ¨\um , ˘ @ ` D @ D p Z “ w2 pEq, Fp˚ rΣs p Z sΣp Fp˚ pE, oE q ” w2 pFp˚ Eq, rΣs 2 2 D @ “ w2 pEq, 0 “ 0. Along with Lemma 6.5, this implies that ˘ ` ˘ ` ˘ ` su1 E, oE . . .sum E, oE “ sΣ F ˚ pE, oE q . Since sE,V is a homotopy class of trivializations of pE, oE q|Y ‘pV, oq over Y , ˘ ` ˘ ` sum su1pE, oE q . . . sumpE, oE q ‘ psu1 . . . p ˘ ` ˘ ` ” su1pE, oE q‘psu1 . . . sumpE, oE q‘psum ˘˘ ` ` “ sΣ F ˚ pE, oE q|Y2 ‘pV, oq ˘ ` ˘ ` “ sΣ F ˚ pE, oE q sΣ F ˚ pV, oq .

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By the last two statements,

` ˘ psu1 . . . psum “ sΣ F ˚ pV, oq .

We conclude that a relative Spin-structure s on pV, oq in the sense of Definition 6.1(b) determines a relative Spin-structure pp su quPLX pY q ” ΘX psq on pV, oq in the sense of Definition 6.3. Suppose η P H 2 pX, Y ; Z2 q. Let m and hη;m be as in (6.69) and u P LX pY q be as above. By the bijectivity of (6.54) and (6.57) with pX, Y q “ pΣ, BΣq, su ph˚η;m r γm q is the mod 2 homotopy class rm q determined by tBuu˚ shη;m if and of trivializations of tBuu˚ ph˚η;m γ only if ˘ @ ˚ D @ D rm q , rΣsZ2 “ 0. u η, rΣsZ2 ” w2 pu˚ ph˚η;m γ (6.80) γm ‘pE, oE qq‘p su of trivialThus, the mod 2 homotopy class su ph˚η;m r izations of ˘ ˘ `` rm ‘pE, oE q |Y ‘pV, oq tBuu˚ h˚η;m γ ` ˘ “ tBuu˚ ph˚η;m q‘ tBuu˚ pE, oE q‘tBuu˚ pV, oq is the mod 2 homotopy class determined by tBuu˚ xxshη;m , sE,V yy‘ if and only if (6.80) holds. Along with (6.70) and (6.26), this implies that ` ˘ x u “ η ¨ psu . η¨s We conclude the construction of ΘX above induces a natural H 2 pX, Y ; Z2 q-equivariant map ΘX : OSpX;2 pV q ÝÑ OSpX;3 pV q

(6.81)

from the set of equivalence classes of relative OSpin-structures on the vector bundle V over Y Ă X in the sense of Definition 6.1(b) to the set of relative OSpin-structures on V in the sense of Definition 6.3. Along with the RelSpinPin 2(b) property, this implies that (6.81) is a bijection. By (6.2) and (6.79), ˘ ` (6.82) w2 ΘX posq “ w2 posq @ os P OSpX;2 pV q. Suppose (5.2) is a Spin-structure on the oriented vector bundle pV, oq over Y in the sense of Definition 1.1(b). Let sr2 be a section of

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SpinpV, oq as above (5.11) and s be the induced Spin-structure on s2 q pV, oq in the sense of Definition 1.2. For each α P LpY q, srα ” α˚ pqV ˝r is then a section of the Spinpnq-bundle α˚ SpinpV, oq in (5.3) and determines a homotopy class sα of trivializations of α˚ pV, oq. With the notation as in (5.4), (5.12), and (6.81), this implies that ` ` ˘˘ ιX Θ3 qV : SpinpV, oq ÝÑ SOpV, oq ˘ ` ˘ ` ” ιX psα qαPLpY q “ ΘX ιX psq ˘˘ ` ` “ ΘX ιX Θ2 pqV : SpinpV, oq ÝÑ SOpV, oqq . Since Θ2 is a bijection, it follows that ˘ ˘˘ ` ` ` ΘX ιX posq “ ιX Θ3 Θ´1 2 posq

@ os P OSp2 pY q

(6.83)

for every vector bundle V of rank at least 3 over Y . Compatibility with RelSpinPin 3 property. Suppose pV, oq is an oriented vector bundle over Y of rank at least 3, ` ˘ ˘ ` and s ” E, oE , sE,V s ” E, oE , sE,V is a relative Spin-structure on pV, oq in the sense of Definition 6.1 and the corresponding relative Spin-structure on pV, oq, respectively. su be the mod 2 Let u P LX pY q, su pE, oE q, and psu be as before and p homotopy class of trivializations of tBuu˚ pV, oq induced by su pE, oE q and the homotopy class sE,V of trivializations of ˘ ` tBuu˚ pE, oE q|Y ‘ pV, oq “ tBuu˚ pE, oE q ‘ tBuu˚ pV, oq ÝÑ BΣ. (6.84) su and su pE, oE q‘p su are the mod 2 homotopy class Thus, su pE, oE q‘p of trivializations of (6.78) determined by tBuu˚ sE,V and the mod 2 homotopy class of trivializations of (6.84) determined by tBuu˚ sE,V , su “psu is the mod 2 respectively. Along with (1.27), this implies that p ˚ homotopy class of trivializations of tBuu pV, oq corresponding to psu . Thus, ` ˘ (6.85) ΘX os “ ΘX posq @ os P OSpX;2 pV q for every vector bundle V of rank at least 3 over Y .



Relative Spin- and Pin-Structures

Compatibility with RelSpinPin 5 property. and s are as above, ΨV 1 ,V 2 is as in (6.74), and

149

Suppose pV, oq

 ˘ (˚` s1E,V ” ΨE|Y ,τY ‘idV StE|Y ‘V psE,V q is the homotopy class of trivializations of pE, oE q|Y ‘StpV, oq induced by s. Let u P LX pY q, su pE, oE q, and psu be as before and ps1u be the mod 2 homotopy class of trivializations of tBuu˚ StpV, oq induced by su pE, oE q and the homotopy class s1E,V of trivializations of ` ` ˘ ˘ tBuu˚ pE, oE q|Y ‘StpV, oq “ tBuu˚ pE, oE q‘tBuu˚ StpV, oq ÝÑ BΣ. (6.86) Thus, su pE, oE q ‘ psu and su pE, oE q ‘ p s1u are the mod 2 homotopy class of trivializations of (6.78) determined by tBuu˚ sE,V and the mod 2 homotopy class of trivializations of (6.86) determined by tBuu˚ s1E,V , respectively. Along with (1.29) and (1.28), this implies that p s1u “ StV ppsu q is the mod 2 homotopy class of trivializations of su . Thus, tBuu˚ StpV, oq corresponding to p ˘ ˘ ` ` ΘX StV posq “ StV ΘX posq

@ os P OSpX;2 pV q

for every vector bundle V of rank at least 3 over Y .

(6.87) 

We define the map ΘX in (6.81) for rank 2 vector bundles V over Y Ă X and then for rank 1 vector bundles V over Y by (6.87). By the RelSpinPin 5 property, the already established properties of (6.81) for vector bundles of ranks at least 3, and Theorem 1.4, the resulting maps ΘX for vector bundles of ranks 1 and 2 are natural H 2 pX, Y ; Z2 q-equivariant bijections which intertwine the identifications of Theorem 1.4 via the maps (6.3) and satisfy (6.82). These maps are compatible with the RelSpinPin 5 property, i.e. satisfy (6.87), by definition. Along with the first equality in (6.13) and (6.85) for vector bundles of rank at least 3, this implies that the maps ΘX satisfy (6.85) for all vector bundles V over Y .

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Compatibility with RelSpinPin 7 property. Suppose pV 1 , o1 q and pV 2 , o2 q are oriented vector bundles over Y of ranks at least 3, ` ` ˘ ˘ s1 ” E 1 , oE 1 , sE 1 ,V 1 and s2 ” E 2 , oE 2 , sE 2 ,V 2 are relative Spin-structures on pV 1 , o1 q and pV 2 , o2 q, respectively, ΨV 1 ,V 2 is as in (6.74), and (˚ ` ˘  sE,V ” idE 1 |Y ‘ΨE 2 |Y ,V 1 ‘idV 2 sE 1 ,V 1 ‘sE 2 ,V 2 is the homotopy class of trivializations of ` ˘ˇ ` ˘ pE, oE q|Y ‘pV, oq ” pE 1 , oE 1 q‘pE 2 , oE 2 q ˇY ‘ pV 1 , o1 q‘pV 2 , o2 q induced by s1 and s2 . For u P LX pY q, let su pE 1 , oE 1 q and su pE 2 , oE 2 q be the mod 2 homotopy class of trivializations of Bu˚ pE 1 , oE 1 q and Bu˚ pE 2 , oE 2 q, respectively, as in (6.1). s2u , and psu be the mod 2 homotopy classes of trivializations Let ps1u , p ˚ of tBuu pV 1 , o1 q induced by su pE 1 , oE 1 q and sE 1 ,V 1 , of tBuu˚ pV 2 , o2 q induced by su pE 2 , oE 2 q and sE 2 ,V 2 , and of tBuu˚ pV, oq induced by ` ˘ ` ˘ ` ˘ su E, oE “ su E 1 , oE 1 ‘su E 2 , oE 2 and the homotopy class sE 1 ,V 1 ‘sE 2 ,V 2 of trivializations of ` ˘ tBuu˚ pE, oE q|Y ‘pV, oq “ tBuu˚ pE 1 , oE 1 q‘tBuu˚ pE 2 , oE 2 q ‘ tBuu˚ pV 1 , o1 q‘tBuu˚ pV 2 , o2 q, (6.88) respectively. Thus, s1u , su pE 2 , oE 2 q‘ps2u , su pE 1 , oE 1 q‘p ˘ ` su pE 1 , oE 2 q‘su pE 2 , oE 2 q ‘psu

and

are the mod 2 homotopy classes of trivializations of tBuW u˚ ppE 1 , oE 1 q|Y ‘ pV 1 , o1 qq determined by tBuu˚ sE 1 ,V 1 , of ˚ 2 2 2 ˚ tBuu ppE , oE 2 q|Y ‘pV , o qq determined by tBuu sE 2 ,V 2 , and of (6.88) determined by tBuu˚ psE 1 ,V 1 ‘sE 2 ,V 2 q, respectively. Along with (1.28), s1u ‘p s2u is the mod 2 homotopy class of trivialthis implies that p su “p ˚ izations of tBuu pV, oq determined by ps1u and ps2u . Thus, DD ` ˘ @@ ΘX xxos1 , os2 yy‘ “ ΘX pos1 q, ΘX pos2 q ‘ @ os1 P OSpX;2 pV 1 q, os2 P OSpX;2 pV 2 q

(6.89)

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151

for all vector bundles V 1 and V 2 of ranks at least 3 over Y . This identity for arbitrary rank bundles then follows from the RelSpinPin 5  property, (6.21), and (6.20). ˘ ˘ ˘ With PX;2 pV q and PX;3 pV q denoting the sets PX pV q of the rela˘ tive Pin -structures on a vector bundle V over Y in the perspectives of Definitions 6.1 and 6.3, respectively, define ` ` ˘ ˘ ˘ ˘ ˘ pV q ÝÑ PX;3 pV q by Co˘ ΘX : PX;2 V ΘX ppq “ ΘX CoV ppq .

(6.90) By the RelSpinPin 6 property, the already established properties of (6.81), and Theorem 1.4, (6.90) is a natural H 2 pX, Y ; Z2 qequivariant bijection which intertwines the identifications of Theorem 1.4 via the maps (6.3). It preserves w2 and is compatible with the RelSpinPin 5 property, i.e. the map ΘX satisfies (6.82) and (6.87) for relative Pin˘ -structures p in place of the relative OSpin-structures os. This map is compatible with the RelSpinPin 6 property, i.e. satisfies the second equality in (6.90), by definition. By the second statement in the RelSpinPin 7(ses3X ) property and (6.89), (6.90) is compatible with the second map in (6.15), i.e. (6.89) holds for relative Pin˘ -structures p2 in place of the relative OSpin-structures os2 . By the RelSpinPin 7(ses6X ) property, (6.90) is compatible with the RelSpinPin 4 property, i.e. ˘ ˘ ` ` ˘ ˘ @ p P PX;2 pV q, o P OpV q ΘX R˘ o ppq “ Ro ΘX ppq for every vector bundle V over Y . This concludes the proof of Theorem 6.4(2).

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Part II

Orientations for Real CR-Operators

The so-called real Cauchy–Riemann operators (or CR-operators for short) are central to Gromov–Witten theory and the Fukaya category literature. As demonstrated in [12, Section 8.1], relative Spin-structures induce orientations on the determinants of such operators. As demonstrated in [39, Section 3], relative Pin-structures induce orientations on certain twisted determinants of real CRoperators. Chapter 7 collects properties of the induced orientations in a ready-to-use format. The construction of these orientations and the verification of their claimed properties are detailed in Chapters 8–10. We formulate all statements in terms of real bundle pairs pV, ϕq over symmetric surfaces pΣ, σq with separating fixed loci Σσ . Analogous statements for vector bundle pairs pE, F q over bordered surfaces pΣb , BΣb q follow immediately by doubling pE, F q to a real bundle pair as in [15, Section 3]. Our construction of orientations in the relative Spin case differs from [12] and is instead inspired by the approach of [30, Section 6]; this approach is now the standard perspective on orienting real CR-operators in this case. Our construction of orientations in the relative Pin case differs from that in [39] and is instead motivated by the approach of [13, Section 3]. This approach combines the Spin and rank 1 cases and eliminates the need for an arbitrary choice of a distinguished Pin-structure on a vector bundle over S 1 in each rank.

153

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Spin/Pin-Structures & Real Enumerative Geometry

However, the way we combine the Spin and rank 1 cases differs from [13]. Chapter 8 constructs orientations of determinants of real CRoperators on rank 1 real bundle pairs over S 2 with the standard orientation-reversing involution τ and on even-degree real bundle pairs pV, ϕq over pS 2 , τ q. As in [13], the orientations opV, ϕ; ox1 q constructed in the former case depend on the choice of orientation ox1 of Vxϕ1 of the real line bundle V ϕ at a point x1 in the τ -fixed locus S 1 and the choice of half-surface D2` of pS 2 , τ q. As in [12, 30], the orientations oos pV, ϕq constructed in the latter case depend on the choice of relative OSpin-structure os on the real vector bundle V ϕ over S 1 Ă S 2 and the choice of half-surface D2` of pS 2 , τ q. We recall the orientation constructions of [13, 30] in these two cases and establish key properties of the resulting orientations. Chapter 8 contains the most technical arguments of Part II. Chapter 9 constructs orientations of determinants of real CRoperators on rank 1 real bundle pairs and on even-degree real bundle pairs pV, ϕq over arbitrary smooth symmetric surfaces pΣ, σq with separating fixed loci Σσ . These orientations are obtained from the orientations constructed in the two corresponding cases in Chapter 8. Similar to Chapter 8, the orientations opV, ϕ; ox q in the former case depend on the choices of orientation oxr of Vxϕr at a point xr in each component Sr1 of the σ-fixed locus Σσ , half-surface Σb of pΣ, σq, and ordering on π0 pΣσ q. The orientations oos pV, ϕq in the latter case depend on the choices of relative OSpin-structure os on V ϕ , half-surface Σb of pΣ, σq, and ordering on π0 pΣσ q. We again recall the orientation constructions of [13, 30] in these two cases and establish key properties of the resulting orientations. These properties are deduced from the analogous properties in Chapter 8 primarily through algebraic considerations involving commutative squares of short exact sequences. Chapter 9 concludes the verification of the properties of orientations of the determinants of real CR-operators induced by relative OSpin-structures stated in Chapter 7. Chapter 10 finally constructs orientations on the determinants and twisted determinants of real CR-operators induced by relative Pin-structures. It uses the RelSpinPin 6 property on page 82 to combine the orientations constructed in Chapter 9. Chapter 10 also deduces the properties of the orientations induced by relative Pinstructures stated in Chapter 7 from the analogous properties established in the two settings of Chapter 9.

Chapter 7

Main Results and Applications of Part II

We assemble the key notions concerning real CR-operators and their determinants and state the main (and only) theorem of Part II in Section 7.1. The properties of orientations of real CR-operators described in Sections 7.2 and 7.3 include the compatibility of the orientations with short exact sequences of real bundle pairs and with flat degenerations of the domains. Some consequences of these properties are deduced in Section 7.4. 7.1

Definitions and Main Theorem

An exact triple (short exact sequence) 0 ÝÑ D 1 ÝÑ D ÝÑ D 2 ÝÑ 0 of Fredholm operators is a commutative diagram 0

/ X1 

0

D1

/Y1

/X 

D

/Y

/ X2 

/0

D2

/Y2

/0

(7.1)

so that the rows are exact sequences of bounded linear homomorphisms between Banach vector spaces (over R) and the columns are Fredholm operators. If X 2 is a finite-dimensional vector space and 155

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Spin/Pin-Structures & Real Enumerative Geometry

Y 2 is the zero vector space, we abbreviate the exact triple (7.1) of Fredholm operators as 0 ÝÑ D 1 ÝÑ D ÝÑ X 2 ÝÑ 0. If X, Y are Banach spaces and D : X ÝÑ Y is a Fredholm operator, let λpDq ” λpker Dq b λpcok Dq˚ denote the determinant of D. An orientation on a Fredholm operator D is an orientation on the one-dimensional real vector space λpDq or equivalently a homotopy class of isomorphisms of λpDq with R. If H is a topological space, a continuous family (  D ” Dt : t P H of Fredholm operators Dt over H determines a line bundle λp Dq over H, called the determinant line bundle of D; see [53]. An exact triple t of Fredholm operators as in (7.1) determines a canonical isomorphism «

Ψt : λpD 1 q b λpD 2 q ÝÑ λpDq.

(7.2)

For a continuous family of exact triples of Fredholm operators, the isomorphisms (7.2) give rise to a canonical isomorphism between the determinant line bundles; see the Exact Triples property in [53, Section 2]. An involution on a topological space X is a homeomorphism φ : X ÝÑ X such that φ ˝ φ “ idX . A nodal surface Σ is a topological r by identifying some pairs space obtained from a smooth surface Σ r of distinct points of Σ (the pairs are disjoint from each other; each r is identified into a node of Σ). The associated pair of points of Σ r ÝÑ Σ is called the normalization of Σ. A (possibly quotient map q : Σ nodal) symmetric surface pΣ, σq consists of a nodal surface Σ and an r of the normalization q of Σ involution σ on Σ so that the domain Σ is a closed oriented surface and r ÝÑ Σ σ˝q “ q˝r σ: Σ r We then call for some smooth orientation-reversing involution σ r on Σ. r pΣ, σ rq the normalization of pΣ, σq. If Σ is smooth, the fixed locus Σσ of σ is a disjoint union of circles. In general, Σσ consists of isolated points (called E nodes in [30, Section 3.2]) and circles identified at

Main Results and Applications of Part II

157

pairs of points (called H nodes in [30]). The remaining nodes of Σ, σ which we call C nodes, come in pairs nd˘ ij R Σ interchanged by the involution σ. A nodal symmetric surface pΣ, σq is obtained from its r σ normalization pΣ, rq by identifying r Σ r σr with σ rpr z q to form an E-node, (E) a point zr P Σ´ r σr to form an H-node, (H) distinct points zr, zr1 P Σ r ´Σ r σr so that zr1 ‰ σ rpr z q and the points (C) distinct points zr, zr1 P Σ 1 σ rpr z q, σ rpr z q to form a pair of C-nodes interchanged by σ. r comA complex structure on pΣ, σq is a complex structure j on Σ r such that σ patible with the orientation of Σ r˚ j “ ´j. A symmetric Riemann surface, possibly nodal, is a triple pΣ, σ, jq so that pΣ, σq is a symmetric surface and j is a complex structure on pΣ, σq. For k P Zě0 , let rks “ t1, . . . , ku. A symmetric surface with k real marked points and l conjugate pairs of marked points is a tuple ` ˘ C ” Σ, σ, pxi qiPrks , pzi` , zi´ qiPrls , (7.3) where pΣ, σq is a symmetric surface and x1 , . . . , xk , z1` , . . . , zl` , and z1´ , . . . , zl´ are distinct smooth points of Σ such that σpxi q “ xi for all i P rks and σpzi` q “ zi´ for all i P rls. There are two basic kinds of symmetric surfaces: connected ones and pairs of connected surfaces interchanged by the involution, called doublets in [20]. Every symmetric surface pΣ, σq is a union of such surfaces, which we then call the real elemental components and the conjugate elemental components of pΣ, σq, respectively. A bordered surface Σb doubles to a symmetric surface pΣ, σq so that Σb Ă Σ,

Σ “ Σb YσpΣb q,

and

Σb XσpΣb q “ Σσ “ BΣb .

In such a case, we call Σb a half-surface of pΣ, σq and orient Σσ as the boundary of Σb as in [43, p. 146]. A smooth symmetric surface pΣ, σq admits a half-surface if and only if Σσ separates Σ into two halves interchanged by σ, i.e. the fixed locus of pΣ, σq is separating. A halfsurface of a doublet pΣ, σq is either of the two connected components of pΣ, σq. If the fixed locus of a connected elemental symmetric surface pΣ, σq is separating, pΣ, σq again contains two half-surfaces. A choice of a half-surface for an arbitrary symmetric surface pΣ, σq corresponds to a choice of a half-surface for each elemental component of pΣ, σq.

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Let pX, φq be a topological space with an involution. A conjugation on a complex vector bundle V ÝÑ X lifting the involution φ is a vector bundle homomorphism ϕ : V ÝÑ V covering φ (or equivalently a vector bundle homomorphism ϕ : V ÝÑ φ˚ V covering idX ) such that the restriction of ϕ to each fiber is anti-complex linear and ϕ ˝ϕ “ idV . A real bundle pair pV, ϕq ÝÑ pX, φq consists of a complex vector bundle V ÝÑ X and a conjugation ϕ on V lifting φ. For example, ˘ ` X ˆCn , φˆc ÝÑ pX, φq, where c : Cn ÝÑ Cn is the standard conjugation on Cn , is a real bundle pair. In general, V ϕ ÝÑ X φ is a real vector bundle with rkR V ϕ “ rkC V . Suppose pΣ, σ, jq is a symmetric Riemann surface, possibly nodal, r σ and pV, ϕq is a real bundle pair over pΣ, σq. Let q : pΣ, rq ÝÑ pΣ, σq be the normalization of pΣ, σq. We denote by ΓpΣ; V q the space of continuous sections ξ of V ÝÑ Σ so that q ˚ ξ is a smooth section of r Define the vector bundle q ˚ V ÝÑ Σ.  ( ΓpΣ; V qϕ “ ξ P ΓpΣ; V q : ξ ˝σ “ ϕ˝ξ , ` ˘ ˚r 0,1 ˚ r Γ0,1 j pΣ; V q “ Γ Σ; pT Σ, jq bC q V ,  ( 0,1 ϕ Γ0,1 σ “ ϕ˝ζ . (7.4) j pΣ; V q “ ζ P Γj pΣ; V q : ζ ˝dr r q ˚ V qq ϕ is the subspace of sections that agree Thus, ΓpΣ; V qϕ Ă ΓpΣ; r that are identified into the nodes of Σ, while at the points of Σ r q ˚ V qq˚ ϕ . Γ0,1 pΣ; V qϕ “ Γ0,1 pΣ; ˚

j

j

A real Cauchy–Riemann (or CR-) operator on a real bundle pair pV, ϕq over a symmetric surface pΣ, σq as above is a linear map of the form ϕ ¯ (7.5) D “ B`A : ΓpΣ; V qϕ ÝÑ Γ0,1 j pΣ; V q , where j is a complex structure on pΣ, σq, B¯ is the holomorphic ¯ B-operator for some holomorphic structure in q ˚ V which is reversed ˚ by q ϕ, and ˘ ` r jq0,1 bC q ˚ V q ϕ r HomR pq ˚ V, pT ˚ Σ, A P Γ Σ;

is a zeroth-order deformation term. The completion of a real CRoperator D on pV, ϕq with respect to appropriate norms on its

Main Results and Applications of Part II

159

domain and target (e.g. Sobolev Lp1 - and Lp -norms, respectively, with p ą 2), which we also denote by D, is Fredholm; see the proof of [15, Proposition 3.6]. If Σ is a smooth connected surface of genus g, then the index of D is given by D @ ind D “ p1´gq rk V ` c1 pV q, rΣs . The norms can be chosen so that the evaluation map ΓpΣ; V qϕ ÝÑ Vx ,

ξ ÝÑ ξpxq,

is continuous for each x P Σ. The space of completions of all real CR-operators on pV, ϕq is contractible with respect to the operator norm. This implies that there is a canonical homotopy class of isomorphisms between the determinant lines of any two real CR-operators on a real bundle pair pV, ϕq; we thus denote any such line by λpDpV,ϕq q. If in addition C is a marked symmetric surface as in (7.3), let ˆ k ˙ k à ϕ â R r CpDq “ λR pV, ϕq˚ bλpDq; Vxi “ λpVxϕi q, λ λ CpV, ϕq “ λ C i“1

i“1

the summands and the factors in the definition of λRCpV, ϕq are not ordered. We call pV, ϕq C-balanced if the parity of the number kr of the real marked points xi carried by every topological component Σσr of Σσ is different from xw1 pV ϕ q, rΣσr sZ2 y. We call a symmetric surface pΣ, σq with separating fixed locus and with choices of a half-surface Σb and of an ordering of the topological components of Σσ a decorated symmetric surface. Theorem 7.1. Suppose pΣ, σq is a smooth decorated symmetric surface and pV, ϕq is a real bundle pair over pΣ, σq. (a) A relative OSpin-structure os on the vector bundle V ϕ over Σσ Ă Σ determines an orientation oos pV, ϕq of λpDpV,ϕq q. (b) If C is a marked symmetric surface with the underlying symmetric surface pΣ, σq and pV, ϕq is C-balanced, a relative Pin˘ structure p on V ϕ determines an orientation o C;p pV, ϕq of r CpDpV,ϕq q. λ These orientations satisfy all properties of Sections 7.2 and 7.3. We denote by λC pV q the top exterior power of a complex vector bundle V . A conjugation ϕ on V induces a conjugation λC pϕq

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Spin/Pin-Structures & Real Enumerative Geometry

on λC pV q. For a real bundle pair pV, ϕq, we define ˘ ` λpV, ϕq “ λC pV q, λC pϕq , ˘ ` pV, ϕq˘ ” V˘ , ϕ˘ “ pV, ϕq ‘ p2˘1qλpV, ϕq.

(7.6)

Let C be a smooth decorated marked symmetric surface and pV, ϕq be a C-balanced real bundle pair over C. By Proposition 8.2, the line r CpDpV,ϕq q has a natural orientation o CpV, ϕq if the rank of pV, ϕq λ is 1. If pV, ϕq is of any rank, a relative Pin˘ -structure p on V ϕ is a relative OSpin-structure os˘ ” Co˘ V ppq on V˘ϕ ; see the RelSpinPin 6 property on page 82. We define the orientation o C;p pDpV,ϕq q of Theorem 7.1(b) at the beginning of Section 10.4 via the isomorphism ˘ ` ˘ ` ˘bp2˘1q ` (7.7) λ DpV,ϕq˘ « λ DpV,ϕq bλ DλpV,ϕq as in (7.2) from the orientation oos˘ ppV, ϕq˘ q as in Theorem 7.1(a) and the orientation o CpλpV, ϕqq; in the plus case, the tensor product of the last two factors of λpDλpV,ϕq q above has a canonical orientation. Most properties of o C;p pV, ϕq then follow directly from the corresponding properties of oos˘ ppV, ϕq˘ q and o CpV, ϕq. 7.2

Properties of Orientations: Smooth Surfaces

Let pΣ, σq be a smooth symmetric surface. If the genera of the connected components of Σ are g1 , . . . , gN , we define gpΣq ” 1 `

N ÿ ˘ ` gq ´1 q“1

to be the genus of Σ. If the fixed locus Σσ Ă Σ is separating, then gpΣq ` 1 has the same parity as the number |π0 pΣσ q| of connected components of Σσ . For a rank n real bundle pair pV, ϕq over pΣ, σq, we denote by D @ deg V ” c1 pV q, rΣsZ P Z and ˇ (ˇ W1 pV, ϕq ” ˇ Sr1 P π0 pΣσ q : w1 pV ϕ q|Sr1 ‰ 0 ˇ P Zě0 the degree of V and the number of connected components of Σσ over which V ϕ is not orientable, respectively. By [4, Proposition 4.1], these

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161

two numbers have the same parity. Thus, the elements gpΣ˚ q`|π0 pΣσ˚ q|´1 2 D deg V |Σ˚ @ ` w2 posq, rΣ˚ sZ2 P Z2 and ` 2 gpΣ˚ q`|π0 pΣσ˚ q|´1

p pΣ˚ q ” n 2 D deg V |Σ˚ ˘W1 ppV, ϕq|Σ˚ q @ ` ` w2 ppq, rΣ˚ sZ2 P Z2 2

os pΣ˚ q ” n

are well defined for every relative OSpin-structure os on the real vector bundle V ϕ over Σσ Ă Σ, relative Pin˘ -structure p on V ϕ , and an elemental component Σ˚ of pΣ, σq. A real CR-operator on a complex vector bundle V over a closed oriented (possibly nodal) surface Σ is a linear map of the form ` ˘ ˚ 0,1 ¯ D “ B`A : ΓpΣ; V q ÝÑ Γ0,1 j pΣ; V q ” Γ Σ; pT Σ, jq bC V , ¯ where B¯ is the holomorphic B-operator for some holomorphic structure in V which lifts a complex structure j on pΣ, σq and ˘ ` A P Γ Σ; HomR pV, pT ˚ Σ, jq0,1 bC V q is a zeroth-order deformation term. The completion of a real CRoperator D on V with respect to appropriate norms on its domain and target, which we also denote by D, is Fredholm; see [32, Theorem C.1.10]. The space of completions of all real CR-operators on V is contractible with respect to the operator norm. The real CR-operator B¯ on V in the notation above is C-linear and its kernel and cokernel have canonical orientations; we call the induced orientation of λpDq for any real CR-operator D on V the complex orientation of λpDq and D. If D is a real CR-operator on a real bundle pair pV, ϕq over a decorated symmetric surface pΣ, σq with Σσ “ H, i.e. pΣ, σq is a union of doublets, and Σb is a half-surface of pΣ, σq, then the homomorphisms ˘ ` ΓpΣ; V qϕ ÝÑ Γ Σb ; V |Σb , ˇ ˇ ˘ 0,1 ` b ϕ ˇ (7.8) ξ ÝÑ ξ ˇΣb , Γ0,1 j pΣ; V q ÝÑ Γj| b Σ ; V |Σb , ζ ÝÑ ζ Σb , Σ

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are isomorphisms. They identify D with its “restriction” to a real CR-operator D ` on V |Σb and thus induce an isomorphism ` ˘ λpDq ÝÑ λ D ` (7.9) between the determinants of real CR-operators. CROrient 1os (Dependence on decorated structure: OSpin). Suppose pΣ, σq is a smooth decorated symmetric surface, pV, ϕq is a rank n real bundle pair over pΣ, σq, and os is a relative OSpinstructure on the real vector bundle V ϕ over Σσ Ă Σ. (1) The orientation oos pV, ϕq of DpV,ϕq determined by os does not depend on the choice of half-surface Σb˚ of an elemental component Σ˚ of pΣ, σq if and only if os pΣ˚ q “ 0. (2) The interchange in the ordering of two components of Σσ preserves the orientation oos pV, ϕq if and only if n P 2Z. (3) If Σσ “ H and Σb Ă Σ is the chosen half-surface, then the isomorphism (7.9) is orientation-preserving with respect to the orientation oos pV, ϕq on the left-hand side and the canonical complex orientation on the right-hand side if and only if xw2 posq, rΣb sZ2 y “ 0. Suppose in addition that C is a decorated marked symmetric surface as in (7.3) and Σb Ă Σ is the chosen half-surface. For a component Sr1 of Σσ , let j1r p Cq, . . . , jkrr p Cq P rks with

j1r p Cq ă j2r p Cq, . . . , jkrr p Cq

be such that the real marked points xj1r p Cq , . . . , xjkr

r

p Cq

P Σσr

(7.10)

are ordered by their position on Sr1 with respect to the orientation of Sr1 induced by Σb . If C1 is a marked symmetric surface obtained from C by interchanging some real marked points, the twisted deterr C1 pDq are the same. We set r CpDq and λ minants λ ˆ ˙ ˆ ˙ ´1 ´1 , ” 0. 1 2 CROrient 1p (Dependence on decorated structure: Pin˘ ). Suppose C is a smooth decorated marked symmetric surface, pV, ϕq is a C-balanced rank n real bundle pair over C, and p is a relative Pin˘ -structure on the real vector bundle V ϕ over Σσ Ă Σ.

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163

r CpDpV,ϕq q determined by p does (1) The orientation o C;p pV, ϕq on λ not depend on the choice of half-surface Σb˚ of an elemental component Σ˚ of pΣ, σq if and only if ÿ ˆˆkr ´1˙ ˆkr ´1˙˙ `n ` 2Z “ p pΣ˚ q P Z2 . 1 2 σ 1 Sr Pπ0 pΣ˚ q

(2) The interchange in the ordering of two consecutive compo1 nents Sr1 and Sr`1 of Σσ preserves the orientation o C;p pV, ϕq if and only if pn`1qpkr kr`1 `1q P 2Z. (3) The interchange of two real marked points xjir p Cq and xj r1 p Cq i on the same connected component Sr1 of Σσ with 2 ď i, i1 ď kr preserves o C;p pV, ϕq. The interchange of the real points xj1r p Cq and xjir p Cq with 2 ď i ď kr preserves o C;p pV, ϕq if and only if pn`1qpkr ´1qpi´1q P 2Z. CROrient 2 (Dependence on Spin/Pin-structure). Suppose pΣ, σq is a smooth decorated symmetric surface, pV, ϕq is a real bundle pair over pΣ, σq, and η P H 2 pΣ, Σσ ; Z2 q. (a) Let os P OSpΣ pV ϕ q. The orientations oos pV, ϕq and oη¨os pV, ϕq are the same if and only if xη, rΣb sZ2 y “ 0. The orientations oos pV, ϕq and oos pV, ϕq are the same if and only if |π0 pΣσ q| is even. (b) Let C be a marked symmetric surface with the underlying symmetric surface pΣ, σq so that pV, ϕq is C-balanced. Let p P PΣ˘ pV ϕ q. The orientations o C;p pV, ϕq and o C;η¨p pV, ϕq r CpDpV,ϕq q are the same if and only if xη, rΣb sZ y “ 0. on λ 2 Let pV, ϕq be a real bundle pair over a decorated marked symmetric 1 be the connected components surface C as in (7.3) and S11 , . . . , SN of Σσ in the chosen order. For each component Sr1 of Σσ , define V Cϕ;r “ Vxϕjr p Cq ‘ ¨ ¨ ¨ ‘Vxϕjr 1

kr

p Cq

.

(7.11)

If o is an orientation of V ϕ , we define the orientation λRCpoq on λRCpV, ϕq to be the orientation induced by the restriction of o to each Vxϕi via the identification ` ˘ ` ˘ λRCpV, ϕq “ λ Vxϕ1 b ¨ ¨ ¨ bλ Vxϕ1 j1 p Cq j p Cq k1 ˘ ` ˘ ` ϕ b ¨ ¨ ¨ bλ VxϕN . b ¨ ¨ ¨ bλ Vx N j p Cq 1

j p Cq kN

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In other words, for each component Sr1 of Σσ , we first orient the direct sum (7.11) from the orientation o based on the listed order of the factors, i.e. the order in which the marked points are positioned on Sr1 . We then orient the direct sum V Cϕ ” V Cϕ;1 ‘ ¨ ¨ ¨ ‘V Cϕ;N

(7.12)

over all components of Σσ based on the ordering of the components. CROrient 3 (Reduction). Suppose C is a decorated marked symmetric surface, pV, ϕq is a C-balanced rank n real bundle pair over C, and o is an orientation on V ϕ . For every p P PΣ˘ pV ϕ q, the orientation r CpDpV,ϕq q corresponds to the homotopy class of isoo C;p pV, ϕq on λ morphisms of λpDpV,ϕq q and λRCpV, ϕq determined by the orientations pV, ϕq and λRCpoq if and only if oR˘ o ppq p P PΣ´ pV ϕ q, n

`|π0 pΣσ q|˘

P 2Z or ` σ q|˘ P 2Z. p P PΣ` pV ϕ q, pn`1q |π0 pΣ 2 2

(7.13)

Suppose pΣ1 , σ1 q and pΣ2 , σ2 q are symmetric surfaces, pV1 , ϕ1 q is a real bundle pair on pΣ1 , σ1 q, and pV2 , ϕ2 q is a real bundle pair on pΣ2 , σ2 q of the same rank as pV1 , ϕ1 q. We denote by ` ˘ (7.14) pV1 , ϕ1 q\pV2 , ϕ2 q ” V1 \V2 , ϕ1 \ϕ2 the induced real bundle pair over the symmetric surface ` ˘ pΣ1 , σ1 q\pΣ2 , σ2 q ” Σ1 \Σ2 , σ1 \σ2 .

(7.15)

For relative OSpin-structures (resp. Pin˘ -structures) os1 (resp. p1 ) on V1ϕ1 and os2 (resp. p2 ) on V2ϕ2 , we denote by os1 \os2 (resp. p1 \p2 ) the relative OSpin-structure (resp. Pin˘ -structure) on the real vector bundle pV1 \V2 qϕ1 \ϕ2 “ V1ϕ1 \V2ϕ2 over

`

Σ1 \Σ2

˘σ1 \σ2

“ Σσ1 1 \Σσ2 2 Ă Σ1 \Σ2 .

(7.16)

If pΣ1 , σ1 q and pΣ2 , σ2 q are decorated symmetric surfaces, we take (7.15) to be the decorated symmetric surface obtained by

Main Results and Applications of Part II

defining

`

Σ1 \Σ2

˘b

165

“ Σb1 \Σb2

and ordering the topological components of (7.16) so that the topological components of Σσ1 1 in their order are followed by the topological components of Σσ2 2 in their order. If in addition C1 and C2 are marked symmetric surfaces with the underlying symmetric surfaces pΣ1 , σ1 q and pΣ2 , σ2 q, respectively, we denote by C1\ C2 the marked symmetric surface with the underlying symmetric surface (7.15) and the same marked points as C1 and C2 which are ordered consistently with their orderings in C1 and C2 (but the marked points of C1 do not necessarily precede the marked points of C2 ). Let ˘ ` ΨRC1 , C2 : λRC1 \ C2 pV1 , ϕ1 q \ pV2 , ϕ2 q « λRC1 pV1 , ϕ1 q b λRC2 pV2 , ϕ2 q (7.17) be the isomorphism induced by the identification ϕ1 ϕ2 2 pV1 \V2 qϕC11\ϕ \ C2 “ pV1 q C1 ‘pV2 q C2 .

If pV1 , ϕ1 q is C1 -balanced and pV2 , ϕ2 q is C2 -balanced, then (7.14) is p C1 \ C2 q-balanced. For every real CR-operator D on the real bundle pair (7.14) over (7.15), there is a canonical splitting ˇ ˇ (7.18) D “ D ˇ ‘D ˇ V1

V2

of D into real CR-operators on pV1 , ϕ1 q and pV2 , ϕ2 q. Along with (7.2), this splitting determines a homotopy class of isomorphisms ˘ ` ˘ ` ˘ ` (7.19) λ DpV1 ,ϕ1 q\pV2 ,ϕ2 q « λ DpV1 ,ϕ1 q bλ DpV2 ,ϕ2 q . r C pDpV ,ϕ q q and r r C pDpV ,ϕ q q correspond o2 of λ Orientations r o1 of λ 1 2 1 1 2 2 to homotopy classes of isomorphisms ` ˘ ˘ ` λ DpV1 ,ϕ1 q « λRC1 pV1 , ϕ1 q and λ DpV2 ,ϕ2 q « λRC2 pV2 , ϕ2 q, respectively. Along with (7.19) and (7.17), these homotopy classes of isomorphisms determine a homotopy class of isomorphisms ˘ ˘ ` ` λ DpV1 ,ϕ1 q\pV2 ,ϕ2 q « λRC1 \ C2 pV1 , ϕ1 q\pV2 , ϕ2 q . r C \ C pDpV ,ϕ q\pV ,ϕ q q, The latter corresponds to an orientation of λ 1 2 1 1 2 2 o2 . which we denote by r o1 \r

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CROrient 4 (Disjoint unions). Suppose pΣ1 , σ1 q and pΣ2 , σ2 q are smooth decorated symmetric surfaces of genera g1 and g2 , respectively, pV1 , ϕ1 q is a rank n real bundle pair on pΣ1 , σ1 q, and pV2 , ϕ2 q is a rank n real bundle pair on pΣ2 , σ2 q. (a) Let os1 P OSpΣ1 pV1ϕ1 q and os2 P OSpΣ2 pV2ϕ2 q. The homotopy class of isomorphisms p7.19q induced by the splitting (7.18) respects the orientations ˘ ` oos1 \os2 pV1 , ϕ1 q\pV2 , ϕ2 q , oos1pV1 , ϕ1 q, and oos2pV2 , ϕ2 q on the three factors. (b) Let C1 and C2 be marked symmetric surfaces with the underlying symmetric surfaces pΣ1 , σ1 q and pΣ2 , σ2 q, respectively, so that pV1 , ϕ1 q is C1 -balanced and pV2 , ϕ2 q is C2 -balanced. If p1 P PΣ˘1 pV ϕ1 q and p2 P PΣ˘2 pV ϕ2 q, then ˘ ` o C1 \ C2 ;p1 \p2 pV1 , ϕ1 q\pV2 , ϕ2 q “ o C1 ;p1pV1 , ϕ1 q\o C2 ;p2pV2 , ϕ2 q if and only if ` ˘` ˘ p P PΣ´ pV ϕ q, 1´g1 `deg V1 p1´g2 qn`deg V2 P 2Z ˘` ˘ ` p P PΣ` pV ϕ q, pn`1q 1´g1 `deg V1 1´g2 P 2Z.

or (7.20)

A short exact sequence 0 ÝÑ pV 1 , ϕ1 q ÝÑ pV, ϕq ÝÑ pV 2 , ϕ2 q ÝÑ 0

(7.21)

of real bundle pairs over pX, φq induces a short exact sequence 1

2

0 ÝÑ V 1ϕ ÝÑ V ϕ ÝÑ V 2ϕ ÝÑ 0

(7.22)

of real vector bundles over X φ . If the first sequence is denoted by e, we denote the second one by eR . By the SpinPinRel 7 property 1 on page 82, a relative OSpin-structure os1 P OSpX pV 1ϕ q on the 1 1ϕ φ over X Ă X and a relative OSpin-structure vector bundle V 2 2 ˘ 2 2ϕ q (resp. Pin˘ -structure p2 P PX pV 2ϕ q) determine a os P OSpX pV relative OSpin-structure xxos1 , os2 yy eR P OSpX pV ϕ q ˘ pV ϕ q). (resp. Pin˘ -structure xxos1 , p2 yy eR P PX

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167

Suppose in addition that pX, φq “ pΣ, σq is a smooth symmetric surface and C is a marked symmetric surface with the underlying symmetric surface pΣ, σq as in (7.3). For each i P rks, the short exact sequence (7.22) determines an isomorphism ` ` ` ˘ 1˘ 2˘ . λ Vxϕi « λ Vx1ϕi bλ Vx2ϕ i Putting these isomorphisms together, we obtain an identification ΨRC : λRCpV, ϕq « λRCpV 1 , ϕ1 qbλRCpV 2 , ϕ2 q.

(7.23)

1

If the real vector bundle V 1ϕ is orientable, then the real bundle pair pV, ϕq is C-balanced if and only if the real bundle pair pV 2 , ϕ2 q is 1 1 C-balanced. If o1 is an orientation on V 1ϕ and os1 P SpinΣ pV 1ϕ , o1 q, let λRCpos1 q ” λRCpo1 q be the orientation on λRCpV 1 , ϕ1 q defined as above the CROrient 3 property. Along with (7.2), the short exact sequence e of real bundle pairs over pΣ, σq as in (7.21) determines a homotopy class of isomorphisms ˘ ` ˘ ` ˘ ` (7.24) λ DpV,ϕq « λ DpV 1 ,ϕ1 q bλ DpV 2 ,ϕ2 q . o2 of Orientations o1D of λpDpV 1 ,ϕ1 q q, λRCpos1 q of λRCpV 1 , ϕ1 q, and r r CpDpV 2 ,ϕ2 q q determine homotopy classes of isomorphisms λ ` ˘ ˘ ` λ DpV 1 ,ϕ1 q « λRCpV 1 , ϕ1 q and λ DpV 2 ,ϕ2 q « λRCpV 2 , ϕ2 q, respectively. Along with (7.24) and (7.23), these homotopy classes of isomorphisms determine a homotopy class of isomorphisms ˘ ` λ DpV,ϕq « λRCpV, ϕq . r CpDpV,ϕq q, which we The latter corresponds to an orientation of λ 1 R 1 2 o . denote by poD λ Cpos qq er CROrient 5 (Exact triples). Suppose pΣ, σq is a smooth decorated symmetric surface, e is a short exact sequence of real bundle 1 pairs over pΣ, σq as in (7.21), and os1 P OSpΣ pV 1ϕ q.

Spin/Pin-Structures & Real Enumerative Geometry

168

2

(a) Let os2 P OSpΣ pV 2ϕ q. The homotopy class of isomorphisms p7.24q induced by e respects the orientations oxxos1 ,os2 yyeRpV, ϕq,

oos1 pV 1 , ϕ1 q,

and

oos2 pV 2 , ϕ2 q

if and only if `

˙ ˆ ˘` ˘ |π0 pΣσ q| 2 P 2Z. rk V rk V 2 1

(7.25)

(b) Let C be a marked symmetric surface with the underlying symmetric surface pΣ, σq so that pV 2 , ϕ2 q is C-balanced. If p2 P 2 PΣ˘ pV 2ϕ q, then ` ˘ o C;xxos1 ,p2 yyeRpV, ϕq “ o C;os1 pV 1 , ϕ1 qλRCpos1 q eo C;p2 pV 2 , ϕ2 q if and only if `

˙ ˆ ˙ ˆ ` ˘` ˘` ˘ k ˘ |π0 pΣσ q| 2 1 2 ` rk V rk V P 2Z. (7.26) rk V rk V `1 2 2 1

There are two equivalence classes of orientation-reversing involutions on the Riemann sphere S 2 “ P1 . They are represented by the involutions τ : P1 ÝÑ P1 ,

z ÝÑ 1{¯ z,

z ÝÑ ´1{¯ z. (7.27) The fixed-point locus of η is empty. The fixed-point locus of τ is the unit circle S 1 Ă C; it separates P1 into the unit disk D2` Ă C and its complement D2´ . A choice of half-surface D2` of pS 2 , τ q is equivalent to a choice of one of the two natural embeddings of C as a subspace of S 2 . Let pV, ϕq be a rank 1 degree a real bundle pair over pS 2 , τ q. If a ě ´1, x1 , . . . , xa`1 P S 1 are distinct points, and DpV,ϕq is a real CR-operator on pV, ϕq, then DpV,ϕq is surjective and the evaluation homomorphism ` ˘ ev : ker DpV,ϕq ÝÑ Vxϕ1 ‘ ¨ ¨ ¨ ‘Vxϕa`1 , ξ ÝÑ ξpx1 q, . . . , ξpxa`1 q , (7.28) and

η : P1 ÝÑ P1 ,

Main Results and Applications of Part II

169

is an isomorphism; see Section 7.4. If a P 2Z, the real line bundle V ϕ over S 1 is orientable. For an orientation o on V ϕ , we then let o0 pV, ϕ; oq ” oιS 2 pos0 pV ϕ ,oqq pV, ϕq

(7.29)

denote the orientation of DpV,ϕq determined by the image ˘ ` ` ˘ ιS 2 os0 pV ϕ , oq P OSpS 2 V ϕ of the canonical OSpin-structure os0 pV ϕ , oq on pV ϕ , oq under the first map in (6.3) with X “ S 2 . If a R 2Z, the real line bundle V ϕ over S 1 is not orientable. ˘ ϕ ϕ The two Pin˘ -structures on V ϕ , p˘ 0 pV q and p1 pV q, are then distinguished by Examples 1.23 and 1.24 from the classical perspective and by Example 5.1 from the trivializations perspectives. The ´ ϕ ϕ distinction between p´ 0 pV q and p1 pV q depends on the choice of 2 2 ϕ half-surface D` of pS , τ q, while the distinction between p` 0 pV q and ` ϕ p1 pV q is independent of such a choice. If C is a marked symmetric surface as in (7.3) so that pΣ, σq is S 2 with the involution τ , k P 2Z, and r “ 0, 1, let o˘ C;r pV, ϕq ” o C;ι

˘ ϕ S 2 ppr pV qq

pV, ϕq

r CpDpV,ϕq q determined by the image denote the orientation on λ ˘ ` ` ϕ˘ ϕ ˘ ιS 2 p˘ r pV q P PS 2 V

(7.30)

ϕ 2 of p˘ r pV q under the second map in (6.3) with X “ S .

CROrient 6 (Normalizations). Let pV, ϕq be a rank 1 real bundle pair over pS 2 , τ q. (a) If deg V “ 0 and o P OpV ϕ q, the isomorphism (7.28) with a “ 0 respects the orientations o0 pV, ϕ; oq and o for every x1 P S 1 . (b) If deg V “ 1, C is a marked symmetric surface with the underlying symmetric surface pS 2 , τ q and k “ 2 real marked points, and D2` is the distinguished half-surface of pS 2 , τ q, then o˘ C;0 pV, ϕq r CpDpV,ϕq q corresponding to the homotopy is the orientation of λ class of the isomorphism (7.28) with a “ 1.

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Spin/Pin-Structures & Real Enumerative Geometry

In the setting of the CROrient 6(a) property, the OSpin-structure os0 pV ϕ , oq on the oriented real line bundle pV ϕ , oq over S 1 is the homotopy class of trivializations of 2τS 1 ‘ V ϕ induced by the unique homotopy classes of trivializations of τS 1 and V ϕ as oriented line bundles. It is thus not surprising that a systematic orienting procedure for CR-operators on even-degree real bundle pairs over pS 2 , τ q with relative OSpin-structures, such as the one described at the beginning of Section 8.4, satisfies the CROrient 6(a) property. This is immediate from the first case of (8.33) and is noted in Proposition 8.9((1)). In the setting of the CROrient 6(b) property, the Pin´ - and Pin` ` ϕ ϕ structures p´ 0 pV q and p0 pV q on the unorientable real line bunϕ 1 dle V over S correspond to the homotopy classes os0 p2γR;1 , o´ γR;1 q ` and os0 p4γR;1 , oγR;1 q of trivializations of the canonically oriented vector bundles τRP1 ‘2γR;1 ÝÑ RP1

and

4γR;1 ÝÑ RP1 ,

respectively. The latter is the canonical homotopy class of trivializations provided by Example 1.13(b). The former is the homotopy class of trivializations determined by the unique homotopy class of trivializations of τRP1 as an oriented line bundle and the homotopy class of trivializations of 2γR;1 represented by the trivialization Φ0 in (1.37). According to [20, Proposition 3.5], the orientation on the line λpD2pV,ϕq q “ λp2DpV,ϕq q “ λpDpV,ϕq qb2 induced by the OSpin-structure os0 p2γR;1 , o´ γR;1 q is the canonical orientation of this line as a square of another line; see also Proposition 8.9((2)). Along with the construction of twisted orientations of Theorem 7.1(b) at the beginning of Section 10.4, this immediately implies both cases of the CROrient 6(b) property. By the CROrient 6 property, the orientations o0 pV, ϕ; oq and o˘ C;0 pV, ϕq match intrinsic evaluation orientations for rank 1 real bundle pairs pV, ϕq over pS 1 , τ q of degrees 0 and 1. Corollary 7.7 extends this property to rank 1 real bundle pairs over pS 1 , τ q of arbitrary degrees.

Main Results and Applications of Part II

7.3

171

Properties of Orientations: Degenerations

The compatibility of the orientations on the determinants of real CRoperators induced by the relative Spin- and Pin˘ -structures with degenerations of smooth domains is essential for any study of the structures in open and real Gromov–Witten theories that depend on node-splitting properties in the spirit of [27, 2.2.6]. We next describe the relevant degeneration settings and characterize the behavior of the induced orientations in these settings by the CROrient 7C and 7H3 properties. Let C” pΣ, pzi qiPrls q be a nodal marked surface, i.e. Σ is a closed oriented (possibly nodal) surface and z1 , . . . , zl are distinct smooth points of Σ. A flat family of deformations of C is a tuple `

˘ π : U ÝÑ Δ, psi : Δ ÝÑ UqiPrls ,

(7.31)

where U is a smooth manifold, Δ Ă CN is a ball around 0, and π, s1 , . . . , sl are smooth maps such that ‚ Σt ” π ´1 ptq is a closed oriented (possibly nodal) surface for each t P Δ and π is a submersion outside the nodes of the fibers of π, ‚ for every t˚ ” pt˚j qjPrN s P Δ and every node z ˚ P Σt˚ , there exist i P t1, . . . , N u with t˚i “ 0, neighborhoods Δt˚ of t˚ in Δ and Uz ˚ of z ˚ in U, and a diffeomorphism Ψ : Uz ˚ ÝÑ

` ˘ ( ptj qjPrN s , x, y P Δt˚ ˆC2 : xy “ ti

such that the composition of Ψ with the projection to Δt˚ equals π| Uz˚, ‚ π ˝si “ idΔ and s1 ptq, . . . , sl ptq P Σt are distinct smooth points for every t P Δ, ‚ pΣ0 , psi p0qqiPrls q “ C. We call an almost complex structure j U on U admissible if the projection π is pjCN , j Uq-holomorphic and for every node z ˚ P Σt˚ the diffeomorphism Ψ above can be chosen to be holomorphic.

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Spin/Pin-Structures & Real Enumerative Geometry

Let C be a nodal marked symmetric surface as in (7.3). A flat family of deformations of C is a tuple ` F ” π : U ÝÑ Δ,rc : U ÝÑ U, ˘ ` ´ (7.32) psR i : Δ ÝÑ UqiPrks , psi : Δ ÝÑ U, si : Δ ÝÑ UqiPrls ` ´ such that pπ, psR i qiPrks , psi , si qiPrls q is a flat family of deformations of C, rc is an involution on U lifting the standard involution c on Δ and restricting to σ over Σ ” Σ0 , and

c ˝sR sR i ˝c “ r i @ i P rks,

s` c ˝s´ i ˝c “ r i @ i P rls.

We call an almost complex structure j U on p U,rcq admissible if j U is admissible for U and rc˚ j U “ ´j U. If F is as in (7.32), let σt “ rc|Σt for each parameter t in ΔR ” ΔXRN . The tuple ` ˘ ` ´ Ct ” Σt , σt , psR i ptqqiPrks , psi ptq, si ptqqiPrls is then a nodal marked symmetric surface. If j U is an admissible almost complex structure on p U,rcq, then jt ” j U|Σt is a complex structure on Σt . We denote by Δ˚R Ă ΔR the subspace of elements t P ΔR so that the surface Σt is smooth. If pΣ0 , σ0 q has only conjugate pairs of nodes, ΔR ´ Δ˚R is a subspace of ΔR of codimension 2; if pΣ0 , σ0 q has real nodes, ΔR ´Δ˚R is a subspace of codimension 1. Let pπ : U ÝÑ Δ,rc : U ÝÑ Uq be a flat family of deformations of a (possibly) nodal symmetric surface and pV, ϕq be a real bundle pair over p U,rcq. For each t P ΔR , ˘ ` pVt , ϕt ” V, ϕq|Σt is then a real bundle pair over pΣt , σt q. Suppose in addition that j U is an admissible almost complex structure on p U,rcq, ∇ is a ϕ-compatible (complex-linear) connection in V , and ˘ϕ ` (7.33) A P Γ U; HomR pV, pT ˚ U, j Uq0,1 bC V q . The restrictions of ∇ and A to each fiber pΣt , σt q of π with t P ΔR then determine a real CR-operator ` ˘ϕ ` ˘ ϕt (7.34) DpV,ϕq;t : Γ Σt ; Vt t ÝÑ Γ0,1 jt Σt ; Vt

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173

on pVt , ϕt q. Let

˘ ğ ˘ ` ` ttuˆλ DpV,ϕq;t ÝÑ ΔR . πpV,ϕq : λ DpV,ϕq ”

(7.35)

tPΔR

The set λp DpV,ϕq q carries natural topologies so that the projection πpV,ϕq is a real line bundle; see [19, Appendix A]. These topologies in particular satisfy the following properties: (D1) A homotopy class of continuous isomorphisms Ψ : pV1 , ϕ1 q ÝÑ pV2 , ϕ2 q of real bundle pairs over p U,rcq|ΔR determines a homotopy class of isomorphisms ` ˘ ` ˘ λp DΨ q : λ DpV1 ,ϕ1 q ÝÑ λ DpV2 ,ϕ2 q of line bundles over ΔR . (D2) The isomorphisms (7.2) determine a homotopy class of isomorphisms ˘ ` ˘ ` ˘ ` λ DpV1 ,ϕ1 q‘pV2 ,ϕ2 q « λ DpV1 ,ϕ1 q b λ DpV2 ,ϕ2 q of line bundles over ΔR for all real bundle pairs pV1 , ϕ1 q and pV2 , ϕ2 q over p U,rcq. For a flat family of deformations of a (possibly) nodal marked symmetric surface as in (7.32) and ∇ and A as above, we similarly define ˙˚ ˆ k ğ â R˚ ` ` ˘ ˘ ` ˘ ϕ r C DpV,ϕq . r si λpV q bR λ DpV,ϕq ” ttuˆ λ λF DpV,ϕq “ t i“1

tPΔR

˘ ` This set inherits a topology from λ DpV,ϕq , which makes it into a real line bundle over ΔR . We describe the behavior of the orientations induced by relative Spin- and Pin-structures under degenerations to two types of nodal surfaces. First, let ˘ ` (7.36) C0 ” Σ0 , σ0 , pxi qiPrks , pzi` , zi´ qiPrls be a marked symmetric surface which contains precisely one conjugate pair pnd` , nd´ q of nodes and no other nodes. Let ˘ ` r 0, σ r0 ” Σ C r0 , pxi qiPrks , pzi` , zi´ qiPrl`2s

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Spin/Pin-Structures & Real Enumerative Geometry

Figure 7.1. A marked symmetric surface C0 with a conjugate pair of nodes nd˘ r0. and its normalization C

r 0 by identibe the normalization of C0 so that C0 is obtained from C ˘ ˘ with zl`2 into the node nd˘ ; see Figure 7.1. If pΣ0 , σ0 q is a fying zl`1 decorated symmetric surface, we take nd` to be the node in the chosen half-surface Σb . In this case, the decorated structure on pΣ0 , σ0 q ` ` r 0, σ induces a decorated structure on pΣ r0 q so that zl`1 and zl`2 lie on b r on the chosen half-surface Σ0 . A real bundle pair pV0 , ϕ0 q over pΣ0 , σ0 q lifts to a real bundle pair r 0, σ r r0 q over pΣ r0 q. A real CR-operator D0 on pV0 , ϕ0 q lifts to a pV0 , ϕ r 0 on pVr0 , ϕ r0 q so that there is a natural exact triple real CR-operator D ˇ ` ˘ ` ˘ r 0 ÝÑ V0 ˇ ` ÝÑ 0, ξr ÝÑ ξr z ` ´ ξr z ` , (7.37) 0 ÝÑ D0 ÝÑ D l`2 l`1 nd of Fredholm operators. It induces an isomorphism ` ˘ r 0 q. λpD0 qbλ V0 |nd` « λpD

(7.38)

The real line λpV0 |nd` q is oriented by the complex orientation r 0 thus induces an orientation of D0 . of V0 |nd`. An orientation of D ` ` and zl`2 are interThis orientation does not change if the points zl`1 changed because the real dimension of V0 |nd` is even. If os0 is a relative OSpin-structure on the real vector bundle V0ϕ0 over Σσ0 0 Ă Σ0 r 0 is its lift to a relative OSpin-structure on the real vector and os r σr0 Ă Σ r 0 , we denote by oos0 pV0 , ϕ0 q the orientabundle Vr0ϕr0 over Σ 0 r r0 q of D r 0 via (7.38). tion of D0 induced by the orientation oos r 0 pV0 , ϕ We call oos0 pV0 , ϕ0 q the intrinsic orientation induced by os0 . If p0

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is a relative Pin˘ -structure on V0ϕ0 and pV0 , ϕ0 q is C0 -balanced, we r C pD0 q similarly define the intrinsic orientation o C0 ;p0 pV0 , ϕ0 q on λ 0 induced by p0 . Suppose in addition (7.32) is a flat family of deformations of C0 and pV, ϕq is a real bundle pair over p U,rcq extending pV0 , ϕ0 q. The decorated structure on pΣ0 , σ0 q determines a subspace Ub Ă U so that Ub is a manifold with boundary, Σbt ” Ub XΣt is a bordered, possibly nodal, surface for every t P Δ,

U “ Ub Yrcp Ubq,

and

Ub Xrcp Ubq “ B Ub .

The ordering of the components of Σσ0 0 likewise induces an ordering of the components of Σσt t for every t P ΔR . Thus, a decorated structure on pΣ0 , σ0 q determines a decorated structure on pΣt , σt q for every t P ΔR in a continuous manner. A relative OSpin-structure os0 on the real vector bundle V0ϕ0 over Σσ0 0 Ă Σ0 extends to a relative OSpin-structure os on the real vector bundle V ϕ over Urc Ă U. The latter in turn restricts to a relative OSpin-structure ost on the real vector bundle Vtϕt over Σσt t Ă Σt and thus determines an orientation oost pVt , ϕt q of DpV,ϕq;t for every t P Δ˚R which varies continuously with t, i.e. an orientation of the restriction of the real line bundle (7.35) to Δ˚R . Since the codimension of ΔR ´Δ˚R in ΔR is 2 in this case, the last orientation extends continuously over the entire base ΔR . We denote the restriction of this orientation to DpV0 ,ϕ0 q by o1os0 pV0 , ϕ0 q and call it the limiting orientation induced by os. If p0 is a relative Pin˘ -structure on V0ϕ0 and pV0 , ϕ0 q is C0 -balanced, we similarly define the limiting orientation r C pDpV ,ϕ q q induced by p0 . These limiting orientao1C0 ;p0 pV0 , ϕ0 q on λ 0 0 0 tions depend only on pV0 , ϕ0 q, os0 , and p0 and not on p U,rcq or pV, ϕq. CROrient 7C (Degenerations: C nodes). Suppose C0 is a decorated marked symmetric surface which contains precisely one conjugate pair pnd` , nd´ q of nodes and no other nodes and pV0 , ϕ0 q is a real bundle pair over C0 . (a) The intrinsic and limiting orientations of DpV0 ,ϕ0 q induced by a relative OSpin-structure on the vector bundle V0ϕ0 over Σσ0 0 Ă Σ0 are the same.

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(b) If pV0 , ϕ0 q is C0 -balanced, the intrinsic and limiting orientations r C pDpV ,ϕ q q induced by a relative Pin˘ -structure on V ϕ0 are on λ 0 0 0 0 the same. Suppose (7.36) is a marked symmetric surface which contains precisely one H node nd (i.e. a non-isolated point of Σσ ) and no other nodes so that nd splits a topological component Σ‚ of Σ into two irreducible surfaces, pΣ‚1 , σ‚1 q and pΣ‚2 , σ‚2 q. Such a node, which we call an H3 node, is one of the three possible types of H nodes a symmetric surface pΣ, σq may have; see [19, Section 3.2]. Let ` ˘ r r 0, σ C0 ” Σ r0 , pnd1 , nd2 , pxi qiPrks q, pzi` , zi´ qiPrls (7.39) r 0 and C0 is obtained be the normalization of C0 so that Σ‚1 , Σ‚2 Ă Σ r from C0 by identifying the real marked points nd1 P Σ‚1 and nd2 P Σ‚2 into the node nd; see Figure 7.2. We view these two points as the r 0 when specifying orientations in the first two real marked points of C ˘ 1 1 ĂΣ r σr0 Pin -case. Let S‚1 Ă Σ‚1 and S‚2 ‚2 be the components of Σ0 containing nd1 and nd2 , respectively. Let k‚1 and k‚2 be the numbers of real marked points (not including nd1 and nd2 ) carried by these r 1 the marked curves obtained components. We denote by r C11 and C 2 C0 by dropping nd1 and nd2 , respectively. If pΣ0 , σ0 q is a decfrom r orated surface. we define a decorated structure on the normalization r 0, σ r0 q by pΣ r 0 to be the preimage rb Ă Σ ‚ taking the distinguished half-surface Σ 0 b of the distinguished half-surface Σ0 Ă Σ0 and ‚ replacing the position r‚ of the topological component of Σσ0 0 con1 Y S 1 ) in the ordering taining the node nd (i.e. the wedge S‚1 nd ‚2 σ0 1 followed by S 1 . of π0 pΣ0 q by S‚1 ‚2

Figure 7.2. A marked symmetric surface C0 with a real H3 node nd and its r0. normalization C

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A real bundle pair pV0 , ϕ0 q over pΣ0 , σ0 q lifts to a real bundle pair r 0, σ r r0 q over pΣ r0 q. A real CR-operator D0 on pV0 , ϕ0 q lifts to a pV0 , ϕ r 0 on pVr0 , ϕ real CR-operator D r0 q so that there is a natural exact triple ˇ r 2 q´ ξpnd r 1 q, (7.40) r 0 ÝÑ V ϕ0 ˇ ÝÑ 0, ξr ÝÑ ξpnd 0 ÝÑ D0 ÝÑ D 0

nd

of Fredholm operators. It induces an isomorphism ` ˘ r 0 q. λpD0 qbλ V0ϕ0 |nd « λpD

(7.41)

r 0 thus induce an orientation on D0 . If Orientations on V0ϕ0 and D ϕ os0 P SpinΣ pV , oq is a relative OSpin-structure on the real vector r 0 is its lift to a relative OSpinbundle V0ϕ0 over Σσ0 0 Ă Σ0 and os r σr0 Ă Σ r 0 , we denote structure on the real vector bundle Vr0ϕr0 over Σ 0 by oos0 pV0 , ϕ0 q the orientation of D0 induced by the orientations ond r r0 q of D r 0 via (7.41). We call oos pV0 , ϕ0 q the of V0ϕ0 |nd and oos r 0 pV0 , ϕ 0 intrinsic orientation induced by os0 . If pV0 , ϕ0 q is C0 -balanced, then r 1 -balanced or C r 1 -balanced (but not both). If the r0 q is either C pVr0 , ϕ 1 2 pVr0 , ϕ r0 q induced by the lift r p0 i-th case holds, the orientation o C r 1 ;r i p0 of a relative Pin˘ -structure p on V0ϕ0 to Vr0ϕr0 determines a homotopy class of isomorphisms ` ˘ r 0 q « λR 1 pVr0 , ϕ r0 q « λR pV0 , ϕ0 qbλ V ϕ0 |nd . (7.42) λpD r C i

C0

0

Combining (7.41) and (7.42), we obtain an orientation o C0 ;p0 pV0 , ϕ0 q r C pD0 q. We call o C ;p pV0 , ϕ0 q the intrinsic orientation induced on λ 0 0 0 by p0 . If (7.32) is a flat family of deformations of C0 , the topological components of the fixed locus of pΣt , σt q for each t P ΔR correspond to the components of Σσ0 0 and thus inherit an ordering from the chosen ordering of π0 pΣσ0 0 q. Let U‚ Ă U be the topological component containing Σ‚ . The restriction of π to p U´ U‚q|ΔR is a topologically trivial fiber bundle of symmetric surfaces and a half-surface of Σ0´Σ‚ determines a half-surface of every fiber of this restriction. For each fiber Σt;‚ of the restriction of π to U‚ |ΔR , we take Σbt;‚ Ă Σt;‚ to be the half-surface so that for every point z in Σbt;‚ ´Σσt t there exists a path α : r0, 1s ÝÑ U|ΔR ´ Urc s.t.

αp0q P Σb‚ ´Σ‚2 ,

αp1q “ z.

Thus, a decorated structure on pΣ0 , σ0 q determines a decorated structure on pΣt , σt q for every t P Δ˚R in a continuous manner. The space

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178

˚ Δ˚R has two topological components. We denote by Δ` R Ă ΔR the σt b topological component so that for every t P Δ` R and z P Σt;‚ ´ Σt there exists a path

α : r0, 1s ÝÑ U|ΔR ´ Urc s.t.

αp0q P Σb‚ ´Σ‚1 ,

αp1q “ z.

Δ´ R.

We denote the other component of Δ˚R by Suppose in addition that pV, ϕq is a real bundle pair over p U,rcq. As above the CROrient 7C property, a decorated structure on pΣ0 , σ0 q and a relative OSpin-structure os0 on the real vector bundle V0ϕ0 over Σσ0 0 Ă Σ0 determine an orientation of the restriction of the real line bundle (7.35) to Δ˚R . The restrictions of this orientation to λp DpV,ϕq q|Δ` and λp DpV,ϕq q|Δ´ extend continuously to orientaR R ´ tions o` os0 pV0 , ϕ0 q and oos0 pV0 , ϕ0 q, respectively, of λpDpV0 ,ϕ0 q q; we call them the limiting orientations induced by os0 . If p0 is a relative Pin˘ structure on V0ϕ0 and pV0 , ϕ0 q is C0 -balanced, we similarly define the ´ r limiting orientations o` C0 ;p0 pV0 , ϕ0 q and o C0 ;p0 pV0 , ϕ0 q of λ C0 pDpV0 ,ϕ0 q q induced by p0 . All four limiting orientations depend only on pV0 , ϕ0 q, os0 , and p0 and not on p U,rcq or pV, ϕq. With the notation as above, let rp C0 q “ 1 if either k‚2 “ 0 or  (  ( 1 1 inf i P rks : xi P S‚1 ă min i P rks : xi P S‚2 ; (7.43) 1 ĂΣ otherwise, we take rp C0 q “ 2. For r “ 1, 2, S‚r ‚r is oriented as b 1 ě0 the boundary of Σ‚r . Let jr p C0 q P Z be the number of real marked 1 between the nodal point nd points that lie on the oriented arc of S‚r r 1 ; and the real marked point xi with the minimal index i P rks on S‚r if k‚r “ 0, we take jr1 p C0 q “ 0. Define ` ˘ 1 δR p C0 q “ jrp C0 q p C0 q`1` rp C0 q´1 k‚1 .

If in addition pV0 , ϕ0 q is a C0 -balanced rank n real bundle pair, let rpV0 , ϕ0 q “

# 1,

1 1 if k‚1 `2Z ‰ xw1 pV0ϕ0 q, rS‚1 sZ2 y, k‚2 `2Z “ xw1 pV0ϕ0 q, rS‚2 sZ2 y;

1 1 sZ2 y, k‚2 `2Z ‰ xw1 pV0ϕ0 q, rS‚2 sZ2 y; if k‚1 `2Z “ xw1 pV0ϕ0 q, rS‚1 ` ˘` 1 ˘ δR pV0 , ϕ0 q “ k‚rpV0 ,ϕ0 q ´1 jrpV0 ,ϕ0 q p C0 q`rpV0 , ϕ0 q .

2,

CROrient 7H3 (Degenerations: H3 nodes). Suppose C0 is a decorated marked symmetric surface which contains precisely one H3 node and no other nodes and pV0 , ϕ0 q is a rank n real bundle pair over C0 .

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(a) The limiting orientation o` os0 pV0 , ϕ0 q of DpV0 ,ϕ0 q induced by a relative OSpin-structure os0 on the real vector bundle V0ϕ0 over Σσ0 0 Ă Σ0 is the same as the intrinsic orientation oos0 pV0 , ϕ0 q if and only if np|π0 pΣσ0 0 q|´r‚ q is even. (b) If pV0 , ϕ0 q is C0 -balanced, the limiting orientation o` C0 ;p0 pV0 , ϕ0 q ˘ r on λ C0 pDpV0 ,ϕ0 q q induced by a relative Pin -structure p0 on V0ϕ0 is the same as the intrinsic orientation o C0 ;p0 pV0 , ϕ0 q if and only if ¨

˛

˚` ˘ ˚ pn`1q ˚ kr‚ ´1 δR p C0 q`δR pV0 , ϕ0 q ` ˝ ˘ ` “ n rpV0 , ϕ0 q´1 `2Z `

ÿ

@ σ

Sr1 Pπ0 pΣ0 0 q

D‹ ‹ w1 pV0ϕ0 q, rSr1 sZ2 ‹ ‚

rąr‚

# nk,

if p0 P PΣ´0 pV0ϕ0 q;

pn`1qk,

if p0 P PΣ`0 pV0ϕ0 q.

Remark 7.2. By CROrient 1os(1) and 7H3(a), the limiting orientation o´ os0 pV0 , ϕ0 q is the same as the intrinsic orientation oos0 pV0 , ϕ0 q if and only if ˘ ` n |π0 pΣσ0 0 q|´r‚ ` 2Z “ os0 pΣ‚2 q P Z2 . By CROrient 1p(1) and 7H3(b), the limiting orientation o´ C0 ;p0 pV0 , ϕ0 q is the same as the intrinsic orientation o C0 ;p0 pV0 , ϕ0 q if and only if ˛ ¨ ˚` ˘ ˚ pn`1q ˚ kr‚ ´1 δR p C0 q`δR pV0 , ϕ0 q ` ˝ ˆˆ

ÿ

`

σ

Sr1 Pπ0 pΣ‚2‚2 q

ÿ

D‹ ‹ w1 pV0ϕ0 q, rSr1 sZ2 ‹ ‚

@ σ

Sr1 Pπ0 pΣ0 0 q rąr‚

˙ ˆ ˙ ˙ ˆ ˆ ˙˙ kr ´1 k‚2 kr ´1 `n ` rpV0 , ϕ0 q k‚2 `n 2 1 2

1 Sr1 ‰S‚2

ˆ ˆ ˙ ˆ ˙ ˙ ˘ k‚2 ´1 k‚2 ´1 ` ` rpV0 , ϕ0 q´1 n 1 2 ˘ ` ` rp C0 q´1 pn`1qpk‚1 ´1qpk‚2 ´1q # ` ˘ nk, “ p0 pΣ‚2 q ` n rpV0 , ϕ0 q´1 ` pn`1qk, `

if p0 P PΣ´0 pV0ϕ0 q;

if p0 P PΣ`0 pV0ϕ0 q.

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The last term on the third line above accounts for the effect of the 1 change in Σb‚2 on jrp C0 q p C0 q if rp C0 q “ 2. The preceding term includes 1 p C0 q if rpV0 , ϕ0 q “ 2. the effect of this change on jrpV 0 ,ϕ0 q 7.4

Some Implications

With the setup as in the CROrient 7C property, suppose pΣ01 , σ01 q and pΣ02 , σ02 q are symmetric surfaces so that Σσ0202 “ H,

`

˘ ` ˘ ` ˘ Σ0 , σ0 “ Σ01 , σ01 \ Σ02 , σ02 ;

(7.44)

see Figure 7.3. We denote by pV1 , ϕ1 q and pV2 , ϕ2 q the restrictions of the real bundle pair pV0 , ϕ0 q to pΣ01 , σ01 q and pΣ02 , σ02 q, respectively, and by os1 (resp p1 ) the restriction of the relative OSpin-structure os0 (resp. Pin˘ -structure p0 ) to relative OSpin-structure (resp. Pin˘ structure) on the real vector bundle V0ϕ0 over Σσ0 0 Ă Σ0 to the real r 0 be as in (7.37) vector bundle V1ϕ1 over Σσ0101 Ă Σ01 . Let D0 and D b r r r r and D01 , D02 , and D02 be the restrictions of D0 to real CR-operators on pV1 , ϕ1 q, pV2 , ϕ2 q, and V2 , respectively. r 01 and the complex orientation of The orientation oos1pV1 , ϕ1 q of D r b determine an orientation oC pVr0 , ϕ r 0 via the isomorphisms D r0 q of D os0 02 ` ˘ ` ˘ ` ˘ ` b ˘ ` ˘ r 01 bλ D r 02 « λ D r 01 bλ D r 02 r0 « λ D λ D

(7.45)

Figure 7.3. A marked symmetric surface C0 with a conjugate pair of nodes nd˘ r0. as in Corollary 7.3 and its normalization C

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as in (7.19) and (7.9) and thus an orientation oC os0 pV0 , ϕ0 q of D0 via C the isomorphism (7.38). We call oos0 pV0 , ϕ0 q the C-split orientation induced by os0 . Suppose in addition that pV0 , ϕ0 q is C0 -balanced and r 01 \ C r 02 r C0 ” C

(7.46)

is the decomposition induced by the decomposition in (7.44). The r r pD r 01 q and the complex orientaorientation o C r 01 ;p1pV1 , ϕ1 q of λ C 01 r r pD r b then determine an orientation oC pVr0 , ϕ r0q tion of D r0 q on λ r 0 ;p0 C

02

C0 C o C0 ;p0 pV0 , ϕ0 q

via the isomorphisms (7.45) and thus an orientation r C pD0 q via the isomorphism (7.38). We call oC pV0 , ϕ0 q the on λ 0 C0 ;p0 C-split orientation induced by p0 . The CROrient 4, 3, and 1os(3) properties compare the intrinsic orientation oos0 pV0 , ϕ0 q (resp. o C0 ;p0 pV0 , ϕ0 q) with the C-split orienC tation oC os0 pV0 , ϕ0 q (resp. o C0 ;p0 pV0 , ϕ0 q). Combining these properties with the CROrient 7C property, we obtain the following conclusion. Corollary 7.3. Suppose C0 and pV0 , ϕ0 q are as in the CROrient 7C property and (7.44) holds. (a) The limiting orientation o1os0 pV0 , ϕ0 q of DpV0 ,ϕ0 q induced by a relative OSpin-structure os0 on the real vector bundle V0ϕ0 over Σσ0 0 Ă Σ0 is the same as the C-split orientation oC os0 pV0 , ϕ0 q if b and only if xw2 pos0 q, rΣ02 sZ2 y “ 0. (b) If pV0 , ϕ0 q is C0 -balanced and (7.46) holds, then the limiting r C pDpV ,ϕ q q induced by a relative orientation o1C0 ;p0 pV0 , ϕ0 q on λ 0 0 0 ϕ0 ˘ Pin -structure p0 on V0 is the same as the C-split orientation oCC0 ;p0 pV0 , ϕ0 q if and only if xw2 pp0 q, rΣb02 sZ2 y “ 0. With the setup as in the CROrient 7H3 property, let pΣ01 , σ01 q and pΣ02 , σ02 q be symmetric surfaces so that Σ‚1 Ă Σ01 ,

Σ‚2 Ă Σ02 ,

` ˘ ` ˘ ` ˘ Σ0 , σ0 “ Σ01 , σ01 \ Σ02 , σ02 (7.47)

and pV1 , ϕ1 q and pV2 , ϕ2 q be as below (7.44). We denote by os1 and os2 (resp. p1 and p2 ) the restrictions of the relative OSpinstructure os0 (resp. Pin˘ -structure p0 ) on the real vector bundle V0ϕ0 over Σσ0 0 Ă Σ0 to relative OSpin-structures (resp. Pin˘ -structures) on

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the real vector bundles V1ϕ1 and V2ϕ2 over Σσ0101 Ă Σ01 and Σσ0202 Ă Σ02 , r 0 be as in (7.40) and D r 01 and D r 02 be respectively. Let D0 and D r 0 to real CR-operators on pV1 , ϕ1 q and pV2 , ϕ2 q, the restrictions of D respectively. The orientations oos1 pV1 , ϕ1 q and oos2 pV2 , ϕ2 q determine an orienr r0 q of D r 0 via the isomorphism tation osp os pV0 , ϕ ` ˘ ` ˘ ` ˘ r0 « λ D r 01 bλ D r 02 λ D

(7.48)

as in (7.19) and thus an orientation osp os0 pV0 , ϕ0 q of D0 via the isosp morphism (7.41). We call oos0 pV0 , ϕ0 q the split orientation induced by os0 . Suppose in addition that pV0 , ϕ0 q is C0 -balanced, i “ rpV, ϕq, and 1 r1 \ r r Ci1 ” C i1 Ci2

is the decomposition induced by the decomposition in (7.47). The orir r 1 pD r r 1 pD r 01 q and o r 1 pV2 , ϕ2 q of λ r 02 q entations o C r 1 ;p1 pV1 , ϕ1 q of λ C Ci2 ;p2 Ci2 i1 i1 r r 1 pD r 0 q via the isomorr0 q on λ determine an orientation osp pVr0 , ϕ r 1 ;p0 C i

Ci

phisms (7.48) and (7.17) and thus an orientation osp C0 ;p0 pV0 , ϕ0 q r C pD0 q via the isomorphisms (7.41) and (7.42). We call on λ 0 sp o C0 ;p0 pV0 , ϕ0 q the split orientation induced by p0 . The CROrient 4 property compares the intrinsic orientation oos0 pV0 , ϕ0 q (resp. o C0 ;p0 pV0 , ϕ0 q) with the split orientation sp osp os0 pV0 , ϕ0 q (resp. o C0 ;p0 pV0 , ϕ0 q) if the decomposition in (7.47) respects the decorated structures, i.e. the elements of π0 pΣσ0 0 q are ordered so that all elements of π0 pΣσ0101 q precede all elements of π0 pΣσ0202 q. Combining the CROrient 4 and 7H3 properties, we obtain the following conclusion. Corollary 7.4. Suppose C0 and pV0 , ϕ0 q are as in the CROrient 7H3 property and the decomposition in (7.47) respects the decorated structures. (a) The limiting orientation o` os0 pV0 , ϕ0 q of DpV0 ,ϕ0 q induced by a relative OSpin-structure os0 on the real vector bundle V0ϕ0 over Σσ0 0 Ă Σ0 is the same as the split orientation osp os0 pV0 , ϕ0 q if and only if np|π0 pΣσ0 0 q|´r‚ q is even.

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(b) If pV0 , ϕ0 q is C0 -balanced, the limiting orientation o` C0 ;p0 pV0 , ϕ0 q ˘ r C pDpV ,ϕ q q induced by a relative Pin -structure p0 on V ϕ0 on λ 0 0 0 0 is the same as the split orientation osp pV , ϕ q if and only if 0 0 C0 ;p0 ¨

˚` ˘ ˚ pn`1q ˚ kr‚ ´1 δR p C0 q`δR pV0 , ϕ0 q ` ˝

“

˛

ÿ

@ σ

Sr1 Pπ0 pΣ0 0 q rąr‚

D‹ ‹ w1 pV0ϕ0 q, rSr1 sZ2 ‹ ‚

` ˘ ` n rpV0 , ϕ0 q´1 # nk`p1´g1 `deg V1 qpp1´g2 qn`deg V2 q`2Z, pn`1qpk`p1´g1 `deg V1 qp1´g2 qq`2Z,

if p0 P PΣ´0 pV0ϕ0 q; if p0 P PΣ`0 pV0ϕ0 q.

The CROrient 4 property and Remark 7.2 similarly yield a comparison between the limiting orientation o´ os0 pV0 , ϕ0 q (resp. sp sp pV , ϕ q) with the split orientation o pV , o´ os0 0 ϕ0 q (resp. o C0 ;p0 C0 ;p0 0 0 pV0 , ϕ0 q). We make a special note of the comparisons of o˘ C0 ;p0 pV0 , ϕ0 q with the split orientation in the case of H3 degenerations of S 2 . In this case, we denote the numbers k‚1 and k‚2 of the CROrient 7H3 property and Remark 7.2 by k1 and k2 , respectively. Corollary 7.5. Let C0 be a decorated marked symmetric surface as in p7.36q so that pΣ0 , σ0 q consists of two copies of pS 2 , τ q joined at a single H3 node. Suppose pV0 , ϕ0 q is a C0 -balanced rank n real bundle pair over pΣ0 , σ0 q and p0 is a relative Pin˘ -structure on the real vector bundle V0ϕ0 over Σσ0 0 Ă Σ0 . r (a) The limiting orientation o` C0 ;p0 pV0 , ϕ0 q on λ C0 pDpV0 ,ϕ0 q q is the sp0 same as the split orientation o C0 ;p0 pV0 , ϕ0 q if and only if ˘ ` ˘ ` pn`1q pk´1qδR p C0 q`δR pV0 , ϕ0 q ` n rpV0 , ϕ0 q´1 # nk`p1`deg V1 qpn`deg V2 q`2Z, if p0 P PΣ´0 pV0ϕ0 q; “ if p0 P PΣ`0 pV0ϕ0 q. pn`1qpk`1`deg V1 q`2Z, r (b) The limiting orientation o´ C0 ;p0 pV0 , ϕ0 q on λ C0 pDpV0 ,ϕ0 q q is the sp same as the split orientation o C0 ;p0 pV0 , ϕ0 q if and only if ˆ ˆ ˙˙ ˘ ` k2 pn`1q pk´1qδR p C0 q`δR pV0 , ϕ0 q `rpV0 , ϕ0 q k2 `n 2

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˙ ˆ ˙˙ ˆ ˆ ` ˘ k2 ´1 k2 ´1 ` ` rpV0 , ϕ0 q´1 n 1` 1 2 ˘ ` ` rp C0 q´1 pn`1qpk1 ´1qpk2 ´1q ˙ ˆ D @ deg V2 “ w2 pp0 q, rΣ˚ sZ2 ` 2 # if p0 P PΣ´0 pV0ϕ0 q; nk`p1`deg V1 qpn`deg V2 q, ` pn`1qpk`1`deg V1 q`deg V2 , if p0 P PΣ`0 pV0ϕ0 q. 7.5

Orientations and Evaluation Isomorphisms

Let C be a marked symmetric surface as in (7.3) and pV, ϕq be a real bundle pair over pΣ, σq. Define ev C : ΓpΣ; V qϕ ÝÑ V C ” ev Cpξq “

``

k À i“1

Vxϕi ‘

l À i“1

Vz ` ,

˘ ` ˘ ˘ ξpxi q iPrks , ξpzi` q iPrls .

i

r CpDpV,ϕq q with a homotopy class of We identify an orientation of λ isomorphisms λpDpV,ϕq q ÝÑ λpV Cq via the complex orientations of the fibers Vz ` of V . The conjugation i ϕ on V induces a conjugation on the complex vector bundle V p´Cq ” V b

k â

OΣ p´xi q b

i“1

l â

OΣ p´zi` ´zi´ q,

i“1

which we denote in the same way. By the twisting construction of [38, Lemma 2.4.1], a real CR-operator D on pV, ϕq induces a real CRoperator D´ C on pV p´Cq, ϕq so that the restriction sequence ev

C V C ÝÑ 0 0 ÝÑ D´ C ÝÑ D ÝÝÑ

(7.49)

of Fredholm operators is exact. We note the following. Lemma 7.6. Suppose C is a marked symmetric surface as in (7.3) with the underlying symmetric surface pS 2 , τ q and D is a real CRoperator on rank 1 real bundle pair pV, ϕq over pS 2 , τ q.

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185

(a) The operator D is surjective if deg V ě ´1 and injective if deg V ď ´1. (b) The homomorphism ev C : ker D ÝÑ V C ”

k À i“1

Vxϕi ‘

l À i“1

Vz ` , i

`` ˘ ` ˘ ˘ ev Cpξq “ ξpxi q iPrks , ξpzi` q iPrls ,

(7.50)

is surjective if k`2l ď deg V `1 and is an isomorphism if k`2l “ deg V `1. Proof. By the proof of [15, Proposition 3.6], the dimensions of the kernel and cokernel of the real CR-operator D on pV, ϕq are halves of the dimensions of the kernel and cokernel of the associated real CR-operator on V (i.e. without restricting to the subspaces of real sections). By [24, Theorem 11 ], the latter is surjective if deg V ě ´1 and injective if deg V ď ´1. This establishes (1). Along with the exactness of the restriction sequence (7.49), (1) in turn  implies (2). If V ϕ is orientable and k R 2Z, let o˘ C;0 pV, ϕq ” o C;ι

˘ ϕ S 2 pp0 pV qq

pV, ϕq

r CpDpV,ϕq q determined by the image denote the orientation on λ ˘ ` ` ϕ˘ ϕ ˘ ιS 2 p˘ 0 pV q P PS 2 V ϕ ϕ as in Remark 1.18 of the canonical Pin˘ -structure p˘ 0 pV q on V 2 under the second map in (6.3) with X “ S . If V ϕ is not orientable, D2` is a half-surface of pS 2 , τ q, and k P 2Z, let o˘ C;0 pV, ϕq be the r orientation of λ CpDpV,ϕq q as above the CROrient 6 property.

A half-surface D2` of pS 2 , τ q determines an orientation of the fixed locus S 1 of pS 2 , τ q (the counterclockwise rotation if D2` is identified with the unit disk around 0 P C). We say that distinct points x1 , . . . , xk P S 1 are ordered by position with respect to D2` if every xi with i P rk ´ 1s directly precedes xi`1 as S 1 is traversed in the direction determined by its orientation as the boundary of D2` .

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Corollary 7.7. Suppose C is a marked symmetric surface as in (7.3) and pV, ϕq is a rank 1 real bundle pair over C so that pΣ, σq “ pS 2 , τ q

and

pS 2 , τ q,

k`2l “ deg V `1. zi` S1

(7.51)

is a half-surface of P for every i P rls, and If are ordered by position with the real marked points x1 , . . . , xk P respect to D2` , then the isomorphism (7.50) lies in the homotopy class r determined by the orientation o˘ C;0 pV, ϕq of λ CpDpV,ϕq q. D2`

D2`

Proof. Let a “ deg V . By (7.51), a ě ´1. The pk, lq “ p1, 0q, p2, 0q cases of this claim are the two statements of the CROrient 6 property. We first show in the following that the claim in the l “ 0 cases implies the claim in all cases. We then give two proofs of the l “ 0 cases. The first proof makes use of Proposition 8.2, which in turn is used in the proof of Theorem 7.1; the second proceeds directly from the properties of the orientations of Theorem 7.1 stated in Sections 7.2 and 7.3. (1) Let C0 ” pΣ0 , σ0 q be the connected symmetric surface which contains precisely one conjugate pair pnd` , nd´ q of nodes and consists of three irreducible components, each of which is isomorphic to S 2 . Let P10 Ă Σ0 be the component preserved by σ0 and D2` , D2´ Ă P10 be the two half-surfaces. We label the two conjugate nodes so that nd` P D2` and denote by P1` the conjugate component of Σ0 also containing nd` ; see the first diagram in Figure 7.4. Let p U,rcq be a flat family of deformations of C0 as in (7.32) over the unit ball Δ Ă C2 around the origin so that ˘ ˘ ` ` (7.52) Σ t 1 , σt 1 “ S 2 , τ

Figure 7.4. The three- and two-component symmetric surfaces C0 of the proof of Corollary 7.7.

Main Results and Applications of Part II

187

for some t1 P ΔR . The decorated structure on pS 2 , τ q then induces a decorated structure on the fiber pΣt , σt q of p U,rcq over every t P ΔR . Let pVr , ϕq r be a rank 1 real bundle pair over p U,rcq so that ˇ ˇ ` ˘ Vrt1 , ϕ rt1 “ pV, ϕq. (7.53) deg Vr ˇ 1 “ k´1, deg Vr ˇ 1 “ l, P0

P`

Such a real bundle pair can be obtained by extending an appropriate real bundle pair from the central fiber and deforming the extension so that it satisfies the last condition in (7.53). The Pin˘ -structure ˘ ϕ ϕ r ϕr r˘ p˘ 0 pV q on V extends to a Pin -structure p on V which restricts r ϕrt r r0 q to p˘ 0 pVt q for every t P ΔR . We denote the restriction of pV0 , ϕ ϕ r r00 q and the restriction of r p˘ to Vr0 0 by r p˘ to P10 by pVr00 , ϕ 0. rt q as Let D” tDt u be a family of real CR-operators on pVrt , ϕ in (7.34) so that Dt1 “ D. By Lemma 7.6(a), each operator Dt is then surjective; their kernels form a vector bundle ğ ` ˘ ttuˆ ker Dt ÝÑ ΔR . (7.54) πK : Kp Dq ” tPΔR

D0`

the restrictions of D0 to real CR-operators We denote by D00 and r r r00 q and V |P1` , respectively. The exact triple on pV00 , ϕ 0 ÝÑ D0 ÝÑ D00 ‘D0` ÝÑ Vrnd` ÝÑ 0, pξ´ , ξ0 , ξ` q ÝÑ pξ0 , ξ` q,

pξ0 , ξ` q ÝÑ ξ` pnd` q´ξ0 pnd` q,

of Fredholm operators then determines an isomorphism ` ˘ ` ˘ ` ˘ λpD0 qbλ Vr ` « λ D00 bλ D ` . 0

nd

(7.55) (7.56)

` rc R R Let s` 1 , . . . , sl be disjoint sections of U´ U and s1 , . . . , sk be disjoint rc sections of U over ΔR so that  ˘( 1 sR s` , si pt1 q “ zi` @ i P rls. i pt1 q “ xi @ i P rks, i p0q P P` ´ nd

By Lemma 7.6(b), the bundle homomorphism ev U : Kp Dq ÝÑ VrUR ‘ VrUC ”

k à i“1

r ϕr sR˚ i V ‘

l à

r s`˚ i V,

i“1

` ` ˘ ` ` ˘ ˘ ev Upt, ξq “ t, ξpsR i ptqq iPrks , ξpsi ptqq iPrls

(7.57)

is then an isomorphism restricting to the isomorphism ev C in (7.50) over t1 . Thus, ev C lies in the homotopy class determined by the orir entation o˘ C;0 pV, ϕq of λ CpDpV,ϕq q if and only if the restriction ev C0

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of ev U over 0 lies in the homotopy class determined by the limitr C pD0 q. By Corollary 7.3, the last ing orientation o1C ;rp˘ pVr0 , ϕ r0 q of λ 0 0

0

r r0 q deterorientation is the same as the C-split orientation oC˘ C0 ;0 pV0 , ϕ ˘ ϕ r ϕr00 mined by the Pin˘ -structure p˘ 0 pV00 q “ p0 pV q. By the first two conditions in (7.53) and Lemma 7.6(b), the evaluation homomorphisms ˇ `` ˘ ˘ ev0C0 pξq “ ξpsR ev0C0 : ker D00 ÝÑ VrUR ˇ0 , i p0qq iPrks , ˇ ` ` ` ˘ ˘ ` ` ` r r Cˇ ev` C0 : ker D0 ÝÑ Vnd` ‘ V U 0 , ev C0 pξq “ ξpnd q, ξpsi p0qq iPrls are isomorphisms. The last isomorphism determines the complex r r0 q is thus orientation of D0` . The C-split orientation oC˘ C0 ;0 pV0 , ϕ determined by the homotopy class of the isomorphism ev` C0 , the ˘ orientation o C00 ;0 pV00 , ϕ00 q of D00 , and the complex orientation of λpVrnd` q via the isomorphism (7.56). Along with the conclusion of the previous paragraph, this implies that the isomorphism ev C in (7.50) lies in the homotopy class determined by the orienta0 tion o˘ C;0 pV, ϕq if and only if the isomorphism ev C0 lies in the homotopy class determined by the orientation o˘ C00 ;0 pV00 , ϕ00 q. Thus, the claim of the corollary holds for pk, lq if and only if it holds for pk, 0q. In light of the CROrient 6 property, this establishes the claim if k “ 1 or k “ 2. (2) By Proposition 8.2, the claim of the corollary holds for pk, lq if and only if it holds for pk ` 2l, 0q. Along with the conclusion of the previous paragraph, this implies that the claim holds if k ą 0 or l ą 0. The pk, lq “ p0, 0q case then follows from the pk, lq “ p0, 1q case via the if and only if statement of the previous paragraph. In the following, we obtain the same conclusions by using Corollary 7.5 instead of Proposition 8.2. We consider the behavior of the relevant orientations under flat degenerations of pS 2 , τ q to a connected symmetric surface C0 ” pΣ0 , σ0 q with one node nd. This node is then of type H3, i.e. it is a non-isolated point of Σσ0 0 separating Σ0 into two copies of pP1 , τ q, which we denote by P11 and P12 . For r “ 1, 2, let D2r˘ Ă P1r be the two distinguished half-surfaces and Sr1 ” D2r` XD2r´ Ă P1r

Main Results and Applications of Part II

189

be the fixed locus of the involution σ0 |P1r ; see the second diagram in Figure 7.4. Let U be a flat family of deformations of C0 as in (7.32) over the unit ball Δ Ă C around the origin satisfying (7.52). We choose a decorated structure on pΣ0 , σ0 q so that t1 P Δ˚R lies in the ˚ topological component Δ` R Ă ΔR defined as above the CROrient 7H3 property. This decorated structure then induces a decorated structure on the fiber pΣt , σt q of p U,rcq over every t P ΔR . Suppose k ě 1. Let pVr , ϕq r be a rank 1 real bundle pair over p U,rcq so that ˇ ˇ ` ˘ deg Vr ˇ 1 “ k`2l´2, deg Vr ˇ 1 “ 1, Vrt , ϕ rt “ pV, ϕq. (7.58) P1

P2

1

1

r0 q to P11 and P12 by pV1 , ϕ1 q and We denote the restrictions of pVr0 , ϕ ˘ ϕ ϕ pV2 , ϕ2 q, respectively. The Pin -structure p˘ 0 pV q on V extends to a ` r ϕrt p˘ on Vr ϕr which restricts to p˘ Pin˘ -structure r 0 pVt q for every t P ΔR ϕ2 ϕ2 ˘ ˘ and to p2 ” p0 pV2 q on V2 . Then, # ´ ϕ ˇ ˇ p0 pV1 1 q, if deg V R 2Z; ϕ1 ´ˇ ` r` ˇ ` r ϕ1 “ ϕ “ p pV ” p ” p p p´ 1 1 0 1 q. V1 V1 1 ϕ1 ´ p1 pV1 q, if deg V P 2Z; (7.59) p˘ . We denote the restriction of r p˘ to Vr0ϕr0 by r 0 rt q as Let D” tDt u be a family of real CR-operators on pVrt , ϕ in (7.34) so that Dt1 “ D. By Lemma 7.6(a), each operator Dt is again surjective; their kernels form a vector bundle Kp Dq as before. We denote by D1 and D2 the restrictions of D0 to real CR-operators on pV1 , ϕ1 q and pV2 , ϕ2 q, respectively. The exact triple ϕ ÝÑ 0, 0 ÝÑ D0 ÝÑ D1 ‘D2 ÝÑ Vnd

pξ1 , ξ2 q ÝÑ ξ2 pndq´ξ1 pndq (7.60) of Fredholm operators then determines an isomorphism ` ϕ˘ (7.61) « λpD1 qbλpD2 q. λpD0 qbλ Vnd ` rc R R Let s` 1 , . . . , sl be disjoint sections of U ´ U and s1 , . . . , sk be disjoint sections of Urc over ΔR so that 1 sR i p0q P S1 ´tndu @ i P rk´1s,

sR i pt1 q “ xi @ i P rks,

1 sR k p0q P S2 ´tndu,

si pt1 q “ zi` @ i P rls,

and the points nd, x1 , . . . , xk´1 of S11 are ordered by position with respect to D21` . By Lemma 7.6(b), the bundle homomorphism ev U as

Spin/Pin-Structures & Real Enumerative Geometry

190

in (7.57) is again an isomorphism restricting to the isomorphism ev C in (7.50) over t1 . Thus, ev C lies in the homotopy class determined r by the orientation o˘ C;0 pV, ϕq of λ CpDpV,ϕq q if and only if the restriction ev C0 of ev U over 0 lies in the homotopy class determined by the r C pD0 q. By Corollary 7.5(a), limiting orientation o` pVr0 , ϕ r0 q of λ 0 C0 ;r p˘ 0 $ &“ osp ´ pVr0 , ϕ r0 q, if deg V R 2Z; C0 ;r p0 ` r o` r0 q pVr0 , ϕ r0 q o C ;rp´ pV0 , ϕ sp C0 ;r p` r 0 0 %‰ o 0 r0 q, if deg V P 2Z; ´ pV0 , ϕ C0 ;r p0

r0 q, “ o C0 ;rp` pVr0 , ϕ

(7.62)

0

r0 q is the split orientation determined by the Pin˘ where o C0 ;rp˘ pVr0 , ϕ 0

ϕ1 ϕ2 structures p˘ and p˘ 1 on V1 2 on V2 . 1 denote the irreducible component P1 of Σ with its invoC11 Let r 0 1 lution and the marked points (k´1 real points and l conjugate pairs 1 denote the irreducible component P1 of Σ with C12 of points). Let r 0 2 its involution, the marked point xk P S21 , and the nodal point nd. By the first two conditions in (7.58) and Lemma 7.6(b), the evaluation homomorphisms

ev C r 1 : ker D1 ÝÑ 11

k´1 à i“1

VrsϕrR p0q ‘ VrUC , i

ev rC1 pξq “ 11

``

˘ ` ` ˘ ˘ ξpsR i p0qq iPrk´1s , ξpsi p0qq iPrls ,

r ϕr r ϕr ev C r 1 : ker D2 ÝÑ VsR p0q ‘ Vnd , 12 k ` ˘ ` ev rC1 pξq “ ξpsR k p0qq, ξpnd q 12

are isomorphisms. By the CROrient 6(b) property, the second isomorphism above lies in the homotopy class determined by the orientation oC r1

˘ 12 ;ιP1 pp2 q 2

pV2 , ϕ2 q “ o˘r 1

C12 ;0

pV2 , ϕ2 q

r r 1 pD2 q. on λ C 12

pVr , ϕ r0 q is thus determined by the The split orientation osp C ;r p˘ 0 0

0

homotopy class of the isomorphism ev C r1

12

and by the orientation

Main Results and Applications of Part II

oC r1

˘ 11 ;ιP1 pp1 q 1

191

r r 1 pD1 q via the isomorphism (7.61). By (7.59) pV1 , ϕ1 q on λ C 11

and the CROrient 2(b) property, $ &“ o´r 1 pV1 , ϕ1 q, if deg V R 2Z; C11 ;0 oC r 1 ;ι 1 pp´ q pV1 , ϕ1 q ´ 1 11 P % pV1 , ϕ1 q, if deg V P 2Z; ‰ o 1 r1 C11 ;0

ˆ oC r1

` 11 ;ιP1 pp1 q 1

pV1 , ϕ1 q “ o`r 1

C11 ;0

pV1 , ϕ1 q.

Along with (7.62) and the conclusion of the previous paragraph, this implies that the isomorphism ev C in (7.50) lies in the homotopy class determined by the orientation o˘ C;0 pV, ϕq if and only if the isomorphism ev C r 1 lies in the homotopy class determined by the orientation o˘r 1

C11 ;0

11

pV1 , ϕ1 q.

Thus, the claim of the corollary holds for pk, lq if and only if it holds for pk ´ 1, lq. Along with the already obtained conclusion for the k “ 1, 2 cases of the claim, this establishes the full statement of  the claim. 2 Ă S 2 be the disk cut out by the fixed locus S 1 Example 7.8. Let D` of the involution τ on S 2 which contains the origin 0 in C Ă S 2 and C be the marked symmetric surface consisting of S 2 with one real marked point. We denote by Br the usual unit outward radial vector field on C˚ and by oS 1 the usual counterclockwise orientation on T S 1 . Let B¯ be the real CR-operator on the rank 1 real bundle pT P1 , dτ q ¯ on T P1 and over pP1 , τ q determined by the holomorphic B-operator 1 o0 pT P , dτ ; oS 1 q be the orientation of

¯ “ λpker Bq ¯ λpBq induced by oS 1 as above the CROrient 6 property in Section 7.2. 1 r ¯ By the CROrient 3 property, the orientation o˘ C;0 pT P , dτ q on λ CpBq ¯ and Tx S 1 corresponds to the homotopy class of isomorphisms of λpBq 1 1 determined by the orientations o0 pT P , dτ ; oS 1 q and oS 1 , respectively. Along with Corollary 7.7, this implies that the isomorphism ` ˘ ker B¯ ÝÑ T0 P1 ‘Tx1 S 1 , ξ ÝÑ ξp0q, ξpx1 q ¯ the complex orienrespects the orientation o0 pT P1 , dτ ; oS 1 q on ker B, 1 tation on T0 P , and the counterclockwise orientation oS 1 on Tx1 S 1 .

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Spin/Pin-Structures & Real Enumerative Geometry

Thus, the homomorphism ¯ pker Bq‘R ÝÑ T0 P1 ‘Tz P1 ,

` ˘ pξ, tq ÝÑ ξp0q, ξpzq´tBr

is an isomorphism which respects the orientation o0 pT P1 , dτ ; oS 1 q on ¯ the standard orientation on R, and the complex orientations ker B, on T0 P1 and Tz P1 for every z P C˚ .

Chapter 8

Base Cases

In Section 8.1, we follow the perspective in [13, Section 3] to describe orientations ok,l pV, ϕ; ox1 q on the determinants of real CR-operators on rank 1 real bundle pairs pV, ϕq over pS 2 , τ q with k, l P Zě0 and k fl deg V mod 2. We establish a number of properties of these orientations in Section 8.1 and study their behavior under degenerations of pS 2 , τ q to nodal symmetric surfaces in Sections 8.2 and 8.3. The main results of Sections 8.2 and 8.3, Propositions 8.5 and 8.7, are the analogs of the CROrient 7H3 and 7C properties for the orientations ok,l pV, ϕ; ox1 q of Section 8.1. They imply in particular that the orientations ok,l pV, ϕ; ox1 q do not depend on the admissible choices of pk, lq; see Proposition 8.2(3). Outside Sections 8.1–8.3, we thus denote these orientations simply by opV, ϕ; ox1 q. In particular, the statements of Propositions 8.2(1), 8.5, and 8.7 apply to the orientations opV, ϕ; ox1 q. In the first part of Section 8.4, we recall the construction of [30, Section 6.5] for orienting the determinants of real CR-operators on even-degree real bundle pairs pV, ϕq over pS 2 , τ q from relative OSpinstructures os on the real vector bundles V ϕ over the τ -fixed locus S 1 Ă S 2 . In the remainder of this section and in Section 8.5, we confirm that the resulting orientations oos pV, ϕq satisfy the applicable properties of Section 7.2. While this can be done through geometric arguments similar to those in Sections 8.1–8.3, we instead deduce some of these properties from the analogous properties of the orientations of Section 8.1 via the CROrient 5(a) property for even-degree real bundle pairs pV, ϕq over pS 2 , τ q, which we establish 193

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Spin/Pin-Structures & Real Enumerative Geometry

directly in Section 8.5. We use the perspective of Definition 6.3 on relative OSpin-structures throughout Sections 8.4 and 8.5.

8.1

Line Bundles over pS 2 , τ q: Construction and Properties

Suppose C is a marked symmetric surface as in (7.3) so that pΣ, σq is S 2 with the involution τ and D is a real CR-operator on a rank 1 real bundle pair pV, ϕq over pS 2 , τ q. By Lemma 7.6, the operator D is surjective and the evaluation homomorphism in (7.50) is an isomorphism if k`2l “ deg V `1 .

(8.1)

If k “ 0, the target V C of this isomorphism is oriented by the complex orientation of V . If k ą 0, an orientation ox1 on Vx1 determines an orientation on V C by transporting ox1 to each xi with 2 ď i ď k along the positive direction of S 1 with respect to the disk D2` Ă S 2 . Along with the complex orientations of the summands Vz ` , ox1 thus deteri mines an orientation on V C. If (8.1) holds, the resulting orientation on V C determines an orientation o Cpox1 q on the determinant ` ˘ λpDq « λ ker D of the real CR-operator D via the isomorphism (7.50). We denote by ok,l pox1 q the orientation o Cpox1 q for a marked symmetric surface C as in (7.3) so that its real points x1 , . . . , xk are ordered by position with respect to D2` and zi` P D2` for every i P rls. Since the space of such C is path-connected, the orientation ok,l pox1 q does not depend on the choice of C in it. We call ok,l pox1 q the pk, lqevaluation orientation of D. It is straightforward to see that this orientation satisfies the following properties. Lemma 8.1. Suppose k, l P Zě0 and pV, ϕq is a rank 1 real bundle pair over pS 2 , τ q so that (8.1) holds. Let ox1 be an orientation of V ϕ at a point x1 in S 1 Ă S 2 . (1) If ox1 is the orientation of Vxϕ1 opposite to ox1 , then # ok,l pox1 q, if deg V R 2Z; ok,l pox1 q “ ok,l pox1 q, if deg V P 2Z.

Base Cases

195

(2) The orientation ok,l pox1 q does not depend on the choice of the half-surface D2` Ă S 2 if and only if deg V – 0, 3 mod 4. The orientation ok,l pox1 q induces an orientation ok,l pV, ϕ; ox1 q of a real CR-operator D on any rank 1 real bundle pair pV, ϕq with k`2l – deg V `1

mod 2

(8.2)

as follows. Let C0 , pnd` , nd´ q, P10 , P1˘ , D2˘ , p U,rcq, and t1 be as above (7.52), a0 “ k`2l´1,

and

a1 “ pdeg V ´a0 q{2.

r be a rank 1 real bundle pair over p U,rcq By (8.2), a1 P Z. Let pVr , ϕq so that ˇ deg Vr ˇP1 “ a0 , 0

ˇ deg Vr ˇP1 “ a1 , `

`

˘ Vrt1 , ϕ rt1 “ pV, ϕq.

(8.3)

For an orientation ox1 of V ϕ at a point x1 in the fixed locus S 1 Ă Σt1 and a point x11 in the fixed locus S 1 Ă Σ0 , let ox11 denote the orientation of Vr ϕr at x11 obtained by transferring ox1 along a path in Urc from x1 to x11 . If the degree of V is even (i.e. the real line bundle V ϕ over S 1 is orientable), then ox11 does not depend on the choice of this path. rt q as Let D” tDt u be a family of real CR-operators on pVrt , ϕ in (7.34) so that Dt1 “ D. As above (7.55), we denote the restricr0 q to P10 by pVr00 , ϕ r00 q and the restrictions of D0 to real tion of pVr0 , ϕ r00 q and Vr |P1` by D00 and D0` , respectively. CR-operators on pVr00 , ϕ The orientation ok,l pox11 q of D00 and the complex orientations of D0` and Vrnd` determine an orientation of D0 via the isomorphism (7.56) and thus an orientation of the line bundle λp Dq over ΔR . The latter restricts to an orientation ok,l pV, ϕ; ox1 q of λpDq, which we call the pk, lq-intrinsic orientation of D. By Lemma 8.1(1), ok,l pV, ϕ; ox1 q does not depend on the choice of the path from x1 to x11 above even if the degree of V is odd. In light of Proposition 8.2(3), which is established in Section 8.3, we denote the orientations ok,l pV, ϕ; ox1 q

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Spin/Pin-Structures & Real Enumerative Geometry

by opV, ϕ; ox1 q after Section 8.3 and call opV, ϕ; ox1 q the intrinsic orientation of D. Proposition 8.2. Suppose pV, ϕq is a rank 1 real bundle pair over pS 2 , τ q and D is a real CR-operator on pV, ϕq. Let ox1 be an orientation of V ϕ at a point x1 in S 1 Ă S 2 . (1) For all k, l P Zě0 satisfying p8.2q, the orientation ok,l pV, ϕ; ox1 q of D satisfies the two properties of Lemma 8.1. (2) For all k, l P Zě0 satisfying p8.1q, the pk, lq-evaluation and pk, lq-intrinsic orientations ok,l pox1 q and ok,l pV, ϕ; ox1 q of D are the same. (3) The orientation ok,l pV, ϕ; ox1 q of D does not depend on the choice of k, l P Zě0 satisfying p8.2q. Proof of Proposition 8.2(1). The property of Lemma 8.1(1)for the orientation ok,l pV, ϕ; ox1 q follows from this property for the orientation ok,l pox1 q because the choice of ox1 affects the orientation only on the first term on the right-hand side of (7.56) and the parˇ ities of the degrees of V and Vr ˇP1 are the same. The change in the 0

choice of the half-surface of pS 2 , τ q acts by the complex conjugation on the complex orientations of Vrnd` and D0` . The complex dimension of Vrnd` is 1; the complex index of D0` is a1 `1. Combining this with Lemma 8.1(2), we find that the orientation ok,l pV, ϕ; ox1 q does not depend on the choice of the half-surface D2` Ă S 2 if and only if the number ´1 ` pa1 `1q `

` pdeg V qpdeg V `1q a0 pa0 `1q “ ´ 2 a0 a1 `a12 q (8.4) 2 2

is even. Thus, the orientation ok,l pV, ϕ; ox1 q satisfies the property of  Lemma 8.1(2). Proposition 8.2(2) is a special case of Lemma 8.4(1). We establish Proposition 8.2(3) in Section 8.3 based on the following observation about orientations of the determinants of real CR-operators on degree 2 rank 1 real bundle pairs over pS 2 , τ q. Lemma 8.3. With the notation as in Lemma 8.1, o3,0 pox1 q “ o1,1 pox1 q.

Base Cases

197

Proof. By [17, Proposition 2.1], every rank 1 degree a real bundle pair pV, ϕq over pP1 , τ q is isomorphic to the holomorphic line bundle OP1 paq with the natural lift τra of τ . Thus, we can assume that pV, ϕq ¯ and is pOP1 p2q, τr2 q, D is the holomorphic B-operator, x1 “ 1,

x2 “ i,

x3 “ ´1,

z1` “ 0.

The evaluation homomorphisms ker D ÝÑ Vxϕ1 ‘Vxϕ2 ‘Vxϕ3

and

ker D ÝÑ Vxϕ1 ‘Vz ` 1

(8.5)

are the isomorphisms determining the orientations o3,0 pox1 q and o1,1 pox1 q, respectively. An orientation ox1 of Vxϕ1 determines an orientation o1 of the target of the first isomorphism; combined with the complex orientation of Vz ` , it also determines an orientation o2 of the 1 target of the second isomorphism. By definition, the first (resp. second) isomorphism above is orientation-preserving with respect to the orientation o3,0 pox1 q (resp. o1,1 pox1 q) on its domain and the orientation o1 (resp. o2 ) on its target. Choose a trivialization of V |D2` so that V ϕ |S 1 is identified with the real line subbundle (  Λ ” pz, vq P S 1 ˆC : v{z P R of S 1 ˆC; see [32, Corollary C.3.9]. We can assume that the orientation ox1 of Vxϕ1 agrees with the standard orientation of R Ă C under this identification. The sets p1, 0, 0q, p0, i, 0q, p0, 0, ´1q P Vxϕ1 ‘Vxϕ2 ‘Vxϕ3 p1, 0q, p0, 1q, p0, iq P Vxϕ1 ‘Vz `

and (8.6)

1

are then oriented bases with respect to the orientations o1 and o2 . The functions z, 1`z 2 , i´iz 2 form a basis for ker D; see Step 2 in the proof of [32, Theorem C.4.1]. With respect to this basis and the bases (8.6), the isomorphisms (8.5) are given by the matrices ¨ ˛ ¨ ˛ 1 2 0 1 2 0 ˝1 0 2‚ ˝0 1 0‚, and 1 ´2 0 0 0 1 respectively. Since these matrices have determinants of the same sign, the orientation o3,0 pox1 q on ker D corresponding to o1 and the orien tation o1,1 pox1 q corresponding to o2 are the same.

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Lemma 8.4. Suppose pV, ϕq is a rank 1 real bundle pair over pS 2 , τ q, D is a real CR-operator on pV, ϕq, and k, l P Zě0 satisfy p8.2q. Let ox1 be an orientation of V ϕ at a point x1 in S 1 Ă S 2 . (1) If k ` 2l ď deg V ` 1, then the orientations ok,l pV, ϕ; ox1 q and ok,pdeg V `1´kq{2 pox1 q are the same. (2) The pk, lq-intrinsic orientation ok,l pV, ϕ; ox1 q does not depend on the choice of l P Zě0 . Proof of Lemma 8.4(1). We continue with the notation above Proposition 8.2, taking C as in the definition of the pk, lq-evaluation orientation above Lemma 8.1. In this case, a1 ě 0 and each operator Dt is surjective. Their kernels form a vector bundle Kp Dq as ` rc R R in (7.54). Let s` 1 , . . . , sl`a1 be disjoint sections of U´U and s1 , . . . , sk be disjoint sections of Urc over ΔR so that sR i pt1 q “ xi

@ i P rks,

 ˘( 1 s` @ i P rls, i p0q P P0 ´ nd

` s` i pt1 q “ zi

@ i P rl`a1 s,

 `( 1 s` @ i P ra1 s. l`i p0q P P` ´ nd

By the assumption that a1 ě 0, the evaluation homomorphisms ev0C0 : ker D00 ÝÑ

k à i“1

ev0C0 pξq “

VrsϕrR p0q ‘ i

l à Vrs` p0q , i“1

i

`` R ˘ ` ˘ ˘ ξpsi p0qq iPrks , ξps` i p0qq iPrls ,

` r ev` C0 : ker D0 ÝÑ Vnd` ‘

1

a à

Vrs`

i“l`1

l`i p0q

,

` ` ` ˘ ˘ ` pξq “ ξpnd q, ξps p0qq , ev` 1 C0 l`i iPra s

(8.7)

are isomorphisms. Along with the sentence containing (7.50), this implies that (7.57) with l replaced by l`a1 ”

deg V `1´k 2

is an isomorphism of vector bundles over ΔR .

Base Cases

199

Since each operator Dt is surjective, the last isomorphism induces an isomorphism λp Dq ÝÑ

k â i“1

r ϕr sR˚ i V

b

1 l`a â

r s`˚ i λpV q

(8.8)

i“1

of line bundles over ΔR . The first product on the right-hand side above is oriented by transferring the orientation ox1 from the point x1 in S 1 Ă Σt1 along the positive direction of S 1 Ă D2` to the points xi with i “ 2, . . . , k and then along the section sR i to an orientation ϕ r oxi ;t of VrsR ptq for each t P ΔR . The second product is oriented by i

the complex orientation of Vr . By definition, the first isomorphism in (8.7) preserves the pk, lq-evaluation ok,l poxi ;0 q. The second isomorphism in (8.7) preserves the complex orientation on its domain. Thus, the restriction of (8.8) to the fiber over t1 is orientation-preserving with respect to the pk, lq-intrinsic orientation ok,l pV, ϕ; ox1 q on its domain. By definition, this restriction is also orientation-preserving with respect to the pk, l`a1 q-evaluation orientation ok,l`a1 pox1 q. This establishes the first claim of the lemma.  Lemma 8.4(2) is proved in Section 8.2. 8.2

Line Bundles over C Degenerations of pS 2 , τ q

We now describe the behavior of the orientations of Proposition 8.2 under flat degenerations of pS 2 , τ q as in the CROrient 7C property on page 121. Suppose k, l P Zě0 , C0 ” pΣ0 , σ0 q is a connected decorated symmetric surface as above (7.52) and in the left diagram in Figure 7.4, pV0 , ϕ0 q is a rank 1 real bundle pair over pΣ0 , σ0 q such that k`2l – deg V0 |P10 `1 – deg V0 `1 mod 2,

(8.9)

and ox1 is an orientation of V0ϕ0 at a point x1 of the fixed locus S 1 “ Σσ0 0 of σ0 . With nd˘ and P10 , P1˘ as above (7.52), we denote the restriction of pV0 , ϕ0 q to P10 by pV00 , ϕ00 q. Let D0 be a real CR-operator on pV0 , ϕ0 q. We denote its restrictions to real CR-operators on pV00 , ϕ00 q and V0 |P1` by D00 and D0` ,

200

Spin/Pin-Structures & Real Enumerative Geometry

respectively. The orientation o0;0 pox1 q ” ok,l pV00 , ϕ00 ; ox1 q of D00 and the complex orientations of D0` and V0 |nd` determine an orientation ` ˘ (8.10) o0 pox1 q ” ok,l V0 , ϕ0 ; ox1 of D0 via the isomorphism (7.56). In an analogy with the C-split orientation of Corollary 7.3, we call (8.10) the pk, lq-split orientation of D0 induced by ox1 . Suppose in addition (7.32) is a flat family of deformations of C0 , pV, ϕq is a real bundle pair over p U,rcq extending pV0 , ϕ0 q, sR 1 is a secrc R tion of U over ΔR with s1 p0q “ x1 , and D” tDt u is a family of real CR-operators on pVt , ϕt q as in (7.34) extending D0 . The decorated structure on C0 induces a decorated structure on the fiber pΣt , σt q of π for every t P ΔR as above the CROrient 7C property. The orienϕ tation ox1 of V0ϕ0 |x1 induces an orientation of the line bundle sR˚ 1 V ϕt over ΔR , which in turn restricts to an orientation ox1 ;t of Vt at sR 1 ptq for each t P ΔR . The orientations ` ˘ t P Δ˚R , ot pox1 q ” ok,l Vt , ϕt ; ox1 ;t , of Dt depend continuously on t and extend to an orientation ` ˘ (8.11) o10 pox1 q ” o1k,l V0 , ϕ0 ; ox1 of D0 . In an analogy with the limiting orientation of the CROrient 7C property, we call (8.11) the limiting orientation of D0 induced by ox1 . It depends only on pV0 , ϕ0 q and ox1 and not on p U,rcq, pV, ϕq, or sR 1. Proposition 8.5. Suppose C0 is a connected decorated symmetric surface with precisely one pair of conjugate nodes and each component isomorphic to S 2 pV0 , ϕ0 q is a rank 1 real bundle pair over pΣ0 , σ0 q. Let k, l P Zě0 be so that p8.9q holds and D0 be a real CR-operator on pV0 , ϕ0 q. The pk, lq-split and limiting orientations, p8.10q and p8.11q, of D0 are the same. We deduce this proposition from Lemma 8.6. It shows that certain orientations o1k,l pV, ϕ; ox1 q of a real CR-operator over a smooth symmetric surface constructed via a flat family of deformations of a

Base Cases

201

Figure 8.1. Deformations of the five-component symmetric surface pΣ0 , σ0 q over a unit ball in C4 in the proof of Proposition 8.5.

symmetric surface with two pairs of conjugate nodes, instead of one as above Proposition 8.2, are in fact the same as ok,l pV, ϕ; ox1 q. Suppose pV, ϕq, k, l, x1 , ox1 , and D are as above Lemma 8.1. Let C0 ” pΣ0 , σ0 q be a connected symmetric surface which contains precisely two conjugate pairs of nodes and consists of one invariant component and two pairs of conjugate components attached to it, all of which are isomorphic to S 2 ; see the left diagram in Figure 8.1. Let P10 Ă Σ0 be the component preserved by σ0 , D2` , D2´ Ă P10 be the two ` 2 2 half-surfaces, and nd` 1 , nd2 P D` be the two nodes on D` . We denote 1 1 by P1` and P2` the other irreducible components containing nd` 1 and nd` , respectively. Let p U ,r c q be a flat family of deformations 2 of C0 as in (7.32) over the unit ball Δ Ă C4 around the origin satisfying (7.52) for some t1 P ΔR . Let a0 “ k`2l´1 be as before and a11 , a12 P Z be such that a0 ` 2pa11 ` a12 q “ deg V.

(8.12)

Let pVr , ϕq r be a rank 1 real bundle pair over p U,rcq so that ˇ deg Vr ˇP1 “ a0 , 0

ˇ deg Vr ˇP1 “ a11 , 1`

ˇ deg Vr ˇP1 “ a12 , 2`

` ˘ Vrt1 , ϕ rt1 “ pV, ϕq

(8.13) rt q as in (7.34) and D” tDt u be a family of real CR-operators on pVrt , ϕ so that Dt1 “ D. The orientation ox1 of V ϕ induces an orientation ox11 of Vr ϕr at a point x1 in the fixed locus S 1 Ă Σ0 as before. 1

Spin/Pin-Structures & Real Enumerative Geometry

202

We denote the restriction of pVr0 , ϕ r0 q to P10 by pVr00 , ϕ r00 q and the r00 q, Vr0 |P11` , and restrictions of D0 to real CR-operators on pVr00 , ϕ Vr0 | 1 by D00 , D ` , and D ` , respectively. The exact triple P2`

01

02

` ` ‘D02 ÝÑ Vrnd` ‘ Vrnd` ÝÑ 0, 0 ÝÑ D0 ÝÑ D00 ‘D01 1 2 ˘ ` ˘ ` ξ2´ , ξ1´ , ξ0 , ξ1` , ξ2` ÝÑ ξ0 , ξ1` , ξ2` , ˘ ˘ ` ` ` ` ` (8.14) ξ0 , ξ1` , ξ2` ÝÑ ξ1` pnd` 1 q´ξ0 pnd1 q, ξ2` pnd2 q´ξ0 pnd2 q

of Fredholm operators then determines an isomorphism ` ˘ ` ˘ ` ˘ ` `˘ ` `˘ λpD0 qbλ Vrnd` bλ Vrnd` « λ D00 bλ D01 bλ D02 . (8.15) 1

2

The pk, lq-evaluation orientation ok,l pox11 q of D00 and the complex orientations of D ` , D ` , Vr ` , and Vr ` determine an orientation of 01

02

nd1

nd2

o1k,l pVr , ϕ; r ox1 q D0 via the isomorphism (8.15) and thus an orientation r of the line bundle λp Dq over ΔR . The latter restricts to an orientation o1k,l pV, ϕ; ox1 q of λpDq. Lemma 8.6. Suppose pV, ϕq is a rank 1 real bundle pair over pS 2 , τ q, D is a real CR-operator on pV, ϕq, k, l P Zě0 satisfy p8.2q, and a11 , a12 P Z satisfy p8.12q. Let ox1 be an orientation of V ϕ at a point x1 in S 1 Ă S 2 . The orientation o1k,l pV, ϕ; ox1 q of D defined above is the same as the pk, lq-intrinsic orientation of Proposition 8.2. Proof of Proposition 8.5. In order to distinguish the nodal symmetric surfaces C0 , their invariant components P10 , and the families of deformations p U,rcq over Δ appearing above the statements of Proposition 8.5 and Lemma 8.6, we denote the objects appearing 1 c1 q, and Δ1 , respectively, in this above Lemma 8.6 by C01 , P11 0 , p U ,r 1 1 proof. We can choose p U ,rc q so that Ct1 0 for some t0 P Δ1R is the nodal symmetric surface C0 as in the statement of the proposition obtained from C01 by smoothing the node nd` 1 ; see Figure 8.1. Let 1R be a section of U1rc1 over Δ1 with s1R pt q “ x . and s t1 P Δ1˚ 0 1 1 1 R R With the notation as above the statement of the proposition, let ` ˘ a11 “ deg V0 |P10 ´a0 {2, a12 “ deg V0 |P1` . a0 “ k`2l´1,

Base Cases

203

For the purposes of applying Lemma 8.6, we choose pVr , ϕq r and D” tDt u so that pVrt0 , ϕ rt0 q “ pV0 , ϕ0 q

and

Dt0 “ D0 .

For each t P Δ1R , let ox1 ;t be the orientation of Vr ϕr at s1R 1 ptq induced 1R˚ ϕ r r by ox1 via the line bundle s1 V . By definition and the evenness of the (real) dimension of Vrnd` , the restriction of the orientation 1 r r ox ;t q of the line bundle λp Dq over Δ1 to λpDt q is the o1 pVr , ϕ; k,l

1

R

1

0

pk, lq-split orientation o0 pox1 q in (8.10). By Lemma 8.6, the restriction of r o1k,l pVr , ϕ; r ox1 ;t1 q to λpDt q for each t P Δ1˚ R is the pk, lq-intrinsic 1 1 r ; ox ;t q. Thus, the restriction of r o1 pVr , ϕ; r ox ;t q orientation ok,l pVr , ϕ t

t

1

k,l

1

1

to λpDt0 q is the limiting orientation o10 pox1 q in (8.11). We conclude  that the orientations (8.10) and (8.11) are the same. Proof of Lemma 8.4(2). Suppose l, r l P Zě0 with r l ě l. Let C0 , 1 1 P0 , P` , p U,rcq, ΔR , and t1 be as above Proposition 8.2 and sR 1 be a rc R r section of U over ΔR with s1 pt1 q “ x1 . We can choose pV , ϕq r and D” tDt u as above Proposition 8.2 so that ˇ ˇ ` ˘ l´1, deg Vr ˇP1 “ deg V ´deg Vr |P10 {2, deg Vr ˇP1 “ k`2r 0 ` ˘ ` rt1 “ pV, ϕq, Dt1 “ D . Vrt1 , ϕ For each t P ΔR , let ox1 ;t be the orientation of Vr ϕr at sR 1 ptq induced R˚ ϕ r r by ox1 via the line bundle s1 V . Since r l ě l, the definitions of the pk, lq- and pk, r lq-split orientations of D0 and Lemma 8.4p1q imply that ` ˘ ` ˘ r0 ; ox1 ;0 “ ok,rl Vr0 , ϕ r0 ; ox1 ;0 . ok,l Vr0 , ϕ Along with Proposition 8.5, this in turn implies that the corresponding limiting orientations of D0 are equal, i.e. ` ˘ ` ˘ r0 ; ox1 ;0 “ o1k,rl Vr0 , ϕ r0 ; ox1 ;0 . o1k,l Vr0 , ϕ rt ; ox1 ;t q and Thus, the pk, lq- and pk, r lq-intrinsic orientations ok,l pVrt , ϕ ˚ r rt ; ox1 ;t q of Dt with t P ΔR are the same. Taking t “ t1 , we ok,rl pVt , ϕ  obtain the claim.

204

Spin/Pin-Structures & Real Enumerative Geometry

Figure 8.2. Deformations of C01 in the family p U1 ,rc1 q over Δ1 Ă C6 of the proof of Lemma 8.6.

Proof of Lemma 8.6. Let C01 ” pΣ10 , σ01 q be a connected symmetric surface which contains precisely three conjugate pairs of nodes and consists of one invariant component P10 and three pairs of conjugate components, all of which are isomorphic to S 2 , so that one pair of the conjugate components is attached to P10 and the other two pairs are attached to the first pair; see the bottom diagram in Figure 8.2. We 2 denote by D2` , D2´ Ă P10 the two half-surfaces, by nd` 0 P D` the node 2 1 1 on D` , by P0` Ă Σ0 the other irreducible component containing nd` 0, ` 1 the other nodes on P1 , and by P1 and P1 the , nd P P by nd` 0` 0` 1` 2` 1 2 ` and nd , respectively. other irreducible components containing nd` 1 2 Let p U1 ,rc1 q be a flat family of deformations of C01 as in (7.32) over the unit ball Δ1 Ă C6 around the origin satisfying (7.52) for some t1 P Δ1R so that ` ` ˘ ˘ Ct1 4 ” Σ1t4 , σt1 4 Ct1 2 ” Σ1t2 , σt1 2 and for some t4 , t2 P Δ1R are a five-component decorated symmetric surface as on the left side of Figure 8.1 obtained from C01 by smoothing

Base Cases

205

the conjugate pair of nodes on P10 and a three-component decorated symmetric surface as on the left side of Figure 7.4 obtained from C01 by smoothing the other two pairs of nodes, respectively; see Figure 8.2. We denote by 1˚ 1 Δ1˚ R;4 , ΔR;2 Ă ΔR

the subspaces parametrizing all five-component symmetric surfaces obtained from C01 by smoothing the conjugate pair of nodes on P10 and all three-component symmetric surfaces obtained from C01 by smoothing the other two pairs of nodes, respectively. Let s1R 1 be a 1 pt q “ x . section of U1rc over Δ1R with s1R 1 1 1 r1 q be a rank 1 real bundle pair over p U1 ,rc1 q so that Let pVr 1 , ϕ ˇ ˇ ˇ deg Vr 1 ˇP1 “ a0 , deg Vr 1 ˇP1 “ 0, deg Vr 1 ˇP1 “ a11 , 0 0` 1` ˇ ` 1 1 ˘ 1 1 rt1 “ pV, ϕq. deg Vr ˇP1 “ a2 , Vrt1 , ϕ 2`

1 For each t P Δ1R , let ox1 ;t be the orientation of Vr 1ϕr at s1R 1 ptq induced 1 1R˚ 1 ϕ r r by ox1 via the line bundle s1 V . r1t q as Let D1 ” tDt1 u be a family of real CR-operators on pVrt1 , ϕ 1 in (7.34) so that Dt1 is the operator D in the statement of the lemma. 1 ,ϕ 1 r100 q the restriction of pVr 1 , ϕ r1 q to P10 and by D00 We denote by pVr00 1 ,ϕ the restriction of D01 to a real CR-operator on pVr00 r100 q. For r “ 1` 1 0, 1, 2, let Dr be the restriction of D0 to a real CR-operator on Vr0 |P1r` . The exact triple

1 1 r 1 ` ‘ Vr 1 ` ÝÑ 0, ‘D01` ‘D11` ‘D21` ÝÑ Vrnd 0 ÝÑ D01 ÝÑ D00 ` ‘V nd1 nd2 0 ˘ ` ˘ ` ξ2´ , ξ1´ , ξ0´ , ξ0 , ξ0` , ξ1` , ξ2` ÝÑ ξ0 , ξ0` , ξ1` , ξ2` , ˘ ` ` ` ξ0 , ξ0` , ξ1` , ξ2` ÝÑ ξ0` pnd` 0 q´ξ0 pnd0 q, ˘ ` ` ` ξ1` pnd` 1 q´ξ0` pnd1 q, ξ2` pnd2 q´ξ0` pnd2 q

of Fredholm operators then determines an isomorphism ` ˘ ` ˘ ` ˘ λpD 1 qbλ Vr 1 ` bλ Vr 1 ` bλ Vr 1 ` 0

`

nd0

1 « λ D00

˘

nd1

nd2

` ˘ ` ˘ ` ˘ bλ D01` bλ D11` bλ D21` .

(8.16)

206

Spin/Pin-Structures & Real Enumerative Geometry

1 and the complex The pk, lq-evaluation orientation ok,l pox1 ;0 q of D00 1 , V r 1 ` , and Vr 1 ` determine an orientations of D01` , D11` , D21` , Vrnd ` nd nd 0

1

2

orientation of D01 via the isomorphism (8.16) and thus an orientation o U1 pox1 q of the line bundle λp D1 q over Δ1R . The restriction of p U1 ,rc1 q to Δ1R;4 is the product of two conjugates pairs of P1 with a family smoothing P10;0 ” P10´ YP10 YP10`

into a single irreducible component P1t;0 Ă Σ1t as above Proposition 8.2; see the top left diagram in Figure 8.2. The other irreducible components of Σ1t with t P Δ1˚ R;4 and its nodes correspond to the conjugate components and conjugate nodes of Σ10 ; we denote them 1 by P1t;r˘ and nd˘ t;r , respectively. For each t P ΔR;4 , we denote the 1 ,ϕ r1t q to P1t;0 by pVrt;0 r1t;0 q and the restrictions of restriction of pVrt1 , ϕ D 1 to real CR-operators on pVr 1 , ϕ r1 q, Vr 1 | 1 , and Vr0 | 1 by D 1 , t 1` , Dt1

t;0

1` and Dt2 , ators with D0

t;0

t Pt;1`

Pt;2`

t;0

respectively. The exact triple (7.55) of Fredholm oper1 replaced by D0;0 and the exact triple (8.14) of Fredholm operators with D0 replaced by Dt1 4 induce isomorphisms ` 1 ˘ ` 1` ˘ ` 1 ˘ ` 1 ˘ bλ Vrnd` « λ D00 bλ D0 , λ D0;0 0

` 1 ˘ ` 1 ˘ ` ˘ ` ˘ ` ˘ bλ Vrnd` « λ Dt1 4 ;0 bλ Dt1`4 1 bλ Dt1`4 2 . λpDt1 4 qbλ Vrnd ` t4 ;1

t4 ;2

(8.17) 1 and the complex The pk, lq-evaluation orientation ok,l pox1 ;0 q of D00 1 1 orientations of D01` and Vrnd ` determine an orientation on D0;0 via 0

1 for the first isomorphism in (8.17) and thus an orientation of Dt;0 1 each t P ΔR;4 . By definition, the latter with t “ t4 is the pk, lq-intrinsic r1t4 ;0 ; ox1 ;t4 q. By Lemma 8.4(1), this orientation orientation ok,l pVrt14 ;0 , ϕ is the same as the pk, lq-evaluation orientation ok,l pox1 ;t4 q. Since the 1 (real) dimension of Vrnd ` is even, this implies that the second isomor0

phism in (8.17) is orientation-preserving with respect to the restriction of the orientation o U1 pox1 q to λpDt1 4 q, the orientation ok,l pox1 ;t4 q of Dt1 4 ;0 , and the complex orientations of the remaining factors.

Base Cases

207

The restriction of p U1 ,rc1 q to Δ1R;2 is the product of pP10 , τ q with a family smoothing the conjugate pair P10;˘ ” P10˘ Y P11˘ Y P12˘ Ă Σ10 into a conjugate pair of irreducible components P1t;˘ Ă Σ1t ; see the last diagram in Figure 8.2. The other irreducible component of Σ1t with t P Δ˚R;2 and its nodes correspond to P10 and the nodes nd˘ 0; ˘ 1 1 we denote them by Pt0 and ndt , respectively. For each t P ΔR;2 , 1 ,ϕ r1t q to P1t0 by pVrt0 r1t0 q and the we denote the restriction of pVrt1 , ϕ 1 1 1 r q and Vr 1 | 1 by restrictions of D to real CR-operators on pVr , ϕ 1 Dt0

and

1` Dt;` ,

t

t0

t0

t Pt;`

respectively. The exact triple

1 1 r 1 ` ÝÑ 0, ÝÑ D01` ‘D11` ‘D21` ÝÑ Vrnd 0 ÝÑ D0;` ` ‘V nd

`

ξ0` , ξ1` , ξ2`

˘

1

2

˘ ` ` ` ` ÝÑ ξ1` pnd` 1 q´ξ0` pnd1 q, ξ2` pnd2 q´ξ0` pnd2 q

of Fredholm operators and the exact triple (7.55) of Fredholm operators with D0 replaced by Dt1 2 induce isomorphisms ` ˘ ` ˘ ` ˘ ` 1 ˘ ` 1 ˘ ` 1 ˘ bλ Vrnd` bλ Vrnd` « λ D01` bλ D11` bλ D21` , λ D0;` 1 2 ` 1 ˘ ` ˘ ` ˘ 1 r (8.18) λpDt2 qbλ Vnd` « λ Dt1 2 0 bλ Dt1 2 ;` . t2

1 The complex orientations of Dr1` for r “ 0, 1, 2 and Vrnd ` for r “ r 1 1, 2 determine an orientation of D0;` via the first isomorphism 1 for each t P Δ1R;2. The latin (8.18) and thus an orientation of Dt;` 1 ter is the complex orientation of Dt;` . This implies that the second isomorphism in (8.18) is orientation-preserving with respect to the restriction of the orientation o U1 pox1 q to λpDt1 2 q, the pk, lq-evaluation orientation ok,l pox1 ;t2 q of Dt1 2 0 , and the complex orientations of the remaining factors. By the conclusion above regarding the second isomorphism in (8.17), the restriction of the orientation o U1 pox1 q to λpDt1 1 q is the orientation o1k,l pV, ϕ; ox1 q constructed above Lemma 8.6. By the conclusion above regarding the second isomorphism in (8.18), the restriction of the orientation o U1 pox1 q to λpDt1 1 q is the pk, lq-intrinsic orientation ok,l pV, ϕ; ox1 q constructed above Proposition 8.2. Thus,  the orientations ok,l pV, ϕ; ox1 q and o1k,l pV, ϕ; ox1 q are the same.

208

8.3

Spin/Pin-Structures & Real Enumerative Geometry

Line Bundles over H3 Degenerations of (S 2 , τ )

We next describe the behavior of the orientations of Proposition 8.2 under flat degenerations of pS 2 , τ q as in the CROrient 7H3 property on page 123. Let C0 ” pΣ0 , σ0 q be a connected decorated symmetric surface as above (7.58) and in the right diagram in Figure 7.4 and nd, P11 , P12 , D21˘ D22˘ , and S11 , S21 be as above (7.58). Suppose k1 , l1 , k2 , l2 P Zě0 and pV0 , ϕ0 q is a rank 1 real bundle pair over pΣ0 , σ0 q such that k1 ` k2 ‰ 0, k1 ` 2l1 – deg V0 |P11 ` 1 mod 2, k2 ` 2l2 – deg V0 |P12 ` 1 mod 2 .

(8.19)

For r “ 1, 2, we denote the restriction of pV0 , ϕ0 q to P1r by pVr , ϕr q. Let ond be an orientation of V0ϕ0 |nd . Let D0 be a real CR-operator on pV0 , ϕ0 q. We denote its restrictions to real CR-operators on pV1 , ϕ1 q and pV2 , ϕ2 q by D1 and D2 , respectively. The orientation ond determines orientations ` ˘ ` ˘ ` ˘ ` ˘ and o0;2 ond ” ok2 ,l2 V2 , ϕ2 ; ond o0;1 ond ” ok1 ,l1 V1 , ϕ1 ; ond (8.20) of D1 and D2 as above Proposition 8.2. These two orientations and the orientation ond induce an orientation ` ˘ (8.21) o0 pond q ” opk1 ,l1 q,pk2 ,l2 q V0 , ϕ0 ; ond of D0 via the isomorphism (7.61). We call it the split orientation of D0 . Suppose in addition (7.32) is a flat family of deformations of C0 , pV, ϕq is a real bundle pair over p U,rcq extending pV0 , ϕ0 q, sR 1 is a section of Urc over ΔR with 1 sR 1 p0q P S1 ´tndu,

and D” tDt u is a family of real CR-operators on pVt , ϕt q as in (7.34) extending D0 . The decorated structure on C0 induces a decorated structure on the fiber pΣt , σt q of π for every t P ΔR and determines open subspaces Δ˘ R Ă ΔR as above the CROrient 7H3 property. The orientation ond determines an orientation ond;0 of V ϕ at sR 1 p0q by translation along the positive direction of S11 Ă D21` and thus an

Base Cases

209

orientation ond;t of V ϕ at sR 1 ptq for every t P ΔR via the line bundle R˚ ϕ ˚ s1 V . For each t P ΔR , let ` ˘ ot pond q ” ok1 `k2 ´1,l1 `l2 Vt , ϕt ; ond;t (8.22) be the intrinsic orientation of Dt as above Proposition 8.2. We denote by ` ˘ ` (8.23) o` 0 pond q ” ok1 `k2 ´1,l1 `l2 V0 , ϕ0 ; ond the orientation of D0 obtained as the continuous extension of the orientations (8.22) with t P Δ` R . We call (8.23) the limiting orientation of D0 . Proposition 8.7. Suppose C0 is a connected decorated symmetric surface consisting of two copies of pS 2 , τ q and one node nd and pV0 , ϕ0 q is a rank 1 real bundle pair over pΣ0 , σ0 q. Let k1 , l1 , k2 , l2 P Zě0 be so that p8.19q holds and D0 be a real CR-operator on pV0 , ϕ0 q. The split and limiting orientations, p8.21q and p8.23q, of D0 are the same if and only if ˘ ` (8.24) degpV0 |P11 q`1 degpV0 |P12 q P 2Z. Proof of Proposition 8.2(3). In light of Lemma 8.4(2), it remains to show that the orientation ok,l pV, ϕ; ox1 q of D does not depend on the choice of k P Zě0 satisfying (8.2). We assume that k ě 2 and show in the following that the orientations ok,l pV, ϕ; ox1 q and ok´2,l`1 pV, ϕ; ox1 q are the same. We can choose C0 , p U,rcq, sR 1 , pV, ϕq, and D ” tDt u as above Proposition 8.7 and just above so that (7.52) holds for some t1 P Δ` R, pt q “ x , sR 1 1 1 pV2 , ϕ2 q ” pV, ϕq|P12 is a degree 2 rank 1 real bundle pair, pVt1 , ϕt1 q is the rank 1 real bundle pair in the statement of Proposition 8.2, and Dt1 “ D. With the notation as before, let ond be the orientation of V0ϕ0 |nd so that ond;t1 “ ox1 . By Proposition 8.7, the orientations ` ˘ ` ˘ ` and o` o0 pond q ” opk´2,lq,p3,0q V0 , ϕ0 ; ond 0 pndq ” ok,l V0 , ϕ0 ; ond (8.25)

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of D0 are the same if and only if the orientations ` ˘ o10 pond q ” opk´2,lq,p1,1q V0 , ϕ0 ; ond and ` ˘ ` o1` 0 pond q ” ok´2,l`1 V0 , ϕ0 ; ond

(8.26)

of D0 are the same. By Proposition 8.2(2) and Lemma 8.3, ` ˘ ` ˘ o3,0 V2 , ϕ2 ; ond “ o1,1 V2 , ϕ2 ; ond . Thus, the split orientations o0 pndq and o10 pndq above are the same. Along with (8.25) and (8.26), this implies that the limiting orienta1` tions o` 0 pndq and o0 pndq of D0 are also the same and so ` ˘ ` ˘ @ t P Δ` ok,l Vt , ϕt ; ond;t “ ok´2,l`1 Vt , ϕt ; ond;t R. Taking t “ t1 , we obtain the claim.



Lemma 8.8. Proposition 8.7 holds if the two congruences in p8.19q are equalities. Proof of Proposition 8.7. k “ k1`k2´1,

l “ l1`l2 ,

By (8.19), the numbers ` ˘L a11 ” deg V0 |P11 ´a1 2

Let

a1 “ k1`2l1´1,

and

a2 “ k2`2l2´1. (8.27)

` ˘L a12 ” deg V0 |P12 ´a2 2

are integers. Let C01 ” pΣ10 , σ01 q be a connected decorated symmetric surface consisting of two copies, C10 and C20 , of the three-component curve as on the left side of Figure 7.4 joined at a real node nd0 ; see the bottom diagram in Figure 8.3. For r “ 1, 2, we denote by P1r0 Ă Σr0 the irreducible component in the r-th copy preserved by the 2 involution σ01 , by D2r` , D2r´ Ă P1r0 the two half-surfaces, by nd` r P Dr` 2 1 1 the node on Dr` , and by Pr` the conjugate component of Σ0 also containing nd` r . Let p U1 ,rc1 q be a flat family of deformations of C01 as in (7.32) over the unit ball Δ1 Ă C5 around the origin so that Ct1 0 ” pΣ1t0 , σt1 0 q for some t0 P Δ1R is the two-component decorated curve C0 of 1 Proposition 8.7. Let Δ1˚ R Ă ΔR be the subspace parametrizing smooth

Base Cases

211

Figure 8.3. Deformations of C01 in the family p U1 ,rc1 q over Δ1 Ă C5 of the proof of Lemma 8.6. 1´ 1˚ symmetric surfaces and Δ1` R , ΔR Ă ΔR be its two topological com1 ponents distinguished as above (7.40). We denote by Δ1˚ R;1 Ă ΔR and 1˚ 1 ΔR;4 Ă ΔR the subspaces parametrizing symmetric surfaces with one real node and two pairs of conjugate nodes, respectively, and by Δ1R;1 1rc1 1 and Δ1R;4 their closures. Let s1R 1 be a section of U over ΔR so that 1 1 1R 1 s1R 1 ptq is not the real node of Σt for all t P ΔR;1 and s1 pt0 q P S1 . r1 q over p U1 ,rc1 q We can choose a rank 1 real bundle pair pVr 1 , ϕ so that ˇ ˇ deg Vr 1 ˇP1 “ a2 , deg Vr 1 ˇP1 “ a1 , ` 1 1 ˘ Vrt0 , ϕ rt0 “ pV0 , ϕ0 q. ˇ 10 ˇ 20 1 1 1 1 deg Vr ˇ 1 “ a2 , deg Vr ˇ 1 “ a1 , P1`

P2`

The orientation ond of V0ϕ0 at the node nd of Σ1t0 induces an orienta1 tion ondt of Vr 1ϕr at the real node ndt of each symmetric surface Σ1t with t P Δ1R;1 . It also determines an orientation of the line bundle r 1ϕr1 over Δ1 as above Proposition 8.7. We denote the restriction s1R˚ 1 V R of this orientation to the fiber over t P Δ1R by ond;t . r1t q as Let D1 ” tDt1 u be a family of real CR-operators on pVrt1 , ϕ 1 in (7.34) so that Dt0 is the operator D0 in the statement of the

Spin/Pin-Structures & Real Enumerative Geometry

212

proposition. For r “ 1, 2, we denote the restriction of pVr01 , ϕ r10 q to 1 1 1 1 Pr0 by pVr0r , ϕ0r q and the restrictions of D0 to real CR-operators on 1 , ϕ1 q and V r 1 | 1 by D 1 and Dr1` , respectively. The exact triple pVr0r 0r 0 Pr` 0r 1ϕ r1 1 1 1 r 1 ` ÝÑ 0, ‘D02 ‘D11` ‘D21` ÝÑ Vrnd ‘ Vrnd 0 ÝÑ D01 ÝÑ D01 ` ‘V 0 nd2 1 ` ˘ ` ˘ ξ1´ , ξ01 , ξ1` , ξ2´ , ξ02 , ξ2` ÝÑ ξ01 , ξ1` , ξ02 , ξ2` , ˘ ` ` ` ξ01 , ξ1` , ξ02 , ξ2` ÝÑ ξ02 pnd0 q´ξ01 pnd0 q, ξ1` pnd` 1 q´ξ01 pnd1 q, ˘ ` ξ2` pnd` 2 q´ξ02 pnd2 q

of Fredholm operators then determines an isomorphism ` 1 ˘ ` ˘ ` 1ϕr1 ˘ r ` b λ Vr 1 ` b λ V λpD01 q b λ Vrnd 0 nd nd ` 1 ˘ 1 ` 1` ˘ 2 ` 1` ˘ ` 1 ˘ « λ D01 b λ D02 b λ D1 b λ D2 . 1 and o 1 The evaluation orientations ok1 ,l1 pond0 q of D01 k2 ,l2 pond0 q of D02 , 1 1ϕ r , and the complex orientations of D11` , the orientation ond0 of Vrnd 0 1 , and V r 1 ` determine an orientation of D 1 via the above isoD21` , Vrnd ` 0 nd 1

2

morphism and thus an orientation o U1 pond q of the line bundle λp D1 q over Δ1R . 1` 1˚ 1˚ Let Δ1` R;4 Ă ΔR;4 be the intersection of ΔR;4 with the closure of ΔR 1 1 1 1 in ΔR . The restriction of p U ,rc q to ΔR;4 is the product of two conjugate pairs of P1 with a family of symmetric surfaces as above Proposition 8.7 smoothing P10;0 ” P110 Ynd0 P120 into a single irreducible component P1t;0 Ă Σ1t for each t P Δ1˚ R;4 ; see the top left diagram in Figure 8.3. The other irreducible components of Σ1t with t P Δ1˚ R;4 and its nodes correspond to the conjugate components and conjugate nodes of Σ10 ; we denote them by P1t;r˘ 1˚ and nd˘ t;r , respectively. For each t P ΔR;4 , we denote the restriction of 1 , ϕ1 q and the restrictions of D 1 to real CRr1t q to P1t;0 by pVrt;0 pVrt1 , ϕ t t;0 operators on pVr 1 , ϕ1 q Vr 1 | 1 by D 1 and D 1` , respectively. The t;0

t;0

t Pt;r`

t;0

t;r

1 exact triple (7.60) of Fredholm operators with D0 replaced by D0;0

Base Cases

213

and the exact triple (8.14) of Fredholm operators with D0 replaced by Dt1 with t P Δ1˚ R;4 induce isomorphisms ` 1 ˘ ` 1ϕr1 ˘ ` 1 ˘ ` 1 ˘ λ D0;0 bλ Vrnd0 « λ D01 bλ D02 , ˘ ` ˘ ` 1 ˘ ` 1` ˘ ` 1` ˘ (8.28) ` bλ Dt;1 bλ Dt;2 , λpDt1 qbλ Vrnd` bλ Vrnd` « λ Dt;0 t;1

t;2

1 , o 1 respectively. The orientations ok1 ,l1 pond0 q of D01 k2 ,l2 pond0 q of D02 , 1 1 ϕ r 1 via the first isoand ond0 of Vrnd0 determine an orientation of D0;0 1 1 for each morphism in (8.28) and thus an orientation ot;0 pond q of Dt;0 1˚ t P ΔR;4 . Let t P Δ1` R;4 . By Proposition 8.2(2) and Lemma 8.8, the last orientation is the same as the evaluation orientation ok,l pond;t q if and only if (8.24) holds. Since the indices of D11` and D21` and the (real) 1 r 1 ` are even, this implies that the second dimensions of Vrnd ` and V nd 1

2

isomorphism in (8.28) with t P Δ1` R;4 is orientation-preserving with respect to the restriction of the orientation o U1 pond q to λpDt1 q, the 1 , and the complex orientations of the orientation ok,l pond;t q of Dt;0 remaining factors if and only if (8.24) holds. Combining this with the definition of the limiting orientation (8.23), we conclude that the restriction of the orientation o U1 pond q to λpDt1 0 q is the limiting orientation (8.23) if and only if (8.24) holds. The restriction of p U1 ,rc1 q to Δ1R;1 is the product of families of symmetric surfaces as above Proposition 8.2 smoothing the threecomponent curves P10;1 ” Σ10 ” P11´ YP110 YP11`

and P10;2 ” Σ20 ” P12´ YP120 YP12`

into irreducible components P1t;1 Ă Σ1t and P1t;2 Ă Σ1t , respectively; see the right diagram in Figure 8.3. For t P Δ1R;1 and r “ 1, 2, we 1 , ϕ1 q and the restricr1t q to P1t;r by pVrt;r denote the restriction of pVrt1 , ϕ t;r 1 1 1 1 . The exact r tion of Dt to a real CR-operator on pVt;r , ϕt;r q by Dt;r 1 for triple (7.55) of Fredholm operators with D0 replaced by D0;r r “ 1, 2 and the exact triple (7.60) of Fredholm operators with D0 replaced by Dt1 with t P Δ1˚ R;1 induce isomorphisms ` 1 ˘ ` 1` ˘ ` 1 ˘ ` 1 ˘ bλ Vrnd` « λ D0r bλ Dr , λ D0;r r ` ϕr ˘ ` 1 ˘ ` 1 ˘ « λ Dt;1 bλ Dt;2 , (8.29) λpDt1 qbλ Vrnd t

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Spin/Pin-Structures & Real Enumerative Geometry

1 and the respectively. The evaluation orientation okr ,lr pond0 q of D0r 1 complex orientations of Dr1` and Vrnd ` determine an orientation r 1 on D0;r via the first isomorphism in (8.29) and thus an orienta1 for each t P Δ1˚ tion ot;r pond q on Dt;r R;1 . By the definition of the pkr , lr q-intrinsic orientation above Proposition 8.2, ` ˘ ` 1 ˘ ,ϕ r1t;r ; ondt . ot;r ond “ okr ,lr Vrt;r 1 r 1 ` are even, this implies that the Since the dimensions of Vrnd ` and V nd 1

2

second isomorphism in (8.29) with t P Δ1˚ R;4 is orientation-preserving with respect to the restriction of the orientation o U1 pond q to λpDt1 q, 1 , and the orithe intrinsic orientations okr ,lr pVrt;r , ϕ rt;r ; ondt q of Dt;r ϕ r . Combining the t “ t0 case of this statement entation ondt of Vrnd t with the definition of the split orientation above Proposition 8.7, we conclude that the restriction of the orientation o U1 pond q to λpDt1 0 q is the split orientation (8.21). The conclusions of the last two paragraphs establish Proposi tion 8.7. Proof of Lemma 8.8. We continue with the notation in (8.27) and above Proposition 8.7. In this case, a1 “ deg V1

and

a2 “ deg V2 .

Since a1 , a2 , a1 `a2 ě ´1, the real CR-operators D1 , D2 , D0 , and Dt with t P Δ˚R are surjective and (7.61) reduces to ` ˘ ` ˘ ` ˘ (8.30) λpker D0 qbλ V0ϕ0 |nd « λ ker D1 bλ ker D2 . By definition of the split orientation, this isomorphism respects the orientations o0 pond q of ker D0 , ond of V0ϕ0 |nd , o0;1 pond q of ker D1 , and o0;2 pond q of ker D2 . ` rc R R Let s` 1 , . . . , sl be disjoint sections of U´ U and s1 , . . . , sk`1 be disjoint sections of Urc over ΔR so that 2 s` i p0q P D1` @ i P rl1 s, 2 s` l1 `i p0q P D2` @ i P rl2 s,

1 xi ” s R i p0q P S1 ´tndu @ i P rk1 s, 1 xi ” s R i p0q P S2 ´tndu @ i P rk`1s ‰ “ ´ maxpk1 , 1q ,

Base Cases

215

the points nd, x1 , . . . , xk1 of S11 are ordered by position with respect to D21` , and the points nd, xk1 `1 , . . . , xk`1 of S21 are ordered by position with respect to D22` . We denote the complex orientations of V1C ”

l1 à i“1

ˇ V0 ˇs` p0q

and

i

V2C ”

l2 à i“1

ˇ V0 ˇs`

l1 `i p0q

C by oC 1 and o2 , respectively. By Proposition 8.2(2), the orientations o0;1 pond q, o0;2 pond q, and o` 0 pond q in (8.20) and (8.23) agree with the orientations obtained via evaluation homomorphisms as in (7.50) at ` ‚ k1 points of S11 and s` 1 p0q, . . . , sl1 p0q, ` ‚ k2 points of S21 and s` l1 `1 p0q, . . . , sl p0q, and σt ` ` 1 ‚ k points of St ” Σt and s1 ptq, . . . , s` l ptq for t P ΔR ,

respectively. Suppose a2 P 2Z. Thus, k2 R 2Z and V0ϕ0 is orientable along S21 . R Let oR 1 and o2 denote the orientations of V1R ”

k1 à ˇ V0ϕ0 ˇx i“1

i

and

V2R ”

k2 à ˇ V0ϕ0 ˇx i“2

k1 `i

(8.31)

obtained by translating the orientation ond;0 of V0ϕ0 |x1 (or equivalently ond of V0ϕ0 |nd ) along the positive direction of S11 and by translating ond along either direction of S21 , respectively. The evaluation homomorphisms ˘ ` ˘ ` C C˘ ` ker D1 , o0;1 pond q ÝÑ V1R , oR 1 ‘ V1 , o1 , ˘ ` ˘ ` ϕ0 ˘ ` C C˘ ` ker D2 , o0;2 pond q ÝÑ V2R , oR 2 ‘ V0 |nd , ond ‘ V2 , o2 , ˘ ` R R˘ ` R R˘ ` C C˘ ` C C˘ ` ker D0 , o` 0 pond q ÝÑ V1 , o1 ‘ V2 , o2 ‘ V1 , o1 ‘ V2 , o2 are then orientation-preserving isomorphisms (since the number of summands in the second sum in (8.31) is even, the orientation of oR 2 in fact does not depend on the choice of ond ). Since the real dimension of V2C is even, this and the sentence following (8.30) imply that o` 0 pond q “ o0 pond q.

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Spin/Pin-Structures & Real Enumerative Geometry

Suppose a2 R 2Z. Thus, a1 – k mod 2, k2 P 2Z, and the orientation o2 pond q does not depend on the choice of ond by Lemma 8.1(1). If R k1 ‰ 0, let oR 1 and o2 denote the orientations of V1R ”

k1 à ˇ V0ϕ0 ˇx i“2

i

and

V2R ”

k2 à i“1

ˇ V0ϕ0 ˇx

k1 `i

obtained by translating ond along the positive direction of S11 and the positive direction of S21 , respectively. The evaluation homomorphisms ` ˘ ` ˘ ` C C˘ ˘ ` ker D1 , o0;1 pond q ÝÑ V0ϕ0 |nd , ond ‘ V1R , oR 1 ‘ V1 , o1 , ˘ ` ˘ ` C C˘ ` ker D2 , o0;2 pond q ÝÑ V2R , oR 2 ‘ V2 , o2 , ˘ ` R R˘ ` R R˘ ` C C˘ ` C C˘ ` ker D0 , o` 0 pond q ÝÑ V1 , o1 ‘ V2 , o2 ‘ V1 , o1 ‘ V2 , o2 are then orientation-preserving isomorphisms. Combining this with the sentence following (8.30) and taking into account the negative sign in (7.60), we conclude that the orientations o` 0 pond q and o0 pond q of D0 are the opposite if k P 2Z (and thus a1 P 2Z) and the same if k R 2Z (and thus a1 R 2Z). If a2 R 2Z and k1 “ 0, then a1 R 2Z and the orientation of V0ϕ0 |nd obtained by translating ox1 pond q along the positive direction of S11 is the opposite ond of ond . We take V2R as in (8.31) and denote by R R oR 2 pond q and o2 pond q the orientations of V2 obtained by translating ond and ond , respectively, along the positive direction of S21 . The evaluation homomorphisms ˘ ` ˘ ` ker D1 , o0;1 pond q ÝÑ V1C , oC 1 , ˘ ` ˘ ` C C˘ ˘ ` ` ker D2 , o0;2 pond q ÝÑ V0ϕ0 |nd , ond ‘ V2R , oR 2 pond q ‘ V2 , o2 , ˘ ` R R ˘ ` C C˘ ` C C˘ ` ker D0 , o` 0 pond q ÝÑ V2 , o2 pond q ‘ V1 , o1 ‘ V2 , o2 are then orientation-preserving isomorphisms. Since the real dimension of V2C is even, this and the sentence following (8.30) imply that  o` 0 pond q “ o0 pond q. 8.4

Even-Degree Bundles over pS 2 , τ q: Construction and Properties

Let pV, ϕq be a real bundle pair over pS 2 , τ q and os P OSpinS 2 pV ϕ q be a relative OSpin-structure on the real vector bundle V ϕ over S 1 Ă S 2 .

Base Cases

217

This implies that V ϕ is orientable and thus the degree of V is even; see [4, Proposition 4.1]. We show in the following that os determines an orientation oos pV, ϕq of a real CR-operator D on pV, ϕq. Let D2` Ă S 2 be a half-surface of pS 2 , τ q as before and C0 , p U,rcq, and t1 P Δ˚R be as in and above (7.52). We first suppose that n “ rk V ě 3. By Definition 6.3, os and the embedding of D2` into S 2 thus determine a homotopy class ospD2` q of isomorphisms V ϕ « S 1 ˆ Rn . A rank n real bundle pair pVr , ϕq r over p U,rcq so that ` ˘ˇ ˘ ` 1 ˘ ` Vr , ϕ r ˇP1 “ P0 ˆCn , τ ˆc and Vrt1 , ϕ rt1 “ pV, ϕq 0

determines a homotopy class of trivializations of V ϕ from the canonical trivialization of Vr ϕr |Σσ0 . Choose such pVr , ϕq r so that this homotopy 0 2 class is ospD` q. Such a real bundle pair can be obtained by extending a trivialization of V ϕ in the homotopy class ospD2` q to a trivialization of V over a τ -invariant tubular neighborhood U of S 1 Ă S 2 and then collapsing two circles in U interchanged by τ to the nodes of Σ0 ; see the proof of [13, Proposition 3.1]. rt q as Let D“ tDt u be a family of real CR-operators on pVrt , ϕ in (7.34) so that Dt1 “ D and the real CR-operator D00 on ¯ pP10 ˆCn , τ ˆcq induced by restricting D0 is the standard B-operator. The exact triple (7.55) of Fredholm operators then induces an isomorphism as in (7.56). The canonical identification of the kernel of the surjective operator D00 with Rn , the complex orientation of the real CR-operator D0` on Vr |P1` induced by D0 , and the complex orir0 q on λpD0 q via entation of Vrnd` determine an orientation oos pVr0 , ϕ this isomorphism and thus an orientation on the line bundle λp Dq over ΔR . The latter restricts to an orientation oos pV, ϕq of λpDq. If n “ 2 and o is an orientation of V ϕ , the group π1 pS 1 q « Z acts freely and transitively on the homotopy classes of trivializations of pV ϕ , oq. The 2Z-orbits of this action are the collections of these homotopy classes that induce the same trivializations of the stabilization StpV ϕ q ” τS 1 ‘V ϕ ÝÑ S 1 . A relative OSpin-structure os and D2` thus determine a collection ospD2` q of homotopy classes of isomorphisms V ϕ « S 1 ˆ R2 . Taking

218

Spin/Pin-Structures & Real Enumerative Geometry

any of these homotopy classes and proceeding as in the n ě 3 case, we obtain an orientation oos pV, ϕq of D. By Corollary 8.14 in Section 8.5, oos pV, ϕq does not depend on the homotopy class in the collection ospD2` q If n “ 1, the orientation o of the real vector bundle St2 pV ϕ q ” τS 1 ‘τS 1 ‘V ϕ ÝÑ S 1 in the pair ospD2` q ” po, sq determines a homotopy class sos pV ϕ q of trivializations of the real line bundle V ϕ over S 1 , which does not depend on the choice of D2` . Let x1 P S 1 and ox1 be the orientation of Vxϕ1 determined by o. The above construction for the pair ˘ ` q “ o, sos pV ϕ q (8.32) os determines the intrinsic orientation ˘ ` ˘ ` o V, ϕ; ox1 ” o1,0 V, ϕ; ox1 of Section 8.1. We take # q opV, ϕ; ox1 q, if ospD2` q “ St2V ϕ posq; oos pV, ϕq “ 2 2 q opV, ϕ; ox1 q, if ospD` q ‰ StV ϕ posq.

(8.33)

In particular, oos pV, ϕq satisfies the property in Proposition 8.9(1). Along with Proposition 8.2(2), this implies that it also satisfies the CROrient 6(a) property on page 117. Suppose pV, ϕq is a rank 1 odd-degree real bundle pair over pS 2 , τ q. In particular, the real line bundle V ϕ over S 1 Ă S 2 is not orientable. Let D be a real CR-operator on pV, ϕq. We denote by ` ˘ ` ˘ ˘ o˘ 0 p3˘1qpV, ϕq ” oι pos pp3˘1qV ϕ ,o˘ qq p3˘1qpV, ϕq S2

0

V

the orientation on ` ˘ λ p3˘1qD « λpDqbp3˘1q

(8.34)

determined by the image ˘ ` ` ˘ ϕ ιS 2 os0 pp3˘1qV ϕ , o˘ V q P OSpS 2 p3˘1qV of the OSpin-structure os0 pp3 ˘ 1qV ϕ , o˘ V q of Examples 3.7 and 5.1 under the first map in (6.3) with X “ S 2 . The right-hand side of (8.34)

Base Cases

219

also carries a canonical orientation; it is obtained by taking any orientation on the first factor of λpDq and the same orientation on the other factor(s) of λpDq. Since @@ DD ϕ ´ ϕ ´ os0 p4V ϕ , o` V q “ os0 p2V , oV q, os0 p2V , oV q ‘ , the orientation o` 0 p4pV, ϕqq is the canonical orientation of λp4Dq. By [20, Proposition 3.5], the orientation o´ 0 p2pV, ϕqq is also the canonical orientation of λp2Dq; for the sake of completion, we recall one of the proofs of this fact in [20] at the end of this section. We thus obtain the following. Proposition 8.9. The orientations oos pV, ϕq of the determinants of real CR-operators on even-degree real bundle pairs pV, ϕq over pS 2 , τ q constructed above satisfy the CROrient 1os(1), 2(a), 5(a), and 6(a) properties of Section 7.2. If pV, ϕq is a rank 1 real bundle pair over pS 2 , τ q and (1) o is an orientation on V ϕ , then the orientation o0 pV, ϕ; oq as in p7.29q is the intrinsic orientation opV, ϕ; ox1 q of Proposition 8.2 for the restriction of o to Vxϕ1 with x1 P S 1 , (2) the degree of V is 1, then the orientation o˘ 0 pp3˘1qpV, ϕqq is the canonical orientation on p8.34q. Remark 8.10. By the pS 2 , τ q case of the CROrient 7C(a) property established in Section 9.3 and Proposition 8.9(2), the conclusion of the latter holds for all rank 1 odd-degree real bundles pV, ϕq over pS 2 , τ q. The orienting construction described above Proposition 8.9 depends on the OSpin-structure ospD2` q on the real vector bundle V ϕ induced by the relative OSpin-structure os, rather than os itself. For an OSpin-structure os on V ϕ , we thus denote by oos pV, ϕq the orientation of D induced by os. The next statement plays a crucial role in establishing Proposition 9.4(1). Proposition 8.11. Suppose pV, ϕq is a real bundle pair over pS 2 , τ q and os and os1 are relative OSpin-structures on V ϕ inducing the same orientation. If ospD2` q ‰ os1 pD2` q, oos pV, ϕq ‰ oos1 pV, ϕq. We prove this proposition and the CROrient 5(a) property for real bundle pairs over pS 2 , τ q in Section 8.5, without referring to the

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CROrient 1os(1) or 2(a) properties. We instead establish these two properties in the following using Proposition 8.11 and the CROrient 5(a) property. Proof of CROrient 1os(1) and 2(a) properties for (S 2 , τ ). Let pV, ϕq be a rank n real bundle pair over pS 2 , τ q, os be a relative OSpin-structure on the real vector bundle V ϕ over S 1 Ă S 2 , and D be a real CR-operator on pV, ϕq. The collections ospD2` q and ospD2´ q of trivializations determined by the two half-surfaces D2` and D2´ of pS 2 , τ q induce the same orientation on V ϕ . By the compatibility condition in Definition 6.3 and (6.2), these two classes are the same if and only if w2 posq vanishes. If ospD2` q “ ospD2´ q, the same reasoning as in the proof of Proposition 8.2(1) implies that the orientations oos pV, ϕq of D induced by D2` and D2´ are the same if and only if the number ` ˘ deg V ´n ` pdeg V q{2`n “ 2 is even. Combining the last two statements with Proposition 8.11, we conclude the orientations oos pV, ϕq induced by D2` and D2´ are the same if and only if @ D deg V ` w2 posq, rS 2 sZ2 ” os pS 2 q 2 vanishes in Z2 . This establishes the CROrient 1os(1) property for pS 2 , τ q. Let η P H 2 pS 2 , S 1 ; Z2 q. The relative OSpin-structures os and η¨os determine the same orientation o on V ϕ . By the RelSpinPin 2 property on page 81, the collections ospD2` q and η¨ospD2` q of trivializations determined by os, η¨os, and D2` are the same if and only if η|D2` vanishes. Along with Proposition 8.11, this implies the first statement of the CROrient 2(a) property for pS 2 , τ q. The collections ospD2` q and ospD2` q of trivializations determined by os, os, and D2` satisfy ospD2` q “ ospD2` q. Suppose first that n “ 1. By (8.35), q ospD2` q “ St2V ϕ posq

ðñ

` ˘ q . ospD2` q “ St2V ϕ os

(8.35)

Base Cases

221

Along with (8.33) and Proposition 8.2(1), this gives oos pV, ϕq “ oos pV, ϕq.

(8.36)

This establishes the second statement of the CROrient 2(a) property for the rank 1 real bundle pairs over pS 2 , τ q. In the following, we deduce the general case from the rank 1 case via the CROrient 5(a) property. We denote by os1 the standard relative OSpin-structure on the trivial line bundle S 1 ˆ R over S 1 Ă S 2 and by B¯ the standard real CR-operator on the real bundle pair pS 2 ˆC, τ ˆsq. Let pL, φq be a rank 1 real bundle pair over pS 2 , τ q of even degree, osL be a relative OSpin-structure on the real bundle Lφ over S 1 Ă S 2 , and B¯L be a real CR-operator on pL, φq. For n P Z` , define ` ˘ pVn;L , ϕn;L q “ Vn´1 , ϕn´1 ‘pL, φq. pVn , ϕn q “ pS 2 ˆCn , τ ˆcq, Let Dn be the real CR-operator on pVn;L , ϕn;L q given by the n-fold direct sum of the operators B¯ on each factor and Dn;L “ Dn´1 ‘ B¯L . Define posL q “ xxos1 , osn´1;L yy‘ . osn;L “ Stn´1 Lφ

(8.37)

By the RelSpinPin 5 and 7 properties in Section 6.2, ` ˘ osn;L “ Stn´1 osL “ xxos1 , osn´1;L yy‘ . Lφ The second equalities in the last two equations hold for n ě 2. Let n ě 2. By the second equality in (8.37) and the CROrient 5(a) property for pS 2 , τ q, the natural isomorphism ¯ λpDn;L q « λpBqbλpD n´1;L q respects the orientations oosn;L pVn;L , ϕn;L q, oos1 pV1 , ϕ1 q, and oosn´1;L pVn´1;L , ϕn´1;L q. It also respects the orientations oosn;L pVn;L , ϕn;L q, oos1 pV1 , ϕ1 q, and oosn´1;L pVn´1;L , ϕn´1;L q. Along with (8.36) with pV, ϕq “ pV1;L , φ1;L q, this implies that oosn;L pVn;L , ϕn;L q “ oosn;L pVn;L , ϕn;L q

@ n P Z` .

By the RelSpinPin 5 property, every relative OSpin-structure on the ϕ real vector bundle Vn;Ln;L over S 1 Ă S 2 equals osn;L for some relative

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OSpin-structure osL on Lφ . By [4, Proposition 4.1], every real bundle pair pV, ϕq over pS 2 , τ q is isomorphic to pVn;L , ϕn;L q for some rank 1 real bundle pair pL, φq over pS 2 , τ q. The last three statements imply  the second statement of the CROrient 2(a) property. Proof of Proposition 8.9(2) for o´ 0 (2(V, ϕ)). Three proofs of this statement appear in [20]. The first one, reproduced in the following, determines the relevant real holomorphic sections and trivializations explicitly. The second proof uses the comparisons of different orientations on the moduli spaces of real lines obtained in [11]. The last one deduces the claim directly from the fixed-edge equivariant contribution determined in [11]. We first note that there is a canonical isomorphism of the real line bundle ` iθ iθ{2 ˘ 1 ( LR P S ˆC : a P R ÝÑ S 1 1 ” e , ae with the real line bundle γR;1 ÝÑ RP1 . The composition of the trivialization (1.37) of 2γR;1 with the direct sum of two copies of this canonical identification is given by ˘ ` ` ˘ 1 r 0 peiθ , aeiθ{2 q, peiθ , beiθ{2 q “ eiθ , pa`ibqeiθ{2 . r 0 : 2LR Φ Φ 1 ÝÑ S ˆC, (8.38) 2 2 Let S‚ “ S ´t1u. The holomorphic map  ( hptq “ eit , h : B ” t P C : |t| ă 1 ÝÑ C Ă S 2 , is injective and intertwines the standard conjugation on B with τ ¯ on the on S 2 . We can assume that D is the standard B-operator holomorphic rank 1 real bundle pair pV, ϕq given by ˘L ` V “ hpBqˆC \ S‚2 ˆC „, ` ˘ ` ˘ ` ˘ hptq, tc „ hptq, c @ pt, cq P B ´t0u ˆC, ` ˘ “ ‰ ϕ rhptq, cs “ hpt¯q, c¯ @ pt, cq P B ˆC, ` ˘ “ ‰ ϕ rz, cs “ τ pzq, c¯ @ pz, cq P S‚2 ˆC. The space ker D of real holomorphic sections of V is then generated by the sections s1 and s2 described by s1 pzq “ 1,

s2 pzq “ i

1`z 1´z

@ z P S‚2 .

Base Cases

223

The canonical orientation for λp2Dq “ λpker 2Dq is then determined by the basis s11 ” ps1 , 0q,

s12 ” ps2 , 0q,

s21 ” p0, s1 q,

s22 ” p0, s2 q

for the kernel of the surjective operator 2D. The trivialization ` ˘ Ψ2 : 2V |S‚2 ´t0,8u ÝÑ S‚2 ´t0, 8u ˆC2 , ˘ ` Ψ2 rz, c1 s, rz, c2 s ˘ ` “ z, ipz´z ´1 qc1 ´z ´1 p1´zq2 c2 , z ´1 p1´zq2 c1 `ipz´z ´1 qc2 r 2 of 2V |S 2 ´t0,8u . Since of 2V |S‚2 ´t0,8u extends to a trivialization Ψ Ψ2 intertwines 2ϕ with the standard lift τ ˆc of τ |S‚2 ´t0,8u to a conr 2 intertwines 2ϕ jugation on the trivial bundle pS‚2 ´t0, 8uqˆC2 , Ψ 2 with the conjugation τˆc on the trivial bundle pS ´t0, 8uqˆC2 . We note that ( ` ˘  r 2 s11 pzq “ iz ´1 pz 2 ´1q, z ´1 p1´zq2 , Ψ ˘ ( `  r 2 s12 pzq “ z ´1 p1`zq2 , iz ´1 p1´z 2 q , Ψ ( ` ˘  r 2 s21 pzq “ ´z ´1 p1´zq2 , iz ´1 pz 2 ´1q , Ψ ( ` ˘  r 2 s22 pzq “ ´iz ´1 p1´z 2 q, z ´1 p1`zq2 . Ψ We define a trivialization Ψ of V over S 2 ´t8u Ą D2` by ˙ ˆ ` ˘ eit ´1 c @ pt, cq P B ˆC, Ψ rhptq, cs “ hptq, 2i t ˘ ` ˘ ` ˘ ` Ψ rz, cs “ z, 2ipz´1qc @ pz, cq P S‚2 ´t8u ˆC. This trivialization satisfies ΨpV ϕ q “ LR 1 . Under the identification of C 2 r 2 , the restriction in the target of (8.38) with R in the target of Ψ ϕ of this trivialization of 2V |S 2 ´t0,8u to 2V is the composition of the trivialization (8.38) with Ψ ‘ Ψ. Thus, this restriction lies in ϕ the homotopy class os0 p2V ϕ , o´ V q of trivializations of 2V . It fol´ lows that the orientation o0 p2pV, ϕqq of λp2Dq is obtained from the isomorphism (  r 2 ξq : ξ P ker 2D , ker 2D ÝÑ R‘R ‘ Resz“0 pΨ ˘ ` r 2 pξqq . r 2 pξqup1q, Res z“0 pΨ ξ ÝÑ tΨ The last space above is a complex subspace of C2 .

Spin/Pin-Structures & Real Enumerative Geometry

224

Under the above isomorphism, the basis s11 , s12 , s21 , s22 is sent to p0, 0; ´i, 1q,

p4, 0; 1, iq,

p0, 0; ´1, ´iq,

p0, 4; ´i, 1q.

Thus, an oriented basis for the target of the above isomorphism is given by p4, 0; 0, 0q,

p0, 4; 0, 0q,

p0, 0; ´i, 1q,

p0, 0; 1, iq.

The change of basis matrix from the first basis to this one is given by ¨ ˛ 0 1 0 0 ˚0 0 0 1‹ ˚ ‹ ˝ 1 0 0 1 ‚. 0 1 ´1 0 The determinant of this matrix is `1. 8.5



Even-Degree Bundles over Degenerations of pS 2 , τ q and Exact Triples

In this section, we establish Proposition 8.11 and the CROrient 5(a) property on page 116 for even-degree real bundle pairs over pS 2 , τ q. The former is immediate from (8.33) for rank 1 real bundle pairs; the latter is straightforward for the short exact sequences of real bundle pairs of rank at least 2, as noted in Step 1 of the proof of the CROrient 5(a) property. The next statement is a key ingredient in establishing Proposition 8.11 for real bundle pairs of rank at least 2. Lemma 8.12. There exist OSpin-structures os and os1 on the trivial oriented rank 2 vector bundle S 1 ˆR2 over S 1 such that ˘ ˘ ` ` oos S 2 ˆC2 , τ ˆc ‰ oos1 S 2 ˆC2 , τ ˆc . Proof. By [12, Proposition 8.1.7], there exist a rank 2 real bundle pair pV, ϕq over S 1 ˆ pS 2 , τ q with c1 pV q “ 0 and w1 pV ϕ q “ 0 and a family D” tDs usPS 1 of real CR-operators on the real bundle pairs ˇ pVs , ϕs q ” pV, ϕqˇtsuˆpS 2 ,τ q so that the determinant line bundle λp Dq over S 1 is not orientable. By the construction in [12] (as well as by [4, Proposition 4.1]), there

Base Cases

225

exists an automorphism Ψ of the trivial rank 2 real bundle pair pS 2ˆ C2 , τ ˆcq covering the identity on S 2 so that ˘L ` ˘ ` ˘ ` 0, Ψpz, vq „ 0, pz, vq . pV, ϕq “ r0, 1sˆpS 2 ˆC2 , τ ˆcq „, This automorphism restricts to an automorphism ΨR of the real vector bundle S 1 ˆR2 over S 1 . Since the vector bundle V ϕ is orientable, ΨR is orientation-preserving. Let o and s be the natural orientation and the natural homotopy class of trivializations, respectively, of the real vector bundle S 1 ˆ R2 over S 1 . Since ΨR is orientation-preserving, the homotopy r ” tD r s usPr0,1s be the class s1 ” Ψ˚R s preserves the orientation o. Let D family of real CR-operators on pS 2 ˆC2 , τ ˆcq induced by the family D. Since the orienting construction at the beginning of Section 8.4 r s q depends continuously on s, it determines a continapplied to λpD r s q from the homotopy class s of uously varying orientation os of λpD 1 2 trivializations of S ˆR . Since the line bundle λp Dq is not orientable, r 1 “ Ψ˚ D r0 the orientations s1 and Ψ˚ s0 of the determinant of D 1 1 induced by the OSpin-structures os ” po, sq and os ” po, s q on the  trivial rank 2 vector bundle S 1 ˆR2 over S 1 are different. We also use Lemma 8.13. It is the Exact Squares property for the determinants of Fredholm operators; its specialization to finite-dimensional vector spaces is a straightforward linear algebra observation. Lemma 8.13 ( [53, (2.27)]). Let Dij with i, j P r3s be Fredholm operators with orientations on their determinants. If the rows and columns in the diagram in Figure 8.4 are exact triples of Fredholm operators and this diagram commutes, then the total number of rows and columns in this diagram for which the associated isomorphism p7.2q respects the orientations is congruent to indpD13 qindpD31 q mod 2. Proof of CROrient 5(a) property for (S 2 , τ ), Step 1. Suppose e is a short exact sequence of real bundle pairs over pS 2 , τ q as in (7.21), os1 and os2 are relative OSpin-structures on the real vector 1 2 bundles V 1ϕ and V 2ϕ over S 1 Ă S 2 , and os ” xxos1 , os2 yy eR

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Spin/Pin-Structures & Real Enumerative Geometry

Figure 8.4. Commutative square of exact rows and columns of Fredholm operators or vector spaces for the statement of Lemma 8.13.

Figure 8.5. Commutative square of exact rows and columns of Fredholm operators for the proof of the CROrient 5(a) property for pS 2 , τ q.

is the induced relative OSpin-structure on V ϕ . Let C0 , p U,rcq, and t1 P Δ˚R be as in the orienting construction at the beginning of Section 8.4, r1 q and pVr 1 , ϕ r2 q be real bundle pairs over p U,rcq associated with pVr 1 , ϕ 1 the collections os pD2` q and os2 pD2` q of trivializations, and D1 ” tDt1 u and D2 ” tDt2 u be associated families of real CR-operators. Let r1 q‘pVr 2 , ϕ r2 q ÝÑ p U,rcq pVr , ϕq r “ pVr 1 , ϕ

(8.39)

and Dt ” Dt1 ‘Dt2 for each t P ΔR . The exact triple (7.55) induces exact triples of Fredholm operators given by the rows in the diagram of Figure 8.5. The splitting (8.39) induces exact triples of Fredholm operators given by the columns in this diagram. The isomorphisms (7.2) induced by the right column and the triple formed by the second summands in the middle column respect the complex orientations of all associated determinants.

Base Cases

227

The isomorphism (7.2) induced by the first summands in the middle column respects the canonical orientations of the associated determinant lines. Since the (real) index of D01` is even, Lemma 8.13 implies that the isomorphism (7.2) induced by the middle column respects the direct sum orientations. Suppose that q 1 q and (1) either rk V 1 ě 2 or rk V 1 “ 1 with os1 pD2` q “ St2V 1ϕ1 pos q 2 q. (2) either rk V 2 ě 2 or rk V 2 “ 1 with os2 pD2` q “ St2V 2ϕ2 pos The real bundle pair (8.39) and the family D ” tDt u are then associated with the collection ospD2` q of trivializations. The rows in the diagram of Figure 8.5 respect the orientations oos1 pVr01 , ϕ r10 q of D01 , 2 2 2 r0 q of D0 , oos2 pVr0 , ϕ r0 q of D0 , and the canonical orientaoos pVr0 , ϕ tions on the remaining terms in this case. Since the (real) dimension 1 of Vrnd ` is even, Lemma 8.13 implies that the isomorphism (7.24) induced by the left column in this diagram respects the orientar10 q, oos pVr0 , ϕ r0 q, and oos2 pVr02 , ϕ r20 q. By the construction tions oos1 pVr01 , ϕ at the beginning of Section 8.4, this in turn implies that the isomorphism (7.24) respects the orientations oos pV, ϕq, oos1 pV 1 , ϕ1 q, and oos2 pV 2 , ϕ2 q under the assumptions 1 and 2 above.  Corollary 8.14. Suppose pV, ϕq is a rank 2 real bundle pair over pS 2 , τ q, os is a relative OSpin-structure on the real vector bundle V ϕ , and D is a real CR-operator on pV, ϕq. The orientation oos pV, ϕq of D does not depend on the choice of the homotopy class in the collection ospD2` q of such classes determined by os and D2` . Proof. We denote by os1 the standard relative OSpin-structure on the vector bundle S 1 ˆR over S 1 Ă S 2 and by B¯ be the standard real CR-operator on the real bundle pair pS 2 ˆC, τ ˆcq over pS 2 , τ q. Let ` ˘ pV 1 , ϕ1 q “ pS 2 ˆ C, τ ˆ cq ‘ pV, ϕq, os1 “ xxos1 , osyy‘ P OSpS 2 τS 1 ‘ V ϕ .

q 1 q, the conclusion of Step 1 of the proof Since os1 pD2` q “ St2V ϕ pos of the CROrient 5(a) property for pS 2 , τ q implies that the natural isomorphism ¯ b λpDq λpB¯ ‘ Dq « λpBq

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Spin/Pin-Structures & Real Enumerative Geometry

respects the orientations oos1 pV 1 , ϕ1 q, oos1 pS 2 ˆC, τ ˆcq, and oos pV, ϕq. Since the first two of these orientations do not depend on the choice of the homotopy class in the collection ospD2` q, neither does the third. 

Proof of Proposition 8.11. This is immediate from (8.33) if rk V “ 1. We can thus assume that rk V ě 2. Let os1 and B¯ be as in the proof of Corollary 8.14, pL, φq be a rank 1 real bundle pair over pP1 , τ q of even degree, osL be a relative OSpin-structure on the q L q, and B¯L real bundle Lφ over S 1 Ă S 2 such that osL pD2` q “ St2Lφ pos ` be a real CR-operator on pL, φq. For n P Z , let ` ˘ pVn;L , ϕn;L q “ Vn´1 , ϕn´1 ‘pL, φq. pVn , ϕn q “ pS 2 ˆCn , τ ˆcq, Let Dn be the real CR-operator on pVn;L , ϕn;L q given by the n-fold direct sum of the operators B¯ on each factor and Dn;L “ Dn´1 ‘ B¯L . By Lemma 8.12, there exist relative OSpin-structures os2 and os12 on the vector bundle S 1 ˆR2 which induce the same orientation, but oos2 pV2 , ϕ2 q ‰ oos12 pV2 , ϕ2 q.

(8.40)

For n ě 3, define ` ˘ osn;L “ Stn´3 ϕ3;L xxos2 , osL yy‘ “ xxos1 , osn´1;L yy‘ , V3;L

` ˘ 1 1 os1n;L “ Stn´3 ϕ3;L xxos2 , osL yy‘ “ xxos1 , osn´1;L yy‘ . V3;L

(8.41)

The second equalities on the two lines above hold for n ě 4. We use the n “ 3 cases of these identities to define relative OSpin-structures os2;L and os12;L on the vector bundle S 1 ˆ R2 . The RelSpinPin 2 and 7 properties in Section 6.2 ensure that this is possible. By (8.41) and the conclusion of Step 1 of the proof of the CROrient 5(a) property for pS 2 , τ q, the natural isomorphisms λpD3;L q « λpD2 qbλpB¯L q

and

¯ λpD3;L q « λpBqbλpD 2;L q

respect the orientations oos3;L pV3;L , ϕ3;L q, oos2 pV2 , ϕ2 q, oosL pL, φq, oos1 pV1 , ϕ1 q, and oos2;L pV2;L , ϕ2;L q. They also respect the orientations oos13;L pV3;L , ϕ3;L q, oos12 pV2 , ϕ2 q, oosL pL, φq, oos1 pV1 , ϕ1 q, and

Base Cases

229

oos12;L pV2;L , ϕ2;L q. Along with (8.40), this implies that oos3;L pV3;L , ϕ3;L q ‰ oos13;L pV3;L , ϕ3;L q, oos2;L pV2;L , ϕ2;L q ‰ oos12;L pV2;L , ϕ2;L q.

(8.42)

For each n ě 4, the natural isomorphism ¯ λpDn;L q « λpBqbλpD n´1;L q respects the orientations oosn;L pVn;L , ϕn;L q, oos1 pV1 , ϕ1 q, and oosn´1;L pVn´1;L , ϕn´1;L q. It also respects the orientations oos1n;L pVn;L , ϕn;L q, oos1 pV1 , ϕ1 q, and oos1n´1;L pVn´1;L , ϕn´1;L q. Along with (8.42), this implies that oosn;L pVn;L , ϕn;L q ‰ oos1n;L pVn;L , ϕn;L q

@ n ě 2.

(8.43)

By [4, Proposition 4.1], every real bundle pair pV, ϕq over pS 2 , τ q is isomorphic to pVn;L , ϕn;L q for some rank 1 real bundle pair pL, φq over pS 2 , τ q. By (8.43), the real vector bundle V ϕ over S 1 Ă S 2 thus admits relative OSpin-structures os and os1 inducing the same orientation such that oos pV, ϕq ‰ oos1 pV, ϕq. Since H1 pS 1 ; Z2 q « Z2 , there are only two possibilities for the collection ospD2` q of trivializations of V ϕ . Since the orientation oos pV, ϕq of D is determined by ospD2` q,  the last three statements establish the claim. Proof of CROrient 5(a) property for (S 2 , τ ), Step 2. We continue with the notation and setup in Step 1 of the proof. Supq 1 q, and 2 holds. Let osV 1 be a pose rk V 1 “ 1, os1 pD2` q ‰ St2V 1ϕ1 pos 1 relative OSpin-structure on the real bundle V 1ϕ over S 1 Ă S 2 such q V 1 q. By the RelSpinPin 2 and 7 properties that osV 1 pD2` q “ St2V 1ϕ1 pos in Section 6.2, ospD2` q ” xxos1 , os2 yy eR pD2` q ‰ xxosV 1 , os2 yy eR pD2` q . Proposition 8.11 then implies that oos1 pV 1 , ϕ1 q ‰ oosV 1 pV 1 , ϕ1 q

and

oos pV, ϕq ‰ oxxosV 1 ,os2 yyeR pV, ϕq. (8.44)

By the conclusion of Step 1 of the proof, the isomorphism (7.24) respects the two orientations on the right-hand sides of the inequalities in (8.44) and the orientation oosV 2 pV 2 , ϕ2 q. Thus, it also

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Spin/Pin-Structures & Real Enumerative Geometry

respects the two orientations on the left-hand sides and the orientation oosV 2 pV 2 , ϕ2 q. q 2 q, and 1 holds, the above arguIf rk V 2 “ 1, os2 pD2` q ‰ St2V 2ϕ2 pos ment applies with the roles of os1 and os2 reversed. Suppose rk V 1 , rk V 2 “ 1,

q 1 q, os1 pD2` q ‰ St2V 1ϕ1 pos

q 2 q. os2 pD2` q ‰ St2V 2ϕ2 pos

By the RelSpinPin 2 and 7 properties, ospD2` q ” xxos1 , os2 yy eR pD2` q “ xxosV 1 , osV 2 yy eR pD2` q. By Proposition 8.11, oos1 pV 1 , ϕ1 q ‰ oosV 1 pV 1 , ϕ1 q and

oos2 pV 2 , ϕ2 q ‰ oosV 2 pV 2 , ϕ2 q. (8.45)

By the conclusion of Step 1 of the proof, the isomorphism (7.24) respects the two orientations on the right-hand sides of the inequalities in (8.45) and the orientation oos pV, ϕq. Thus, it also respects the two orientations on the left-hand sides and the orientation oos pV, ϕq.



Chapter 9

Intermediate Cases

In Section 9.1, we use the intrinsic orientations opV, ϕ; ox1 q of real CR-operators on rank 1 real bundle pairs pV, ϕq over pS 2 , τ q constructed in Section 8.1 to define orientations opV, ϕ; ox q of real CR-operators on rank 1 real bundle pairs pV, ϕq over arbitrary decorated smooth symmetric surfaces pΣ, σq. We deduce a number of properties of the orientations opV, ϕ; ox q in Sections 9.1 and 9.2 from the properties of the orientations opV, ϕ; ox1 q established in Sections 8.1–8.3. In Section 9.3, we similarly use the orientations oos pV, ϕq of real CR-operators on real bundle pairs pV, ϕq over pS 2 , τ q induced by relative OSpin-structures os on the real vector bundles V ϕ over the τ -fixed locus S 1 Ă S 2 to define orientations oos pV, ϕq on the determinants of real CR-operators on real bundle pairs pV, ϕq over arbitrary decorated smooth symmetric surfaces pΣ, σq from relative OSpin-structures os on the real vector bundles V ϕ over the σ-fixed locus Σσ Ă Σ. We then deduce all properties of these orientations stated in Sections 7.2 and 7.3 from the already established properties of the orientations opV, ϕ; ox q of Section 9.1 and the orientations oos pV, ϕq of Section 8.4. This concludes the proof of the part of Theorem 7.1 concerning orientations induced by relative OSpin-structures. 9.1

Orientations for Line Bundle Pairs

Suppose pΣ, σq is a smooth decorated symmetric surface, Sr1 Ă Σσ is the r-th connected component of Σσ with respect to the chosen order

231

232

Spin/Pin-Structures & Real Enumerative Geometry

on π0 pΣσ q, and x ” pxr qSr1 Pπ0 pΣσ q is a tuple of points so that xr P Sr1 . Let pV, ϕq be a rank 1 real bundle pair over pΣ, σq and ` ˘ ox ” oxr S 1 Pπ0 pΣσ q r

(9.1)

be a tuple of orientations of Vxϕr . We then define an orientation opV, ϕ; ox q of a real CR-operator D on pV, ϕq as follows. Let N ” |π0 pΣσ q| be the number of connected components of the fixed locus of pΣ, σq. For each Sr1 , choose a σ-invariant tubular neighborhood Ur Ă Σ of Sr1 so that the closures Ur of such neighborhoods are disjoint closed tubular neighborhoods of Sr1 . Let C0 ” pΣ0 , σ0 q be the nodal symmetric surface obtained from pΣ, σq by collapsing each of the boundary components of each Ur to a single point; see Figure 9.1. We denote by P1r Ă Σ0 the irreducible component containing Sr1 and set τr “ σ0 |P1r . Let ΣC 0 be the union of the remaining irreducible components of Σ0 and σ0C “ σ0 |ΣC . The choice 0 of half-surface Σb of Σ determines a distinguished half-surface D2r` C C of pP1r , τr q for each r and a distinguished half-surface Σ` 0 of pΣ0 , σ0 q so that all nodal points of the latter are shared with the nodal points of the former. We denote the complementary half-surfaces by D2r´ ˘ 2 and Σ´ 0 , respectively, and the unique nodal point on Dr˘ by ndr .

Figure 9.1. A decorated smooth symmetric surface pΣ, σq and its associated degeneration as below (9.1).

Intermediate Cases

233

Let p U,rcq be a family of deformations of C0 over Δ Ă C2N , rc R r r be a rank 1 real sR 1 , . . . , sN be sections of U over ΔR , and pV , ϕq bundle pair over p U,rcq so that ˘ ` and Σt1 , σt1 “ pΣ, σq, sR r pt1 q “ xr @ r P rN s, ˘ ` rt1 “ pV, ϕq (9.2) Vrt1 , ϕ for some t1 P ΔR . For each t P ΔR , ox determines a tuple ` ˘ ox;t ” oxr ;t S 1 Pπ0 pΣσ q r

R˚ r ϕ r over Δ . of orientations of Vr ϕr at sR R r ptq via the line bundles sr V ϕ If w1 pV q|Sr1 “ 0, then oxr ;t does not depend on the choice of the section sR r. rt q as Let D” tDt u be a family of real CR-operators on pVrt , ϕ r r0r q in (7.34) so that Dt1 “ D. For each r P rN s, we denote by pV0r , ϕ r0 q to P1r and by D0r the real CR-operator on the restriction of pVr0 , ϕ pVr0r , ϕ r0r q induced by D0 . Let D0` be the real CR-operator on Vr0 |Σ` 0 induced by D0 . The exact triple à à Vrnd` 0 ÝÑ D0 ÝÑ D0r ‘D0` ÝÑ ÝÑ 0, r

`

Sr1 Pπ0 pΣσ q

˘

Sr1 Pπ0 pΣσ q

` ˘ ξ´ , pξr qSr1 Pπ0 pΣσ q , ξ` ÝÑ pξr qSr1 Pπ0 pΣσ q , ξ` , ˘ ˘ ` ` ` pξr qSr1 Pπ0 pΣσ q , ξ` ÝÑ ξ` pnd` r q´ξr pndr q S 1 Pπ0 pΣσ q r

(9.3) of Fredholm operators then determines an isomorphism â ` â ` ˘ ˘ ` `˘ « bλ D0 . λ Vrnd` λ D λpD0 qb 0r r Sr1 Pπ0 pΣσ q

(9.4)

Sr1 Pπ0 pΣσ q

r0r ; oxr ;0 q be the intrinsic orientaFor each Sr1 P π0 pΣσ q, let opVr0r , ϕ tion of D0r as above Proposition 8.2. Along with the chosen order , these orion π0 pΣσ q and the complex orientations of D0` and Vrnd` r r0 ; ox q of λpD0 q via the entations determine an orientation opVr0 , ϕ isomorphism (9.4) and thus an orientation of the line bundle λp Dq over ΔR . The latter restricts to an orientation opV, ϕ; ox q of λpDq. By Proposition 8.2(1), opV, ϕ; ox q does not depend on the choice of

234

Spin/Pin-Structures & Real Enumerative Geometry

ϕ the section sR r above even if w1 pV q|Sr1 ‰ 0. We call opV, ϕ; ox q the intrinsic orientation of D.

Proposition 9.1. Suppose pΣ, σq is a smooth decorated symmetric surface, pV, ϕq is a rank 1 real bundle pair over pΣ, σq, and D is a real CR-operator on pV, ϕq. Let ox be a tuple of orientations of V ϕ at points xr in Sr1 Ă Σσ as in (9.1). p1q The intrinsic orientation opV, ϕ; ox q of D does not depend on the choice of the real bundle pair pVr , ϕq r over p U,rcq satisfying the last condition in (9.2). p2q The orientation opV, ϕ; ox q does not depend on the choice of halfsurface Σb˚ of an elemental component Σ˚ of pΣ, σq if and only if gpΣ˚ q`|π0 pΣσ˚ q|´1 deg V |Σ˚ `W1 ppV, ϕq|Σ˚ q ` P 2Z. 2 2 p3q The interchange in the ordering of two consecutive compo1 of Σσ reverses the orientation opV, ϕ; ox q if nents Sr1 and Sr`1 ϕ 1 “ 0. and only if w1 pV q|Sr1 , w1 pV ϕ q|Sr`1 p4q Reversing the component orientation oxr in (9.1) preserves the orientation opV, ϕ; ox q if and only if w1 pV ϕ q|Sr1 ‰ 0. p5q Suppose pΣ, σq “ pΣ1 , σ1 q\pΣ2 , σ2 q is a decomposition of pΣ, σq into decorated symmetric surfaces as below (7.16), pV1 , ϕ1 q and pV2 , ϕ2 q are the restrictions of pV, ϕq to Σ1 and Σ2 , respecp1q p2q tively, and ox “ ox ox is the associated decomposition of ox . If D1 and D2 are the real CR-operators on pV1 , ϕ1 q and pV2 , ϕ2 q induced by D, then the isomorphism ` ˘ ` ˘ λpDq « λ D1 bλ D2 induced by (7.18) respects the orientations opV, ϕ; ox q, opV1 , ϕ1 ; p1q p2q ox q, and opV2 , ϕ2 ; ox q. We establish Proposition 9.1(1), as well as Propositions 9.2 and 9.3 stated in Section 9.2. Proof of Proposition 9.1(2)–(5). We continue with the notation above the statement of the proposition. For each Sr1 P π0 pΣσ q, let # 0, if w1 pV ϕ q|Sr1 “ 0; r pV, ϕq “ 1, if w1 pV ϕ q|Sr1 ‰ 0.

Intermediate Cases

235

With the notation as in (2), let Σ0˚ Ă Σ0 be the elemental component corresponding to Σ˚ Ă Σ and ` Σ` 0˚ “ Σ0˚ XΣ0 .

We note that gpΣ` ˚0 q “

gpΣ˚ q´|π0 pΣσ˚ q|`1 , 2 ř deg Vr0r deg V |Σ˚ ´ Sr1 Pπ0 pΣσ ˚q

deg Vr0 |Σ` “

2

˚0

,

r indC D0` |Σ` “ 1´gpΣ` ˚0 q`deg V0 |Σ` . ˚0

˚0

(9.5)

The change in the choice of the half-surface Σb˚ of pΣ˚ , σq acts by the complex conjugation on the complex orientations of D0` |Σ` and each ˚0 Vr ` with Sr1 P π0 pΣσ˚ q. By Proposition 8.2(1), this change preserves ndr

the orientation opVrr , ϕ rr ; oxr ;0 q of D0r with Sr1 P π0 pΣσ˚ q if and only if deg Vr0r `r pV, ϕq P 2Z . 2

r0 ; ox q and opV, ϕ; ox q do not depend on Thus, the orientations opVr0 , ϕ the choice of half-surface Σb˚ of pΣ˚ , σq if and only if ˇ ˇ indC D0` |Σ` ´ ˇπ0 pΣσ˚ qˇ ` ˚0

ÿ

deg Vr0r `r pV, ϕq P 2Z. 2 σ

Sr1 Pπ0 pΣ˚ q

Combining this with (9.5), we obtain (2). Since the index of D0r is odd if and only if w1 pV ϕ q|Sr1 “ 0, (3) follows from the Direct Sum property for the determinants of Fredholm operators; see [53, Section 2]. Proposition 8.2(1) immediately implies 4. r1 q, pVr2 , ϕ r2 q, It remains to establish 5. Let p U1 ,rc1 q, p U2 ,rc2 q, pVr1 , ϕ p1q p2q ox;t , ox;t , D1;t , and D2;t be as in the construction of the orientations

Spin/Pin-Structures & Real Enumerative Geometry

236 p1q

p2q

opV1 , ϕ1 ; ox q and opV2 , ϕ2 ; ox q above Proposition 9.1. The orientation opV, ϕ; ox q on D is obtained via this construction applied with ˘ ` ˘ ` ˘ ` ˘ ` ˘ ` ˘ ` U,rc “ U1 ,rc1 \ U2,rc2 , Vr , ϕr “ Vr1 , ϕr1 \ Vr2 , ϕr2 , p1q p2q

ox;t “ ox;t ox;t , Dt “ D1;t \D2;t . The exact triple (9.3) of Fredholm operators induces the exact triples of Fredholm operators given by the rows in the diagram of Figure 9.2. The decompositions ` ` Σ0 “ Σ1;0 \Σ2;0 , Σ` 0 “ Σ1;0 \Σ2;0 , ` ˘ ` ˘ ` ˘ π0 Σσ “ π0 Σσ1 1 \π0 Σσ2 2

and

induce the exact triples of Fredholm operators given by the columns in this diagram. By definition, the rows in this diagram respect the orientations ` ˘ ` ` p1q ˘ p2q ˘ r1;0 ; ox , o Vr0 , ϕ r0 ; ox , and o Vr2;0 , ϕ r2;0 ; ox (9.6) o Vr1;0 , ϕ of the operators in the left column, the orientations # ` p1q ˘ σ1 1 r0r , ϕ ˘ ` o V r ; o 0r xr ;0 , if Sr P π0 pΣ1 q; r0r ; oxr;0 “ o Vr0r , ϕ ˘ ` p2q r0r ; oxr ;0 , if Sr1 P π0 pΣσ2 2 q; o Vr0r , ϕ

(9.7)

of the operators in the direct sums in the middle column, and the complex orientations of the remaining terms.

Figure 9.2. Commutative square of exact rows and columns of Fredholm operators for the proof of Proposition 9.1(5).

Intermediate Cases

237

The right column and the exact triple formed by the last summands in the middle column respect the complex orientations of the corresponding operators. Since the decomposition of pΣ, σq into pΣ1 , σ1 q and pΣ2 , σ2 q respects the orderings of the components of the fixed loci, the exact triple formed by the direct sums in the middle column respects the ordered direct sum orientations. Since the ` is even, Lemma 8.13 and the last two statements (real) index of D1;0 imply that the middle column respects the direct sum orientations. p1q Since the (real) dimensions of Vrnd` are even, Lemma 8.13 again, the r last sentence in the previous paragraph, and the conclusions in this paragraph imply that the left column in Figure 9.2 respects the orientations (9.6). The claim in 5 now follows from the continuity of the isomorphisms ` ˘ ` ˘ ` ˘ λ Dt « λ Dt;1 bλ Dt;2 with respect to t P ΔR .



We next describe the behavior of the orientations of Proposition 9.1 under flat degenerations of pΣ, σq to nodal symmetric surfaces as in the CROrient 7C and 7H3 properties of Section 7.3. We first suppose that C0 is a decorated symmetric surface with one conjugate pair of nodes nd˘ as in (7.36) and in the top left diagram of Figure 9.4 on page 246 so that xr P Sr1 for each r P rN s. We also suppose that pV0 , ϕ0 q is a rank 1 real bundle pair over pΣ0 , σ0 q and ox r 0 , pΣ r 0, σ is a tuple of orientations of V0ϕ0 |xr as in (9.1). Let C r0 q, and b r be as below (7.36), pVr0 , ϕ r q be the lift of pV , ϕ q to a real bundle Σ 0 0 0 0 r r0 q, and r ox be the lift of ox to a tuple of orientations pair over pΣ0 , σ ϕ r0 r of fibers of V0 . Let D0 be a real CR-operator on pV0 , ϕ0 q. We denote its lift to a r 0 . The orientation real CR-operator on pVr0 , ϕ r0 q by D ` ˘ ` ˘ r r0 ; r ox (9.8) o0 ox ” o Vr0 , ϕ r 0 and the complex orientation of V0 | ` determine an orientation of D nd ` ˘ ` ˘ (9.9) o0 ox ” o V0 , ϕ0 ; ox of D0 via the isomorphism (7.38). In an analogy with the intrinsic orientation of the CROrient 7C property, we call (9.9) the intrinsic orientation of D0 induced by ox .

238

Spin/Pin-Structures & Real Enumerative Geometry

Suppose in addition that p U,rcq is a flat family of deformations of C0 as in (7.32), pV, ϕq is a real bundle pair over p U,rcq extendrc R R ing pV0 , ϕ0 q, sR 1 , . . . , sN are sections of U over ΔR with sr p0q “ xr for all r P rN s, and D” tDt u is a family of real CR-operators on pVt , ϕt q as in (7.34) extending D0 . The decorated structure on C0 induces a decorated structure on the fiber pΣt , σt q of π for every t P ΔR as above the CROrient 7C property. The orientation oxr of ϕ over Δ , V0ϕ0 |xr induces an orientation of the line bundle sR˚ R r V ϕ R which in turn restricts to an orientation oxr ;t of V at sr ptq for each t P ΔR . For each t P Δ˚R , the tuple ` ˘ (9.10) ox;t ” oxr ;t S 1 Pπ0 pΣσ0 q r

of orientations of

0

Vtϕt |sRr ptq

determines an orientation ` ˘ ot ” o Vt , ϕt ; ox;t

of Dt as above Proposition 9.1. These orientations vary continuously with t and extend to an orientation ` ˘ (9.11) o10 pox q ” o1 V0 , ϕ0 ; ox of D0 as above the CROrient 7C property on page 121. In an analogy with the limiting orientation of the CROrient 7C property, we call (9.11) the limiting orientation of D0 induced by ox . Proposition 9.2. Suppose C0 is a decorated marked symmetric surface as in (7.36) which contains precisely one conjugate pair pnd` , nd´ q of nodes and no other nodes and carries precisely one real marked point xr on each connected component Sr1 of Σσ0 0 . Let pV0 , ϕ0 q be a rank 1 real bundle pair over pΣ0 , σ0 q, ox be a tuple of orientations of V0ϕ0 |xr as in (9.1), and D0 be a real CR-operator on pV0 , ϕ0 q. The intrinsic and limiting orientations, (9.9) and (9.11), of D0 are the same. We next suppose that C0 is a decorated symmetric surface with one H3 node nd as above (7.39) and in the top left diagram of Figure 9.5 on page 250 so that xr P Sr1 for each r P rN s different from the index r‚ of the singular topological component of Σσ0 0 and xr‚ “ nd (in this case, we allow a “marked point” to be a node). We also suppose that pV0 , ϕ0 q is a rank 1 real bundle pair over pΣ0 , σ0 q and ox

Intermediate Cases

239

is a tuple of orientations of V0ϕ0 at points xr P Sr1 as in (9.1). Let r 0 , pΣ r 0, σ r b , nd1 P S 1 , and nd2 P S 1 be as in and below (7.39), C r0 q, Σ 0 ‚1 ‚2 r 0, σ r0 q be the lift of pV0 , ϕ0 q to a real bundle pair over pΣ r0 q, and pVr0 , ϕ ` ` ˘ ˘ r (9.12) ox ” ond1 , ond2 , oxr S 1 Pπ0 pΣσ0 q,r‰r‚ r

0

be the tuple of orientations of the fibers of Vr0ϕr0 at the points nd1 , nd2 r σr0 . and xr of Σ 0 Let D0 be a real CR-operator on pV0 , ϕ0 q. We denote its lift to a r 0 . The orientation real CR-operator on pVr0 , ϕ r0 q by D ` ˘ ` ˘ r o0 ox ” o Vr0 , ϕ r0 ; r ox (9.13) r 0 and the orientation ond ” oxr of V ϕ0 |nd determine an orientaof D 0 ‚ tion ` ˘ ` ˘ (9.14) o0 ox ” o V0 , ϕ0 ; ox of D0 via the isomorphism (7.41). We call (9.14) the intrinsic orientation of D0 induced by ox . Suppose in addition that p U,rcq is a flat family of deformations of C0 as in (7.32), pV, ϕq is a real bundle pair over p U,rcq extendrc R ing pV0 , ϕ0 q, sR 1 , . . . , sN are sections of U over ΔR such that  (  ( 1 (9.15) and sR sR r p0q “ xr @ r P rN s´ r‚ r‚ p0q P S‚1 ´ nd , and D” tDt u is a family of real CR-operators on pVt , ϕt q as in (7.34) extending D0 . The decorated structure on C0 induces a decorated structure on the fiber pΣt , σt q of π for every t P ΔR and determines open subspaces Δ˘ R Ă ΔR as above the CROrient 7H3 property on page 123. For each r ‰ r‚ and t P ΔR , oxr determines an orientation oxr ;t of VsϕR ptq as above Proposition 9.2. The orienr

tation ond determines an orientation oxr‚ ;t of V ϕ at sR r‚ ptq as above 1 1 Proposition 8.7 (with S1 there replaced by S‚1 ). For each t P Δ˚R , the resulting tuple (9.10) of orientations of Vtϕt |sRr ptq determines an orientation ˘ ` (9.16) ot ” o Vt , ϕt ; ox;t of Dt as above Proposition 9.1. We denote by ` ˘ ` ˘ ` o` 0 ox ” o0 V0 , ϕ0 ; ox

(9.17)

Spin/Pin-Structures & Real Enumerative Geometry

240

the orientation of D0 obtained as the continuous extension of the orientations (9.16) with t P Δ` R . We call (9.17) the limiting orientation of D0 . Define ˇ (ˇ W1 pV0 , ϕ0 qr‚ “ ˇ Sr1 P π0 pΣσ q : r ą r‚ , w1 pV0ϕ0 q|Sr1 ‰ 0 ˇ. (9.18) Proposition 9.3. Suppose C0 is a decorated marked symmetric surface as in (7.36) which contains precisely one H3 node nd and no other nodes and carries precisely one real marked point xr on each smooth connected component Sr1 of Σσ0 0 and a marked point xr‚ “ nd on the nodal connected component of Σσ0 0 . Let pV0 , ϕ0 q be a rank 1 real bundle pair over pΣ0 , σ0 q, ox be a tuple of orientations of V0ϕ0 |xr as in (9.1), and D0 be a real CR-operator on pV0 , ϕ0 q. The intrinsic and limiting orientations, (9.14) and (9.17), of D0 are the same if and only if ˇ ˇ ˘ ` 1 1 sZ2 y`1 xw1 pV0ϕ0 q, rS‚2 sZ2 y ` ˇπ0 pΣσ0 0 qˇ ´r‚ xw1 pV0ϕ0 q, rS‚1 ´W1 pV0 , ϕ0 qr‚ “ 0 P Z2 . 9.2

(9.19)

Proofs of Propositions 9.1(1), 9.2, and 9.3

In all three proofs, we continue with the notation in the statement of the corresponding proposition and just above. Proof of Proposition 9.1(1). The substance of this claim is that the orientation opV, ϕ; ox q does not depend on the choice of the restriction of pVr , ϕq r to each real component pP1r , τr q of pΣ0 , σ0 q. In light of Proposition 9.1(3), it is sufficient to show that this is the case for r “ 1. Thus, suppose that pVp , ϕq p is a rank 1 real bundle pair over p U,rcq so that ` ` ˘ˇ ˘ˇ ˘ ` ˘ˇ ` pt ˇ “ pV, ϕq and Vp0r , ϕ p0r ” Vp , ϕ r ˇ 1 p ˇ 1 “ Vr , ϕ Vpt , ϕ 1

1

Σ

`

” Vr0r , ϕ r0r

˘

Pr

Pr

@ r ě 2.

Let U11 Ă U1 be a σ-invariant tubular neighborhood of S11 so that U11 Ă U1 is a closed tubular neighborhood of S11 . Let ` ` ˘ ˘ p p 0, σ C0 ” Σ p0 and C01 ” Σ10 , σ01 (9.20)

Intermediate Cases

241

Figure 9.3. Deformations of C01 in the family p U1 ,rc1 q over Δ1 Ă C2N`2 for the 1 1 proof of Proposition 9.1(1); t1 P Δ1˚ R does not lie in the span of ΔR;1 and ΔR;2 unless N “ 1.

be the nodal symmetric surfaces obtained from pΣ, σq by collapsing each of the boundary components of U11 , U2 , . . . , UN and U11 , U1 , U2 , . . . , UN , respectively, to a single point; see the left diagrams in Figure 9.3. We define P1r Ă Σ10 and τr1 as below (9.1). The choice of half-surface Σb of Σ again determines a distinguished halfsurface D2r` of pP1r , τr q for r P rN s. Let P1˘ Ă Σ10 be the irreducible component sharing a node nd˘ with D21˘ . We label the remaining ˘ ˘ 1 2 nodes of Σ10 by nd˘ r so that nd1 P P˘ and ndr P Dr˘ for r ě 2. We 1C denote by Σ0 the union of the remaining irreducible components 1C b of Σ10 and by Σ1` 0 Ă Σ0 its half-surface distinguished by Σ . 1 1 1 Let p U ,rc q be a flat family of deformations of C0 as in (7.32) over the unit ball Δ1 Ă C2N `2 around the origin satisfying the first condition in (9.2) for some t1 P Δ1R so that ` ˘ Ct1 0 ” Σ1t0 , σt1 0 and

` ˘ Ct1 1 ” Σ1t1 , σt1 1 0

0

0

Spin/Pin-Structures & Real Enumerative Geometry

242

for some t0 , t10 P Δ1R are the symmetric surfaces C0 as below (9.1) and p 0 as in (9.20) obtained from C1 by smoothing the conjugate pair C 0 of nodes nd˘ and the conjugate pair of nodes nd˘ 1 , respectively. We denote by 1˚ 1 Δ1˚ R;1 , ΔR;2 Ă ΔR

the subspaces parametrizing all symmetric surfaces obtained from C01 by smoothing the conjugate pair of nodes nd˘ and the conjugate pair 1 1 1 of nodes nd˘ 1 , respectively, and by ΔR;1 , ΔR;2 Ă ΔR their closures. 1 1R 1rc 1 r 1 r1 q be a rank 1 Let s1R 1 , . . . , sN be sections of U over ΔR and pV , ϕ 1 1 real bundle pair over p U ,rc q so that s1R r pt1 q “ xr @ r P rN s, ˘ ` 1 1 ˘ ` ˘ ` 1 1 ˘ ` rt0 “ Vr0 , ϕ r0 , rt1 “ Vp0 , ϕ p0 , Vrt1 , ϕ Vrt0 , ϕ 0

` 1 1 ˘ rt1 “ pV, ϕq. Vrt1 , ϕ

0

(9.21) For each t P Δ1R , ox determines a tuple ` ˘ o1x;t ” o1xr ;t S 1 Pπ0 pΣσ q r

1 1R˚ r 1ϕ r1 over Δ1 . of orientations of Vr 1ϕr at s1R r ptq via the line bundles sr V R r1t q as Let D1 “ tDt1 u be a family of real CR-operators on pVrt1 , ϕ in (7.34) so that Dt1 1 is the operator D in the statement of the 1 ,ϕ r10r q the restricproposition. For each r P rN s, we denote by pVr0r 1 the real CR-operator on pV 1 r1 ,ϕ tion of pVr01 , ϕ r10 q to P1r and by D0r 0r r0r q ` ` 1 1 induced by D0 . Let D01 and D0 be the real CR-operators on Vr0 |P1` and Vr 1 | 1` , respectively, induced by D 1 . The exact triple

0 Σ0

0 ÝÑ D01 ÝÑ

0

à

1` 1 D0r ‘D01 ‘D01` ÝÑ

rPrN s

à

1 r 1 ` ÝÑ 0, Vrnd ` ‘V nd

rPrN s

r

˘ ` ˘ ` ξΣ1´ , ξP1´ , pξr qrPrN s , ξP1` , ξΣ1` ÝÑ pξr qrPrN s , ξP1` , ξΣ1` , 0 0 0 ˘ ` ` pξr qrPrN s , ξP1` , ξΣ1` ÝÑ ξP1` pnd` q´ξ1 pnd` q, 0

` ξΣ1` pnd` 1 q´ξP1` pnd1 q, ˘ ˘ ` 0 q´ξr pnd` q rą1 ξΣ1` pnd` r r 0

Intermediate Cases

243

of Fredholm operators then determines an isomorphism â ` 1 ˘ â ` 1 ˘ ` 1` ˘ ` 1` ˘ ` 1 ˘ λ Vrnd` bλ Vrnd λ D0r bλ D01 bλ D0 . « λpD01 qb ` r

rPrN s

rPrN s

1 ,ϕ 1 and the complex orientar10r ; o1xr ;0 q of D0r The orientations opVr0r 1` 1 , and V r 1 ` determine an orientation of D 1 , D01` , Vrnd tions of D01 ` 0 nd r via the above isomorphism and thus an orientation o U1 pox q of the line bundle λp D1 q over Δ1R . The restriction of p U1 ,rc1 q to Δ1R;1 is the product of a family smoothing

P10;1 ” P1´ YP11 YP1` into a single irreducible component P1t;1 Ă Σ1t as above Proposition 8.2 with the union of remaining irreducible components of Σ10 ; see the right diagram in Figure 9.3. The other components of Σ1t with t P Δ˚R;1 and its nodes correspond to the components P1r of Σ10 with r ě 2 ˘ 1 and Σ1˘ 0 and to the nodes ndr with r ě 1; we denote them by Pt;r , 1˘ ˘ 1 Σt , and ndt;r , respectively. For t P ΔR;1 and r P rN s, we denote 1 ,ϕ 1 r1t;r q the restriction of pVrt1 , ϕ r1t q to P1t;r and by Dt;r the real by pVrt;r ` 1 1 1 rt;r q induced by Dt . Let Dt be the real CRCR-operator on pVrt;r , ϕ 1 operator on Vr | 1` induced by D 1 . The exact triple (7.55) of Fredholm t Σt

t

1 and the exact triple (9.3) of operators with D0 replaced by D0;1 Fredholm operators with D0 replaced by Dt1 0 induce isomorphisms ` 1 ˘ ` 1` ˘ ` 1 ˘ ` 1 ˘ bλ Vrnd` « λ D01 bλ D01 , λ D0;1 â â ˘ ˘ ` ˘ ` ` (9.22) « bλ D 1` . λ Vr 1 ` λ D1 λpD 1 qb t0

rPrN s

ndt

0 ;r

rPrN s

t0 ;r

t0

1 ,ϕ 1 and the complex orientations r101 ; o1x1 ;0 q of D01 The orientation opVr01 1` 1 and Vrnd of D01 ` determine an orientation of D0;11 via the first isomor1 for each t P Δ1 ; the phism in (9.22) and thus an orientation of Dt;1 R;1 1 induced orientation of D0;1 is the C-split orientation in the terminology of Proposition 8.5. By this proposition and the first assumption in (9.21), the induced orientation of Dt1 0 ;1 is thus the intrinsic orientation ˘ ` ˘ ` r1t0 ;1 ; o1x1 ;t0 “ o Vr01 , ϕ r01 ; ox1 ;0 . o Vrt10 ;1 , ϕ

Spin/Pin-Structures & Real Enumerative Geometry

244

1` 1 Since the (real) index of D01 and the (real) dimension of Vrnd ` are even, this implies that the second isomorphism in (9.22) is orientation-preserving with respect to the restriction of the orientation o U1 pox q to λpDt1 0 q, the intrinsic orientations ˘ ` ˘ ` r1t0 ;r ; o1xr ;t0 “ o Vr0r , ϕ r0r ; oxr ;0 o Vrt10 ;r , ϕ 1 of Dt1 0 ;r “ D0r , and the complex orientations of Dt1`0 and Vrnd ` . t0 ;r

The restriction of p U1 ,rc1 q to Δ1R;2 is the product of a family smoothing the conjugate pair 1 Σ10;˘ ” P1˘ Y Σ1˘ 0 Ă Σ0

into a conjugate pair Σ1t;˘ Ă Σ1t with the union of remaining irreducible components of Σ10 ; see the top left diagram in Figure 9.3. The other components of Σ1t with t P Δ˚R;2 and its nodes correspond to the components P1r of Σ10 with r ě 1 and to the nodes nd˘ and ˘ ˘ 1 nd˘ r with r ě 2; we denote them by Ptr , ndt;1 , and ndt;r , respectively. r1tr q the restriction of For t P Δ1R;2 and r P rN s, we denote by pVrtr1 , ϕ 1 the real CR-operator on pV 1 r1 ,ϕ r1t q to P1tr and by Dtr pVrt1 , ϕ tr rtr q induced by D 1 . Let Dt;` be the real CR-operator on Vr 1 |Σ1 induced by D 1 . t

t

The exact triple 1 1` 1 ÝÑ D01 ‘D01` ÝÑ Vrnd 0 ÝÑ D0;` ` ÝÑ 0, 1

ÝÑ

t

t;`

`

ξP1` , ξΣ1`

˘

0

` ξΣ1` pnd` 1 q´ξP1` pnd1 q 0

of Fredholm operators and the exact triple (9.3) of Fredholm operators with D0 replaced by Dt1 1 induce isomorphisms 0

` 1` ˘ ` 1` ˘ ` 1 ˘ ` 1 ˘ bλ Vrnd` « λ D01 bλ D0 , λ D0;` 1 â ` 1 â ` 1 ˘ ˘ ` ˘ λ Vrnd` λ Dt1 r bλ Dt1 1 ;` . « λpDt1 1 qb 0

rPrN s

t1 ;r 0

rPrN s

0

(9.23)

0

1` 1 , D01` , and Vrnd The complex orientations of D01 ` determine an 1

1 via the first isomorphism in (9.23) and thus an orientation on D0;` 1 orientation of Dt;` for each t P Δ1R;2 . The latter is the complex orientation of this real CR-operator. Since the dimension of Vr 1 ` is even, nd1

Intermediate Cases

245

this statement and the second assumption in (9.21) imply that the second isomorphism in (9.23) is orientation-preserving with respect to the restriction of the orientation o U1 pox q to λpDt1 1 q, the intrinsic 0 orientations ˘ ` r1t1 r ; o1xr ;t1 q “ o Vp0r , ϕ p0r ; oxr ;0 opVrt11 r , ϕ 0

0

0

1 of Dt1 1 r , and the complex orientations of Dt1 1 ;` and Vrnd ` . 0

0

t1 ;r 0

By the conclusion above regarding the second isomorphism in (9.22) and the last assumption in (9.21), the restriction of the orientation o U1 pox q to λpDt1 1 q “ λpDq is the orientation opV, ϕ; ox q of D constructed above Proposition 9.1 from the real bundle pair pVr , ϕq r over the nodal curve C0 below (9.1). By the conclusion above regarding the second isomorphism in (9.23) and the last assumption in (9.21), this restriction is the orientation opV, ϕ; ox q constructed above Proposition 9.1 from the real bundle pair pVp , ϕq p over the nodal r and curve C0 below (9.1). Thus, the rank 1 real bundle pairs pVr , ϕq p pV , ϕq p over p U,rcq determine the same orientation of D.  Proof of Proposition 9.2. We recall that the nodal symmetric surface pΣ0 , σ0 q with one conjugate pair of nodes nd˘ in this case is r 0, σ obtained from its normalization pΣ r0 q by identifying z1` with z2` ´ ´ ` into the node nd and z1 with z2 into the node nd´ . In particular, the complement of nd` , nd´ in Σ is canonically identified with the r complement of z1` , z1´ , z2` , z2´ in Σ. r 0 be a σ-invariant tubular neighFor each r P rN s, let Ur Ă Σ0 , Σ 1 borhood of Sr so that the closures Ur of such neighborhoods are disjoint closed tubular neighborhoods of Sr1 not containing the points nd` , nd´ and z1` , z1´ , z2` , z2´ . Let C01 ” pΣ10 , σ01 q be the nodal symmetric surface obtained from pΣ0 , σ0 q by collapsing each of the boundary components of each Ur to a single point; see the bottom left diagram in Figure 9.4. We define 1 P1r , Σ1C 0 Ă Σ0 ,

D2r˘ Ă P1r ,

1C Σ1˘ 0 Ă Σ0 ,

2 1˘ and nd˘ r P Dr˘ , Σ0

r 1` r1 as below (9.1). In this case, nd˘ P Σ1˘ 0 . We denote by Σ0 and Σ0 the 1 closed surfaces obtained from Σ1` 0 and Σ0 , respectively, by replacing ˘ ˘ ˘ and z0;2 the conjugate pair of nodes nd with two conjugate pairs z0;1 of marked points.

246

Spin/Pin-Structures & Real Enumerative Geometry

Figure 9.4. Deformations of C01 in the family p U1 ,rc1 q over Δ1 Ă C2N`2 for the proof of Proposition 9.2.

Let p U1 ,rc1 q be a flat family of deformations of C01 as in (7.32) over the unit ball Δ1 Ă C2N `2 around the origin so that ` ` ˘ ˘ Ct1 0 ” Σ1t0 , σt1 0 and Ct1 1 ” Σ1t1 , σt1 1 0

0

0

for some t0 , t10 P Δ1R are the symmetric surfaces C0 below (9.1) and in the statement of the proposition obtained from C01 by smoothing the conjugate pair of nodes nd˘ and the conjugate pairs of nodes nd˘ r , respectively. We denote by 1˚ 1 Δ1˚ R;1 , ΔR;2 Ă ΔR

the subspaces parametrizing all symmetric surfaces obtained from C01 by smoothing the conjugate pair of nodes nd˘ and the conjugate pairs 1 1 1 of nodes nd˘ r , respectively, and by ΔR;1 , ΔR;2 Ă ΔR their closures.

Intermediate Cases

247

1

1R 1rc 1 1 1 Let s1R 1 , . . . , sN be sections of U over ΔR and pV , ϕ q be a rank 1 1 1 real bundle pair over p U ,rc q so that ` 1 1 ˘ 1 Vt1 , ϕt1 “ pV0 , ϕ0 q. s1R r pt0 q “ xr @ r P rN s and 0

For each

t P Δ1R ,

0

ox determines a tuple ˘ ` o1x;t ” o1xr ;t S 1 Pπ0 pΣσ q r

1

1

1R˚ 1ϕ over Δ1 . of orientations of V 1ϕ at s1R r ptq via the line bundles sr V R 1 1 1 1 1 1 r ,σ r0 q the lift of pV0 , ϕ0 q to pΣ r q. We denote by pVr0 , ϕ 0 0 Let D1 “ tDt1 u be a family of real CR-operators on pVt1 , ϕ1t q as in (7.34) so that Dt1 1 is the operator D0 in the statement of the 0 1 , ϕ1 q the restricproposition. For each r P rN s, we denote by pV0r 0r 1 1 1 1 1 , ϕ1 q tion of pV0 , ϕ0 q to Pr and by D0r the real CR-operator on pV0r 0r 1` 1 1 induced by D0 . Let D0 be the real CR-operator on V0 |Σ1` induced 0 r 1 and D r 1` be the lifts of D 1 and D 1` , respectively, to by D01 and D 0 0 0 0 r1 q and Vr 1 | r 1` . The exact triple real CR-operators on pVr 1 , ϕ 0

0 ÝÑ D01 ÝÑ

à

0

0 Σ0

1 r 1` ÝÑ D0r ‘D 0

rPrN s

à

1 1 Vnd ÝÑ 0, ` ‘V nd`

rPrN s

r

˘ ` ˘ ` ξ´ , pξr qrPrN s , ξ` ÝÑ pξr qrPrN s , ξ` , ˘ ˘ ˘ `` ` ` ` ` pξr qrPrN s , ξ` ÝÑ ξ` pnd` r q´ξr pndr q rPrN s , ξ` pz2 q´ξ` pz1 q of Fredholm operators then determines an isomorphism â ` 1 ˘ ` 1` ˘ â ` 1 ˘ ` 1 ˘ r . « λ Vnd` bλ Vnd λ D0r bλ D λpD01 qb ` 0 rPrN s

r

rPrN s

1 , ϕ1 ; o1 1 The orientations opV0r 0r xr ;0 q of D0r and the complex orienta1 , V 1 , and D r 1` determine an orientation of D 1 via tions of Vnd ` 0 0 nd` r the above isomorphism and thus an orientation o U1 pox q of the line bundle λp D1 q over Δ1R . The restriction of p U1 ,rc1 q to Δ1R;1 is the product of a family 1˘ 1 smoothing the conjugate pair Σ1˘ 0 into a conjugate pair Σt Ă Σt 1 1 with the union of remaining irreducible components pPr , τr q of Σ0 . The other components of Σ1t with t P Δ˚R;1 and its nodes correspond to the components P1r of Σ10 and to the nodes nd˘ r ; we denote them

Spin/Pin-Structures & Real Enumerative Geometry

248

1 by P1tr and nd˘ t;r , respectively. For t P ΔR;1 and r P rN s, we denote 1 the real by pVtr1 , ϕ1tr q the restriction of pVt1 , ϕ1t q to P1tr and by Dtr 1` 1 1 1 CR-operator on pVtr , ϕtr q induced by Dt . Let Dt be the real CRoperator on V01 |Σ1` induced by D01 . The exact triple t

r 1` ÝÑ V 1 ` ÝÑ 0, 0 ÝÑ D01` ÝÑ D 0 nd

` `˘ ` `˘ ξ ÝÑ ξ z2;0 ´ξ z1;0

of Fredholm operators and the exact triple (9.3) of Fredholm operators with D0 replaced by Dt1 0 induce isomorphisms ˘ ` 1 ˘ ` 1` ˘ ` r , «λ D λ D01` bλ Vnd ` 0 â â ` ` ˘ ˘ ` ˘ 1 1 (9.24) r λpDt qb λ V ` λ Dt1 r bλ D 1` . « 0

rPrN s

ndt

0 ;r

rPrN s

t0

0

r 1` and V 1 ` determine an orientation The complex orientations of D 0 nd 1` of D0 via the first isomorphism in (9.24) and thus an orientation of Dt1` for each t P Δ1R;1 . The latter is the complex orientation of Dt1` . This implies that the second isomorphism in (9.24) is orientationpreserving with respect to the restriction of the orientation o U1 pox q 1 1 1 r 1 1 q, the intrinsic orientations opVr 1 , ϕ to λpD t0 r rt0 r ; oxr ;t0 q of Dt0 r , and t0 the complex orientations of D 1` and Vr 1 ` . Thus, the restrict0

λpDt1 q

ndt

0 ;r

with t P Δ1˚ tion of o U1 pox q to R is the intrinsic orientation 1 1 1 opVt , ϕt1 ; ox;t q defined above Proposition 9.1. This in turn implies that the restriction of o U1 pox q to λpDt1 1 q is the limiting orienta0 tion o10 pox q of Dt1 1 “ D0 in (9.11). 0 The restriction of p U1 ,rc1 q to Δ1R;2 is a family deforming pΣ10 , σ01 q to the symmetric surface in the statement of the proposition and 1 r1 , σ in (7.36). For each t P Δ1R;2 , we denote by pΣ t rt q the symmetric surface obtained from pΣ1t , σt1 q by replacing the conjugate pair ˘ ˘ with two conjugate pairs zt;1 of nodes nd˘ t corresponding to nd ˘ 1 r1 , σ and zt;2 of marked points, by pVrt1 , ϕ r1t q the lift of pVt1 , ϕ1t q to pΣ t rt q, r 1 the real CR-operator on pV 1 , ϕ1 q induced by D 1 . The exact and by D t t t t r 1 induces triple (9.3) of Fredholm operators with D0 replaced by D 0 an isomorphism â ` 1 ˘ â ` 1 ˘ ` 1` ˘ r . r 1 qb λ Vnd` « λ D0r bλ D (9.25) λpD 0 0 rPrN s

r

rPrN s

Intermediate Cases

249

1 , ϕ1 ; o1 1 The orientations opV0r 0r xr ;0 q of D0;r and the complex r 1 via the r 1` and V 1 ` determine an orientation of D orientations of D 0 0 ndr 1 r isomorphism (9.25) and thus an orientation of D for each t P Δ1 . t

R;2

o0 pox q of By definition, this orientation for t “ t10 is the orientation r 1 1 r r Dt1 “ D0 in (9.8). Since the dimension of Vnd` is even, this implies 0 that the restriction of o U1 pox q to λpDt1 1 q is the intrinsic orienta0 tion o0 pox q of Dt1 1 “ D0 in (9.9). Combining this with the conclusion 0 of the previous paragraph, we conclude that o0 pox q “ o10 pox q. 

Proof of Proposition 9.3. In this case, the nodal symmetric surface pΣ0 , σ0 q with one H3 node is obtained from its normalization 1 and nd P S 1 into the r 0, σ r0 q by identifying marked points nd1 P S‚1 pΣ 2 ‚2 node nd. In light of Proposition 9.1(3), we can assume that r‚ “ N . For r “ 1, 2, let U‚r Ă Σ‚r be a σ-invariant tubular neighbor1 so that its closure U hood of S‚r ‚r is a closed tubular neighborhood 1 of S‚r . Let C01 ” pΣ10 , σ01 q be the nodal symmetric surface obtained from pΣ0 , σ0 q by collapsing each of the boundary components of U‚1 and U‚2 to a single point; see the bottom left diagram in Figure 9.5. 1 , We denote by P1‚r Ă Σ10 the irreducible component containing S‚r 1c 1 by Σ0 Ă Σ0 the union of the remaining irreducible components, and 1 and σ 1c the restrictions of σ 1 to P1 and Σ1c , respectively. by τ‚r ‚r 0 0 0 The decorated structure on pΣ0 , σ0 q induces decorated structures 1 q and pΣ1c , σ 1c q. We denote by on the symmetric surfaces pP1‚r , τ‚r 0 0 ` 1 nd‚r P P‚r the node contained in the interior of the distinguished 1 half-surface D2‚r of P1‚r and by nd´ ‚r P P‚r its conjugate. Let p U1 ,rc1 q be a flat family of deformations of C01 as in (7.32) over the unit ball Δ1 Ă C5 around the origin so that ` ` ˘ ˘ Ct1 0 ” Σ1t0 , σt1 0 and Ct1 1 ” Σ1t1 , σt1 1 0

0

0

for some t0 , t10 P Δ1R are a symmetric surface obtained from C01 by smoothing the real node nd and the symmetric surface C0 in the statement of the proposition obtained from C01 by smoothing the con1˚ 1 jugate pairs of nodes nd˘ ‚r , respectively. Let ΔR Ă ΔR be the subspace 1` 1˚ parametrizing smooth symmetric surfaces and ΔR , Δ1´ R Ă ΔR be its two topological components distinguished as above (7.40). Denote by 1˚ 1 Δ1˚ R;1 , ΔR;2 Ă ΔR

Spin/Pin-Structures & Real Enumerative Geometry

250

Figure 9.5. Deformations of C01 in the family p U1 ,rc1 q over Δ1 Ă C5 for the proof of Proposition 9.3.

the subspaces parametrizing all symmetric surfaces obtained from C01 by smoothing the real node nd and the conjugate pairs of nodes nd˘ ‚r , respectively, and by Δ1R;1 , Δ1R;2 Ă Δ1R their closures. For each t P Δ1R;2 , let ndt be the real node of Σ1t . 1R 1rc1 1 1 1 Let s1R 1 , . . . , sN be sections of U over ΔR and pV , ϕ q be a rank 1 real bundle pair over p U1 ,rc1 q so that 1 s1R r pt0 q “ xr

1 1 @ r P rN ´1s, sR N pt0 q P S‚1 , ` 1 1 ˘ ` ˘ @ t P Δ1R;2 , Vt1 , ϕt1 “ V0 , ϕ0 . 0

sR N ptq ‰ ndt

0

For each t P Δ1R and r P rN s, the orientation oxr of V0ϕ0 |xr induces 1 1R an orientation o1xr ;t of V 1ϕ at s1R r ptq via the section sr as above Proposition 9.1. For t P Δ1R;2 , oxN also induces an orientation o1ndt 1

1ϕ of Vnd so that o1xN ;t is obtained from o1ndt by translation along the t

Intermediate Cases

251

1 Ă D2 . Let positive direction of S‚1 ‚1 ` 1 ˘ ` 1 ˘ 1 1 ox;t “ oxr ;t rPrN s , o1c x;t “ oxr ;t rPrN ´1s @ t P ΔR , ˘ ˘ ` ` r @ t P Δ1R;2 . o1x;t “ o1ndt , o1ndt , o1xr ;t rPrN ´1s 1 , ϕ1 q, pV 1 , ϕ1 q, and pV 1c , ϕ1c q the restrictions of We denote by pV‚1 ‚1 ‚2 ‚2 0 0 1 1 1 pV0 , ϕ0 q to P‚1 , P1‚2 , and Σ1c , respectively. 0 Let D1 “ tDt1 u be a family of real CR-operators on pVt1 , ϕ1t q as in (7.34) so that Dt1 1 is the operator D0 in the statement of the 0 proposition. We denote by ` ˘ ` ˘ (9.26) o1t0 o1x;t0 ” o1 Vt10 , ϕ1t0 ; o1x;t0

the orientation of Dt0 continuously extending the orientations opVt1 , ϕ1t ; ox;t q with t P Δ1˚ R as above the statement of Proposition 9.2; this is the analog of the limiting orientation (9.11) for the rank 1 real bundle pVt10 , ϕ1t0 q over the symmetric surface Ct1 0 with two conjugate pairs of nodes. 1 , D 1 , and D 1c be the real CR-operators on pV 1 , ϕ1 q, Let D‚1 ‚2 0 ‚1 ‚1 1 1 1 pV‚2 , ϕ‚2 q, and pV01c , ϕ1c 0 q, respectively, induced by D0 . The exact triple 1

1ϕ 1 1 1 1 ‘D‚2 ÝÑ Vnd 0 ÝÑ D01 ÝÑ D01c ‘D‚1 ` ‘V ` ‘Vnd ÝÑ 0, 0 nd ‚1 ‚2 ` ˘ ` ` ` ` pξ, ξ1 , ξ2 q ÝÑ ξpnd‚1 q´ξ1 pnd‚1 q, ξpnd‚2 q´ξ2 pnd‚2 q, ξ2 pndq´ξ1 pndq

of Fredholm operators then determines an isomorphism ` ˘ ` 1 ˘ ` 1 ˘ ` 1 ˘ ` 1 ˘ ` 1ϕ1 ˘ bλ Vnd0 « λ D01c bλ D‚1 bλ D‚2 . λpD01 qbλ Vnd ` bλ V nd` ‚1

The orientations 1ϕ1

‚2

1c opV01c , ϕ1c 0 ; ox;0 q

1 , ϕ1 ; o1 q of D 1 , and of D01c , opV‚r ‚r nd0 ‚r

1 o1nd0 of Vnd0 and the complex orientations of Vnd ` determine an ‚r 1 orientation of D0 via the above isomorphism and thus an orientation o U1 pox q of the line bundle λp D1 q over Δ1R . 1` 1˚ 1˚ Let Δ1` R;1 Ă ΔR;1 be the intersection of ΔR;1 with the closure of ΔR 1 1 1 1 in ΔR . The restriction of p U ,rc q to ΔR;1 is the product of a family smoothing

P10;‚ ” P1‚1 YP1‚2 1 into an irreducible component P1t;‚ Ă Σ1t with Σ1c 0 . For t P ΔR;1 , 1c 1 we denote by Σt Ă Σt the union of the irreducible components

252

Spin/Pin-Structures & Real Enumerative Geometry

˘ other than P1t;‚ and by nd˘ t;‚1 , ndt;‚2 the nodes corresponding to ˘ 1 1 1c 1c nd˘ ‚1 , nd‚2 , respectively. Let pVt;‚ , ϕt;‚ q and pVt , ϕt q be the restric1 1c tions of pVt1 , ϕ1t q to P1t;‚ and Σ1c t , respectively, and Dt;‚ and Dt 1 1 1c 1c be the real CR-operators on pVt;‚ , ϕt;‚ q and pVt , ϕt q, respectively, induced by Dt1 . The exact triple (7.60) of Fredholm operators with 1 D0 replaced by D0;‚ and the exact triple 1 1 ‘Vnd ÝÑ 0, 0 ÝÑ Dt1 0 ÝÑ Dt1c0 ‘Dt1 0 ;‚ ÝÑ Vnd ` ` t0 ;‚1 t0 ;‚2 ` ˘ ` ` ` pξ, ξ‚ q ÝÑ ξpnd` t0 ;‚1 q´ξ‚ pndt0 ;‚1 q, ξpndt0 ;‚2 q´ξ‚ pndt0 ;‚2 q

of Fredholm operators induce isomorphisms ` 1 ˘ ` 1ϕ1 ˘ ` 1 ˘ ` 1 ˘ λ D0;‚ bλ Vnd0 « λ D‚1 bλ D‚2 , ` 1 ˘ ` 1 ˘ ` 1c ˘ ` 1 ˘ 1 bλ Vnd` « λ Dt0 bλ Dt0 ;‚ . λpDt0 qbλ Vnd` t0 ;‚1

t0 ;‚2

(9.27)

1

1ϕ 1 , ϕ1 ; o1 q of D 1 and o1 The orientations opV‚r ‚r nd0 ‚r nd0 of Vnd0 determine 1 via the first isomorphism in (9.27) and thus an orientation of D0;‚ 1 1 for every t P Δ1 ; the induced orientation an orientation o‚;t on Dt;‚ R;1 1 is the split orientation in the terminology of Proposition 8.7. of D0;‚ By this proposition and the first assumption in (9.21), the orientation o1‚;t with t P Δ1` R;1 agrees with the intrinsic orientation

` 1 ˘ , ϕ1t;‚ ; o1xN ;t o‚;t ” o Vt;‚ if and only if ˘@ D ` 1 1 sZ2 y`1 w1 pV ϕ q, rS‚2 sZ2 “ 0 P Z2 . xw1 pV ϕ q, rS‚1

(9.28)

We now assume that t0 P Δ1` R;1 . The conclusion of the previous paragraph then implies that the second isomorphism in (9.27) respects the restriction ot0 of the orientation o U1 pox q to λpDt1 0 q, the 1c 1c 1 orientations opVt1c0 , ϕ1c t0 ; ox;t0 q of Dt0 and o‚;t0 of Dt0 ;‚ , and the com1 plex orientations of Vnd` if and only if (9.28) holds. By the assumpt0 ;‚r

1 tion that r‚ “ N , the decomposition Σ1c t0 \Pt0 ;‚ respects the orderings of the topological components of the fixed loci, i.e. Sr1 with r P rN´1s and St10 ;r‚ . Combining these two statements with two applications of

Intermediate Cases

253

Proposition 9.2, we conclude that the orientation ot0 of Dt1 0 is the analog of the intrinsic orientation ` ˘ ` ˘ o0 o1x;t0 ” o Vt10 , ϕ1t0 ; o1x;t0 in (9.9) for the rank 1 real bundle pVt10 , ϕ1t0 q over the symmetric surface Ct1 0 with two conjugate pairs of nodes if and only if (9.28) holds. Along with two applications of Proposition 9.2, this implies that the orientation ot0 of Dt1 0 agrees with (9.26) if and only if (9.28) holds. Thus, the restriction of o U1 pox q to λpDt1 q with t P Δ1` R is the intrinsic orientation opVt1 , ϕ1t1 ; o1x;t q defined above Proposition 9.1 if and only if (9.28) holds. This in turn implies that the restriction of o U1 pox q to λpDt1 1 q is the limiting orientation o` 0 pox q in (9.17) if 0 and only if (9.28) holds. The restriction of p U1 ,rc1 q to Δ1R;2 is a family deforming pΣ10 , σ01 q to the symmetric surface in the statement of the proposition and 1 r1 , σ above (7.39). For each t P Δ1R;2 , we denote by pΣ t rt q the symmetric surface obtained from pΣ1t , σt1 q by replacing the real node ndt with 1 and nd P S 1 , by pV 1 r 1, ϕ real marked points ndt;1 P S‚1 t;2 t rt q the pullback ‚2 1 1 1 1 1 r ,σ r r 1 r1t q of pVt , ϕt q to pΣ t rt q, and by Dt the real CR-operator on pVt , ϕ induced by Dt1 . The exact triple 1 1 1 1 r 01 ÝÑ D01c ‘D‚1 ‘D‚2 ÝÑ Vnd ÝÑ 0, 0 ÝÑ D ` ‘V nd` ‚1 ‚2 ` ˘ ` ` ` pξ, ξ1 , ξ2 q ÝÑ ξpnd` ‚1 q´ξ1 pnd‚1 q, ξpnd‚2 q´ξ2 pnd‚2 q

of Fredholm operators then determines an isomorphism ` ˘ ` ˘ ` ˘ ` 1 ˘ ` 1 ˘ r 01 qbλ V 1 ` bλ V 1 ` « λ D01c bλ D‚1 bλ D‚2 . λpD nd nd ‚1

‚2

1c 1c 1 r1 ; o1 1 The orientations opV01c , ϕ1c ‚r nd;0 q of D‚r 0 ; ox;0 q of D0 and opV‚r , ϕ r1 and the complex orientations of V 1 ` determine an orientation of D nd‚r

0

r 1 for every via the above isomorphism and thus an orientation r ot of D t 1 t P ΔR;2 . 1 1 By the assumption that r‚ “ N , the decomposition Σ1c 0 \P‚1 \P‚2 respects the orderings of the topological components of the fixed 1 , and S 1 . Along with r 0, σ r0 q, i.e. Sr1 with r P rN ´ 1s, S‚1 loci of pΣ ‚2

254

Spin/Pin-Structures & Real Enumerative Geometry

Proposition 9.1(5) applied twice, this implies that the orientation r o0 1 r of D0 above is the analog of the intrinsic orientation ` 1 ˘ ` ˘ o0 r o1x;0 ox;0 ” o V01 , ϕ10 ; r r10 q over the symmetric surface in (9.9) for the rank 1 real bundle pVr01 , ϕ 1 1 r ,σ pΣ 0 r0 q with two conjugate pairs of nodes. Combining this with two applications of Proposition 9.2, we conclude that r ot10 is the orientation ` ˘ ` ˘ ` ˘ r r1t0 ; r o1x;t1 “ o Vr0 , ϕ r0 ; r ox o0 ox;t10 ” o Vrt10 , ϕ 0

r 0 in (9.13). Thus, the restriction of o U1 pox q to λpD 1 1 q is r1 “ D of D t0 t0 the intrinsic orientation o0 pox q of Dt1 0 “ D0 in (9.14). Along with the conclusion regarding the limiting orientation o` 0 pox q above, this po implies that o0 pox q and the limiting orientation o` x q of D0 are the 0 same if and only if (9.28) holds and establishes the r‚ “ N case of the proposition.  9.3

Orientations from OSpin-Structures

Suppose pΣ, σq is a smooth decorated symmetric surface, pV, ϕq is a real bundle pair over pΣ, σq, and os P OSpinΣ pV ϕ q is a relative OSpinstructure on the real vector bundle V ϕ over Σσ Ă Σ. We show in the following that os determines an orientation oos pV, ϕq of every real CR-operator D on pV, ϕq. ˘ C cq, t1 P ΔR , pVr , ϕq, r Let C0 , pP1r , τr q, Sr1 , D2r˘ , Σ` 0 Ă Σ0 , ndr , p U,r ` r r0r q, D” tDt u, D0r , and D0 be as below (9.1). By the compatpV0r , ϕ ibility condition in Definition 6.3, a tuple ź ˘ ` ˘ ` r 0 ” pos0;r qSr1 Pπ0 pΣσ q , os0;C P OSpP1r Vr ϕr |Sr1 os ` ˘ ˆ OSpΣC Vr ϕr|H 0

Sr1 Pπ0 pΣσ q

(9.29)

of relative OSpin-structures on the restrictions of Vr0ϕr0 to the σ0 -fixed loci contained in P1r and ΣC 0 determines a relative OSpin-structure os0 ϕ r0 r on V0 and thus a relative OSpin-structure ost on the restriction

Intermediate Cases

255

r 0 as in (9.29) so that of Vr ϕr to Σσt t Ă Σt for each t P ΔR . Fix a tuple os the induced relative OSpin-structure ost1 is os. The exact triple (9.3) of Fredholm operators again induces an isomorphism (9.4) of the determinants of the associated real CRoperators. For each Sr1 P π0 pΣσ q, let oos0;r pVr0r , ϕ r0r q be the orientation of D0r as above Proposition 8.9. We denote by oos0;C pVr |Σ` q the com-

plex orientation of D0` if

0

D @ ` os0;C pΣ` 0 q ” w2 pos0;C q, rΣ0 sZ2 P Z2

vanishes and the opposite orientation otherwise. Along with the chosen order on π0 pΣσ q and the complex orientations of Vrnd` , these r r r0 q of D0 via the isoorientations determine an orientation oos pV0 , ϕ morphism (9.4) and thus an orientation oos pV, ϕq of D “ Dt1 . If the real bundle pair pV, ϕq is of rank 1 and o is an orientation on V ϕ , we let o0 pV, ϕ; oq ” oιΣ pos0 pV ϕ ,oqq pV, ϕq

(9.30)

denote the orientation of D determined by the image ˘ ` ` ˘ ιΣ os0 pV ϕ , oq P OSpΣ V ϕ of the canonical OSpin-structure os0 pV ϕ , oq on pV ϕ , oq under the first map in (6.3) with X “ Σ. Proposition 9.4. Suppose pV, ϕq is a real bundle pair over a smooth decorated symmetric surface pΣ, σq, os P OSpinΣ pV ϕ q is a relative OSpin-structure on the real vector bundle V ϕ over Σσ Ă Σ, and D is a real CR-operator on pV, ϕq. p1q The orientation oos pV, ϕq of D constructed above does not depend the choice of an admissible tuple (9.29). p2q If pV, ϕq is of rank 1 and o is an orientation on V ϕ , the orientation (9.30) of D is the same as the intrinsic orientation opV, ϕ; ox q of Proposition 9.1 for a tuple ox of orientations as in (9.1) induced by the orientation o. p3q If pΣ, σq “ pS 2 , τ q, the orientation oos pV, ϕq of D is the same as the orientation constructed above Proposition 8.9.

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The orientations oos pV, ϕq of real CR-operators on real bundle pairs pV, ϕq over smooth decorated symmetric surfaces pΣ, σq constructed above satisfy all applicable CROrient properties of Sections 7.2 and 7.3. Proof of Proposition 9.4(1)–(3). (1) Suppose ź ˘ ` ˘ ` r 10 ” pos10;r qSr1 Pπ0 pΣσ q , os10;C P OSpP1r Vr ϕr |Sr1 os ` ˘ ˆ OSpΣC Vr ϕr|H

Sr1 Pπ0 pΣσ q

0

is another tuple of relative OSpin-structures inducing the relative OSpin-structure os as below (9.29). For each Sr1 P π0 pΣσ q, let os0;r pD2r` q and os10;r pD2r` q be the collections of trivializations of Vr ϕr induced by the relative OSpin-structures os0;r and os10;r , respectively, and by the inclusion D2r` Ă P1r . Define # 0 P Z2 , if os0;r pD2r` q “ os10;r pD2r` q; r “ 1 P Z2 , if os0;r pD2r` q ‰ os10;r pD2r` q; 1 ` C “ os0;C pΣ` 0 q´os0;C pΣ0 q P Z2 .

r 0 and os r 10 induce the same relative OSpin-structure on Since os the restriction of Vr ϕr to Σσt t Ă Σt for each t P Δ˚R , the equivalence classes os0 pΣb0 q and os10 pΣb0 q of trivializations of the restriction of Vr ϕr to Σσ0 0 Ă Σ0 determined by os0 and os10 , respectively, and the normalization map ğ rb ” D2` \Σ` u: Σ 0 0 ÝÑ Σ0 Sr1 Pπ0 pΣσ q

are the same. Thus,

ÿ

r ` C Sr1 Pπ0 pΣσ q

“ 0 P Z2 .

Along with Proposition 8.11, this implies that the number of factors on the right-hand side of the isomorphism (9.4) for which the r 10 differ is even. Thus, the orir 0 and os orientations determined by os r 0 and os r 10 via the entations of D0 and D “ Dt1 determined by os isomorphism (9.4) are the same.

Intermediate Cases

257

(2) For the purposes of constructing the orientation (9.30), we can choose the tuple in (9.29) so that ` ` ˘˘ @ Sr1 P π0 pΣσ q and os0;C pΣ` os0;r “ ιP1r os0 pV ϕ , oq|Sr1 0 q “ 0. The definition of the orientation (9.30) above Proposition 9.4 then becomes identical to that of the intrinsic orientation of Proposition 9.1 for a tuple ox of orientations as in (9.1) induced by the orientation o of V ϕ . (3) In this case, |π0 pΣσ q| “ 1. There are canonical homotopy classes of identifications of pP11 , τ1 , D21` q with pΣ, σ, Σb q ” pS 2 , τ, D2` q and of the restriction of Vr ϕr to Σσ0 0 with the real vector bundle V ϕ over S 1 Ă Σ. We can choose the tuple in (9.29) so that os0;1 is identified with os via these homotopy classes and 1 os0;C pΣ` 0 q ” os0;C pP` q “ 0 P Z2 .

The definition of the orientation oos pV, ϕq above Proposition 9.4 then becomes identical to that above Proposition 8.9.  Proof of CROrient 1os, 2(a), and 6(a) properties. We continue with the notation above the statement of Proposition 9.4. With the notation as in the statement of the CROrient 1os(1) property and in the proof of Proposition 9.1(2), the first two identities in (9.5) still apply; the third becomes ` ˘ r (9.31) indC D0` “ n 1´gpΣ` ˚0 q `deg V |Σ` . ˚0

The change in the choice of the half-surface Σb˚ of pΣ˚ , σq acts by the complex conjugation on the complex orientations of D0` |Σ` and each ˚0 1 P π pΣσ q. By the CROrient 1os(1) property for pP1 , τ q Vrnd` with S 0 r ˚ r provided by Proposition 8.9, this change preserves the orientation r0r q of D0r with Sr1 P π0 pΣσ˚ q if and only if os0;r pP1r q vanoos0;r pVr0r , ϕ r0 q and oos pV, ϕq do not ishes in Z2 . Thus, the orientations oos pVr0 , ϕ depend on the choice of half-surface Σb˚ of pΣ˚ , σq if and only if ˇ ˇ @ D indC D0` |Σ` ´ nˇπ0 pΣσ˚ qˇ ` w2 pos0;C q, rΣC 0˚ sZ2 ` ˚0

ÿ

os0;r pP1r q

Sr1 Pπ0 pΣσ ˚q

Spin/Pin-Structures & Real Enumerative Geometry

258

ˇ ˇ “ indC D0` |Σ` ´ nˇπ0 pΣσ˚ qˇ ` ˚0

D @ ` w2 posq, rΣ˚ sZ2

ÿ deg Vr0r 2 σ

Sr1 Pπ0 pΣ˚ q

vanishes in Z2 . Combining this with (9.31) and the first two identities in (9.5), we obtain the CROrient 1os(1) property. Since the parity of the index of D0r is the same as the parity of n, the CROrient 1os(2) property follows from the Direct Sum property for the determinants of Fredholm operators; see [53, Section 2]. The CROrient 1os(3) property holds by definition. Propositions 9.4(3) and 8.9 imply the CROrient 6(a) property. The second statement of the CROrient 2(a) property follows immediately from its pS 2 , τ q case provided by Proposition 8.9. Let η P H 2 pΣ, Σσ ; Z2 q. Since the composition of the retraction isomorphism and the restriction homomorphism ` ˘ ` ˘ ` ˘ H 2 Σ0 , Σσ0 0 ; Z2 « H 2 U, Urc; Z2 ÝÑ H 2 Σ, Σσ ; Z2 is surjective, some η0 P H 2 pΣ0 , Σσ0 0 ; Z2 q is mapped to η under this composition. The tuple ` ˘ r 0 ” pη0 |pP1r ,Sr1 q ¨os0;r qS 1 Pπ0 pΣσ0 q , η0 |ΣC ¨os0;C η0 ¨ os r 0

0

then induces the relative OSpin-structure η ¨os as below (9.29). By the first statement of the CROrient 2(a) property for pS 2 , τ q provided by Proposition 8.9 and the definition of the orientation oos0;C pVr |Σ` q 0 r0 q and oη¨os pV, ϕq are the same as of D0` , the orientations oη¨os pVr0 , ϕ r0 q and oos pV, ϕq, respectively, if and only if oos pVr0 , ϕ ÿ @ D @ D @ D b η0 |pP1r ,Sr1 q , rD2r` sZ2 ` η0 |ΣC , rΣ` 0 sZ2 “ η, rΣ sZ2

Sr1 Pπ0 pΣσ ˚q

0

vanishes in Z2 . This establishes the first statement of the CROri ent 2(a) property.

Intermediate Cases

Proof of CROrient 4(a) property. pV1 , ϕ1 q, pV2 , ϕ2 q, os1 , os2 , D1 ” DpV1 ,ϕ1 q , pV, ϕq ” pV1 , ϕ1 q\pV2 , ϕ2 q,

259

Suppose pΣ1 , σ1 q, pΣ2 , σ2 q,

D2 ” DpV2 ,ϕ2 q ,

os ” os1 \os2 ,

and

D ” D1 \D2

are as in the statement of this property on page 114. Let p U1 ,rc1 q, p U2 ,rc2 q, pVr1 , ϕ r1 q, pVr2 , ϕ r2 q, ` p1q ` p2q p1q ˘ p2q ˘ σ σ r p2q r p1q os os 0 ” pos0;r qSr1 Pπ0 pΣ 1 q , os0;C , 0 ” pos0;r qSr1 Pπ0 pΣ 2 q , os0;C , 1

2

D1;t , and D2;t be as in the construction of the orientations oos1pV1 , ϕ1 q and oos2pV2 , ϕ2 q above Proposition 9.4. The orientation oos pV, ϕq of D is obtained via this construction applied with ˘ ` ˘ ` ˘ ` ˘ ` ˘ ` ˘ ` U,rc “ U1 ,rc1 \ U2,rc2 , r1 \ Vr2 , ϕ r2 , Vr , ϕ r “ Vr1 , ϕ Dt “ D1;t \D2;t , ` p2q p1q p2q ˘ σ σ r 0 “ posp1q os 0;r qSr1 Pπ0 pΣ 1 q , pos0;r qSr1 Pπ0 pΣ 2 q , os0;C \os0;C . 1

2

In particular, p1q ` ` ˘ p2q ` ` ˘ os0;C pΣ` 0 q “ os0;C Σ1;0 `os0;C Σ2;0 P Z2 .

The proof now proceeds via the diagram of Figure 9.2, as in the proof of Proposition 9.1(5), with the orientations (9.6) replaced by ` ˘ ` ˘ ` ˘ r1;0 , oos Vr0 , ϕ r0 , and oos2 Vr2;0 , ϕ r2;0 , oos1 Vr1;0 , ϕ respectively, the orientations (9.7) replaced by $ &o p1q pVr0r , ϕ r0r q, if Sr1 P π0 pΣσ1 1 q; os 0;r r0r q “ oos0;r pVr0r , ϕ %o p2q pVr0r , ϕ r0r q, if Sr1 P π0 pΣσ2 2 q; os 0;r

` , D0` , and and the complex orientations on the determinants of D1;0 ` replaced by D2;0

` ˘ oosp1q Vr1 |Σ` , 0;C

respectively.

1;0

` ˘ oos0;C Vr |Σ` , 0

and

` ˘ oosp2q Vr2 |Σ` , 0;C

2;0



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Spin/Pin-Structures & Real Enumerative Geometry

Proof of CROrient 5(a) property. os ” xxos1 , os2 yy eR ,

D 1 ” DpV 1 ,ϕ1 q ,

Suppose pΣ, σq, e, os1 , os2 , and D 2 ” DpV 2 ,ϕ2 q

are as in the statement of this property on page 116. Let p U,rcq, pVr 1 , ϕ r1 q, pVr 2 , ϕ r2 q, ` ` ˘ ˘ r 20 ” pos20;r qSr1 Pπ0 pΣσ q , os20;C , r 10 ” pos10;r qSr1 Pπ0 pΣσ q , os10;C , os os Dt1 , and Dt2 be associated objects as in the construction above Proposition 9.4. The orientation oos pV, ϕq of D 1 ‘ D 2 is obtained via this construction applied with ˘ ` ˘ ` ˘ ` r1 ‘ Vr 2 , ϕ r2 , Dt “ Dt1 ‘Dt2 , Vr , ϕ r “ Vr 1 , ϕ ` ˘ r 0 “ pos0;r ” xxos10;r , os20;r yy‘ qSr1 Pπ0 pΣσ q , os0;C ” xxos10;C , os20;C yy‘ . os The exact triple (9.3) of Fredholm operators induces the exact triples of Fredholm operators given by the rows in the diagram of Figure 9.6. The short exact sequence ˘ ` ˘ ` ˘ ` r10 ÝÑ Vr0 , ϕ r0 ÝÑ Vr02 , ϕ r20 ÝÑ 0 0 ÝÑ Vr01 , ϕ of real bundle pairs induces the exact triples of Fredholm operators given by the columns in this diagram.

Figure 9.6. Commutative square of exact rows and columns of Fredholm operators for the proof of the CROrient 5(a) property.

Intermediate Cases

261

By definition, the rows in this diagrams respect the orientations ` ˘ ` ˘ ` ˘ r10 , oos0 Vr0 , ϕ r0 , and oos20 Vr02 , ϕ r20 (9.32) oos10 Vr01 , ϕ of the operators in the left column, the orientations ` 1 1 ˘ ` ˘ ` 2 2 ˘ oos10;r Vr0r ,ϕ r0r , oos0;r Vr0r , ϕ r0r , and oos20;r Vr0r ,ϕ r0r

(9.33)

of the operators in the direct sums in the middle column, the orientations ` ˘ ` ˘ ` ˘ (9.34) oos10;C Vr 1 |Σ` , oos0;C Vr |Σ` , and oos20;C Vr 2 |Σ` 0

0

0

on the last summands in middle column, and the complex orientations of the vector spaces in the right column. The last column respects the complex orientations; the exact triple formed by the last summands in the middle column respects the orientations (9.34), since 1 ` 2 ` os0;C pΣ` 0 q “ os0;C pΣ0 q`os0;C pΣ0 q.

The exact triple formed by the r-th summands in the middle column respects the orientations (9.33). Since 1 “ rk V 1 , ind D0r

2 ind D0r “ rk V 2 ,

and ind D01` P 2Z,

the last two statements and Lemma 8.13 imply that the middle column respects the direct sum orientations if and only if (7.25) holds. 1 Since the (real) dimensions of Vrnd ` are even, another application of r Lemma 8.13 combined with the statements implies that the left column respects the orientations (9.32) if and only if (7.25) holds. The claim now follows from the continuity of the isomorphisms ` ˘ ` ˘ ` ˘ λ Dt « λ Dt1 bλ Dt2 with respect to t P ΔR .



Proof of CROrient 7C(a) property. Let C0 be as in (7.36) with k, l “ 0. We denote by os1 the standard relative OSpin-structure on the trivial line bundle Σσ0 0 ˆR over Σσ0 0 Ă Σ0 and by B¯ the standard real CR-operator on the real bundle pair pΣ0ˆC, σ0ˆcq over pΣ0 , σ0 q. Let pL, φq be a rank 1 real bundle pair over pΣ0 , σ0 q so that the

262

Spin/Pin-Structures & Real Enumerative Geometry

real line bundle Lφ over Σσ0 0 is orientable, osL be a relative OSpinstructure on the real line bundle Lφ over Σσ0 0 Ă Σ0 , and B¯L be a real CR-operator on pL, φq. For n P Z` , let pVn , ϕn q “ pΣ0 ˆCn , σ0 ˆcq,

osn;L “ Stn´1 posL q Lφ

“ xxos1 , osn´1;L yy‘ , ˘ ` ˘ ` pVn;L , ϕn;L q “ Vn´1 , ϕn´1 ‘pL, φq “ V1 , ϕ1 ‘pVn´1;L , ϕn´1;L q. (9.35) Let Dn be the real CR-operator on pVn;L , ϕn;L q given by the n-fold direct sum of the operators B¯ on each factor and Dn;L “ Dn´1 ‘ B¯L . We first note that the CROrient 7C(a) holds for os1 and os1;L ” osL , i.e. ` ˘ ` ˘ and oosL pL, φq “ o1osL pL, φq. (9.36) oos1 V1 , ϕ1 “ o1os1 V1 , ϕ1 The first equality follows immediately from Propositions 9.4(2) and 9.2 so does the second if osL “ ιΣ0 pos0 pLφ , oqq for some orientation o on Lφ . In general, ˘ ` osL “ η0 ¨ιΣ0 os0 pLφ , oq for some orientation o on V ϕ and some ` ˘ ` ˘ ` ˘ r σr0 ; Z2 , H 2 U, Urc; Z2 ; r 0, Σ η0 P H 2 Σ0 , Σσ0 0 ; Z2 « H 2 Σ 0 see the RelSpinPin 2 property on page 81. By the CROrient 2(a) property, ` oosL pL, φq “ oιΣ pos0 pLφ ,oqq pL, φq resp. o1osL pL, φq 0 ˘ 1 “ oιΣ pos0 pLφ ,oqq pL, φq 0

if and only if xη, Σb0 y “ 0. Along with the second statement in (9.36) with osL replaced by ιΣ0 pos0 pLφ , oqq, this implies the second statement itself. The exact triple (7.37) induces exact triples of Fredholm operators given by the rows in the diagram of Figure 9.7. The splitting in (9.35) induces exact triples of Fredholm operators given by the columns in

Intermediate Cases

263

Figure 9.7. Commutative square of exact rows and columns of Fredholm operators for the proof of the property of the CROrient 7C(a) property.

this diagram. The right column respects the complex orientations of the associated vector spaces. By the CROrient 5(a) property, the left (resp. middle) column respects the orientations ` ˘ ` ˘ ` ˘ o1os1 V1 , ϕ1 , o1osn;L Vn;L , ϕn;L , and o1osn´1;L Vn´1;L , ϕn´1;L , ` ˘ ` ˘ ` r r1 , oos r rn;L , and resp. oos r 1 V1 , ϕ r n;L Vn;L , ϕ ` ˘˘ r rn´1;L oos r n´1;L Vn´1;L , ϕ σ0 ˘ ` if and only if pn´1q |π0 pΣ2 0 q| is even. By definition, the rows in Figure 9.7 respect the orientations r r and the complex orientation on V | ` with oos pV, ϕq, oos r pV , ϕq, nd ˘ ` ˘ ` ˘ ` V, ϕ, os “ V1 , ϕ1 , os1 , Vn;L , ϕn;L , osn;L , ˘ ` ˆ Vn´1;L , ϕn´1;L , osn´1;L ,

depending on the row. Along with the conclusion of the previous paragraph, (9.36), Lemma 8.13, and the evenness of the real dimension of V1 |nd` , this implies that ` ˘ ` ˘ @ n P Z` . oosn;L Vn;L , ϕn;L “ o1osn;L Vn;L , ϕn;L By the RelSpinPin 5 property, every relative OSpin-structure on the ϕ real vector bundle Vn;Ln;L over Σσ0 0 Ă Σ0 equals osn;L for some relative OSpin-structure osL on Lφ . By [17, Theorem 1.1], every real

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Spin/Pin-Structures & Real Enumerative Geometry

bundle pair pV0 , ϕ0 q over pΣ0 , σ0 q with V0ϕ0 orientable is isomorphic to pVn;L , ϕn;L q for some rank 1 real bundle pair pL, φq over pΣ0 , σ0 q with Lφ orientable. The last three statements imply the CROrient 7C(a) property.  Proof of CROrient 7H3(a) property. Let C0 and r‚ be as in the statement of this property. In light of the CROrient 1os(2) ¯ pL, φq, property, we can assume that r‚ “ |π0 pΣσ0 0 q|. We take os1 , B, ¯ osL , and BL as in the proof of the CROrient 7C(a) property and again define pVn , ϕn q, pVn;L , ϕn;L q, osn;L , Dn , and Dn;L as in (9.35) and just above and below. We denote by o1;nd the orientation of V1ϕ1 |nd ϕ induced by os1 and by on;L;nd the orientation of Vn;Ln;L |nd induced by osn;L . By the same reasoning as below (9.36), with Proposition 9.3 used in place of 9.2, ` ˘ ` ˘ oos1 V1 , ϕ1 “ o` os1 V1 , ϕ1

and oosL pL, φq “ o` osL pL, φq.

(9.37)

The exact triple (7.40) induces exact triples of Fredholm operators given by the rows in the diagram of Figure 9.8. The splitting in (9.35) induces exact triples of Fredholm operators given by the columns in this diagram. The right column respects the orientations o1;nd , on;L;nd , and osn´1;L;nd of the associated vector spaces. By the CROrient 5(a) property, the left (resp. middle) column respects the

Figure 9.8. Commutative square of exact rows and columns of Fredholm operators for the proof of the property of the CROrient 7H3(a) property.

Intermediate Cases

265

orientations ` ˘ ` ˘ ` ˘ o` o` and o` os1 V1 , ϕ1 , osn;L Vn;L , ϕn;L , osn´1;L Vn´1;L , ϕn´1;L ` ` ˘ ` ˘ r r1 , oos r rn;L , and resp. oos r 1 V1 , ϕ r n;L Vn;L , ϕ ` ˘˘ r rn´1;L oos r n´1;L Vn´1;L , ϕ σ0 ˘ σ0 ` ˘ ` if and only if pn´1q |π0 pΣ2 0 q| (resp. pn´1q |π0 pΣ02 q|`1 ) is even. By definition, the rows in Figure 9.8 respect the orientations r r and the orientation oV ;nd on V ϕ |nd with oos pV, ϕq, oos r pV , ϕq, ` ˘ ` ˘ ` ˘ V, ϕ, os, oV ;nd “ V1 , ϕ1 , os1 , o1;nd , Vn;L , ϕn;L , osn;L , on;L;nd , ˘ ` Vn´1;L , ϕn´1;L , osn´1;L , on´1;L;nd ,

depending on the row. Since ˙ ˆ ˙ ˆ |π0 pΣσ0 0 q| ` 1 |π0 pΣσ0 0 q| ` pn´1q pn´1q 2 2 ˘` ˘ ` ` dim V1ϕ1 |nd ind Dn´1;L ˇ ˇ ˇ ˇ – pn´1qˇπ0 pΣσ0 0 qˇ ` pn´1qˇπ0 pΣσ0 0 qˇ – 0 mod 2, the last two statements, (9.37), and Lemma 8.13 imply that ` ˘ ` ˘ @ n P Z` . oosn;L Vn;L , ϕn;L “ o` osn;L Vn;L , ϕn;L Similar to the end of the proof of the CROrient 7C(a) property, this  in turn implies the CROrient 7H3(a) property.

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Chapter 10

Orientations for Twisted Determinants

Let C be a smooth decorated marked symmetric surface as in (7.3) and pV, ϕq be a real bundle pair over C. In Section 9.1, we define an orientation o CpV ϕ , ox q of ˆ k ˙ k à ϕ â R Vx i “ λpVxϕi q. (10.1) λ CpV, ϕq ” λ i“1

i“1

This orientation generally depends on a tuple ox of orientations ox˚r of fibers of V ϕ at one point x˚r of each topological component Sr1 of the fixed locus Σσ of C and on the decorated structure of C. In Section 10.2, we define an orientation op pV, ϕ; ox q of every real CRoperator D on the real bundle pair pV, ϕq. This orientation a priori depends on a relative Pin-structure p on the real vector bundle V ϕ over Σσ Ă Σ, on a tuple ox of orientations as above, and on the decorated structure of C. We establish properties of the orientations o CpV ϕ , ox q and op pV, ϕ; ox q, analogous to the CROrient properties of Sections 7.2 and 7.3, in Section 10.1 and in Sections 10.2 and 10.3, respectively. The orientations o CpV ϕ , ox q and op pV, ϕ; ox q determine an orientation o C;p pV, ϕ; ox q of the twisted determinant r CpDq ” λR pV, ϕq˚ bλpDq λ C

(10.2)

of D. In Section 10.4, we readily deduce properties of the orientations o C;p pV, ϕ; ox q from the properties of the orientations o CpV ϕ , ox q and 267

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op pV, ϕ; ox q established in Sections 10.1–10.3. By Corollary 10.9(4), the orientation o C;p pV, ϕ; ox q does not depend on the choice of ox if pV, ϕq is C-balanced; we then denote it by o C;p pV, ϕq. In this case, the properties established in Section 10.4 reduce to the CROrient properties of Sections 7.2 and 7.3 concerning orientations induced by Pin-structures. This completes the proof of Theorem 7.1(b). Given a symmetric surface pΣ, σq, possibly nodal, we denote by N the number of the topological components of its fixed locus Σσ . 10.1

Orientations of the Twisting Target

Suppose C is a smooth decorated marked symmetric surface as in (7.3) and pV, ϕq is a real bundle pair over C. For each topological component Sr1 of the σ-fixed locus Σσ , let kr P t0u\rks

and j1r p Cq, . . . , jkrr p Cq P rks

be as in (7.10). We take x˚r P Sr1 to be the real marked point xj1r p Cq if kr ą 0 and any point in Sr1 if kr “ 0. Fix a tuple ` ˘ (10.3) ox ” ox˚r S 1 Pπ0 pΣσ q r

x˚r .

of orientations ox˚r of V at For each real marked point xi P Sr1 , the orientation ox˚r induces an orientation oxi of Vxϕi by the transfer along the positive direction of Sr1 determined by the chosen half-surface Σb of Σ. Along with the ordered decompositions (7.11) and (7.12), the orientations oxi in turn determine an orientation o CpV ϕ , ox q of (10.1). If e is a short exact sequence of real bundle pairs over pΣ, σq and ` ˘ ` ˘ (10.4) o1x ” o1x˚r S 1 Pπ0 pΣσ q and o2x ” o2x˚r S 1 Pπ0 pΣσ q ϕ

r

r

1 , ϕ1 q

and pV 2 , ϕ2 q, respecare tuples of orientations for fibers of pV tively, as in (10.3), we denote by ` ˘ xxo1x , o2x yy eR ” ox˚r S 1 Pπ0 pΣσ q r

the tuple of orientations for fibers of pV, ϕq so that the restriction of the induced exact sequence eR in (7.22) to each x˚r respects the orientations o1x˚ , ox˚r , and o2x˚ . The following observations are straightr r forward.

Orientations for Twisted Determinants

269

Lemma 10.1. Let pV, ϕq be a rank n real bundle pair over a smooth decorated marked symmetric surface C and ox be a tuple of orientations of V ϕ at points x˚r in Sr1 Ă Σσ as in (10.3). (1) The orientation o CpV ϕ , ox q of λRCpV, ϕq does not depend on the choice of half-surface Σb˚ of an elemental component Σ˚ of pΣ, σq if and only if ÿ

ˆ @

Sr1 Pπ0 pΣσ ˚q

w1 pV

ϕ

q, rSr1 sZ2

˙ ˆ ˙ ˙ ˆ D kr ´1 kr ´1 `n “ 0 P Z2 . 1 2

(2) The interchange in the ordering of two consecutive compo1 of Σσ preserves the orientation o CpV ϕ , ox q nents Sr1 and Sr`1 if and only if nkr kr`1 P 2Z. (3) The interchange of two real marked points xjir p Cq and xj r1 p Cq on i the same connected component Sr1 of Σσ with 2 ď i, i1 ď kr preserves o CpV ϕ , ox q. The combination of the interchange of the real points xj1r p Cq and xjir p Cq with 2 ď i ď kr and the replacement of the component ox˚r ” oxjr p Cq in (10.3) by oxjr p Cq preserves o CpV ϕ , ox q 1 i if and only if @ D˘ ` npkr ´1q` w1 pV ϕ q, rSr1 sZ2 pi´1q “ 0 P Z2 .

(10.5)

(4) The reversal of the component orientation ox˚r in (10.3) preserves o CpV ϕ , ox q if and only if kr P 2Z. Lemma 10.2. Suppose C is a smooth decorated marked symmetric surface, e is a short exact sequence of real bundle pairs over C as in (7.21), and o1x and o2x are tuples of orientations for fibers of pV 1 , ϕ1 q and pV 2 , ϕ2 q, respectively, as in (10.4). The 1 isomorphism (7.23) respects the orientations o CpV 1ϕ , o1x q, o CpV` ϕ˘, 2 xxo1x , o2x yy eR q, and o CpV 2ϕ , o2x q if and only if prk V 1 qprk V 2 q k2 is even. Suppose C0 is a decorated marked symmetric surface as in (7.36) which contains precisely one conjugate pair pnd` , nd´ q of nodes and no other nodes and pV0 , ϕ0 q is a real bundle pair over C0 . The fixed locus Σσ0 0 Ă Σ0 in this case is a disjoint union of circles. Given a tuple ox of orientations for fibers of V0ϕ0 as in (10.3), we define an

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orientation o C0 pV0ϕ0 , ox q of ˆ k ˙ k ` à ϕ0 â ˘ R V0 |xi “ λ V0ϕ0 |xi λ C0 pV0 , ϕ0 q ” λ i“1

(10.6)

i“1

as above Lemma 10.1. We call it the intrinsic orientation of λRC0 pV0 , ϕ0 q induced by ox . We define the limiting orientation o1C0 pV0ϕ0 , ox q of (10.6) induced by ox similar to the construction ˚ of the limiting orientation above Proposition 9.2 with sR r p0q “ xr . ϕ0 In this case, it is immediate that the orientations o C0 pV0 , ox q and o1C0 pV0ϕ0 , ox q are the same. Suppose C0 is a decorated marked symmetric surface as in (7.36) which contains precisely one H3 node nd and no other nodes and 1 Ă Σ , S1 Ă Σ , pV0 , ϕ0 q is a real bundle pair over C0 . Let S‚1 ‚1 ‚2 ‚2 1 1 r0, C r ,C r , and pVr0 , ϕ r‚ , k‚1 , k‚2 , nd1 , nd2 , C r q be as in the para0 1 2 graph containing (7.39) and just below and rp C0 q, j11 p C0 q, j21 p C0 q, and δR p C0 q be as in the sentence containing (7.43) and just below. The fixed locus Σσ0 0 Ă Σ0 in this case is a disjoint union of circles 1 Y S 1 of two circles. We take Sr1 with r ‰ r‚ and of the wedge S‚1 nd ‚2 x˚r P Sr1 for r ‰ r‚ as above (10.3) and x˚r‚ “ nd. For r “ 1, 2, let #  ( 1 ă 8; if i ” inf j P rks : xj P S‚r xi , x‚r “ ndr , otherwise. Given a tuple of orientations of V0ϕ0 |x˚r as in (10.3), denote by ond1 and ond2 the induced orientations of Vr0ϕr0 at nd1 and nd2 , respectively. Let o‚1 be the orientation of V0ϕ0 |x‚1 obtained by transferring ond 1 determined by the chosen halfalong the positive direction of S‚1 b surface Σ of Σ. We denote by o‚2 (resp. o1‚2 ) the orientation of V0ϕ0 |x‚2 obtained by transferring ond along the positive direction 1 (resp. transferring o 1 of S‚2 nd first around S‚1 back to an orientaϕ0 1 1 tion ond of V0 |nd and then transferring ond along the positive direc1 ). tion of S‚2 The tuples ` ˘ r and ox;1 ” o‚1 , ond2 , pox˚r qSr1 Pπ0 pΣσ0 q,r‰r‚ 0 ˘ ` r ox;2 ” ond1 , o‚2 , pox˚r qSr1 Pπ0 pΣσ0 q,r‰r‚ 0

Orientations for Twisted Determinants

of orientations of fibers of Vr0ϕr0 determine orientations ` ˘ ` ϕr0 ˘ r r ox;1 and o C0 ;1 V0ϕ0 , ox ” o C r 1 V0 , r 1 ` ˘ ` ϕr0 ˘ r r ox;2 o C0 ;2 V0ϕ0 , ox ” o C r 1 V0 , r

271

(10.7)

2

of the lines

` ˘ r r0 q « λRC pV0 , ϕ0 qbλ V ϕ0 |nd λRC and r 1 pV0 , ϕ 0 0 1 ` ˘ r r0 q « λRC pV0 , ϕ0 qbλ V ϕ0 |nd , λRC r 1 pV0 , ϕ 0 0 2

(10.8)

respectively. We denote the orientations of λRC0 pV0 , ϕ0 q induced by the orientations (10.7) and ond via the isomorphisms (10.8) by o C0 ;1 pV0ϕ0 , ox q and o C0 ;2 pV0ϕ0 , ox q, respectively. Suppose in addition that p U,rcq is a flat family of deformations ˚ of C0 as in (7.32) over a disk Δ Ă C, Δ` R Ă ΔR is as above (7.40), and pV, ϕq is a real bundle pair over p U,rcq extending pV0 , ϕ0 q. Let rc R sR 1 , . . . , sN be sections of U over ΔR so that ˚ sR r p0q “ xr @ r ‰ r‚ ,

sR r‚ p0q “ x‚rp C0 q .

For each t P ΔR and r ‰ r‚ , the orientation ox˚r induces an orientaR˚ ϕ tion oxr ;t of V ϕ at sR r ptq via the vector bundle sr V . If rp C0 q “ 1 1 (resp. rp C0 q “ 2), the orientation o‚1 (resp. o‚2 ) similarly induces an orientation ox˚r‚ ;t of V ϕ at sR r‚ ptq. Let ` ˘ ` ˘ ox;t “ ox˚r S 1 Pπ0 pΣσ q , ot “ o Ct Vtϕt , ox;t . r

We denote by

` ˘ ` ϕ0 ˘ ` o` C0 ox ” o C0 V0 , ox

(10.9)

the orientation of λRC0 pV0 , ϕ0 q obtained as the continuous extension of the orientations ot with t P Δ` R. We define ` ˘ @ D` 1 ˘ 1 1 R C0 ; V0ϕ0 “ w1 pV0ϕ0 q, rS‚1 sZ2 `rS‚2 sZ2 jrp C0 q p C0 q`prp C0 q´1qk‚1 .

For r ˚ “ 1, 2, let

` ˘` ˘ δr˚ p C0 q “ k‚r˚ ´1 jr1 ˚ p C0 q`r˚ , @ “ 1 ` ˘ @ D 1 D 1 w1 pV0ϕ0 q, S‚1 sZ2 k‚2 . r˚ C0 ; V0ϕ0 “ w1 pV0ϕ0 q, rS‚r ˚ sZ2 jr ˚ p C0 q `

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272

Lemma 10.3. Suppose C0 is a decorated marked symmetric surface which contains precisely one H3 node and no other nodes, pV0 , ϕ0 q is a rank n real bundle pair over C0 , and ox is a tuple of orientations of V0ϕ0 at points x˚r P Sr1 for r ‰ r‚ and x˚r‚ “ nd. Let r ˚ “ 1, 2. The intrinsic orientation o C0 ;r˚ pV0ϕ0 , ox q and the limiting orientation (10.9) of λRC0 pV0 , ϕ0 q are the same if and only if ¨

˛

˚` ÿ ˘ ˚ n˚ kr‚ ´1 δR p C0 q`δr˚ p C0 q ` ˝ 1

σ

Sr Pπ0 pΣ0 0 q rąr‚

‹ ‹ kr‹ ‚

` ˘ ` ˘ ` ˘ ` R C0 ; V0ϕ0 `r˚ C0 ; V0ϕ0 “ n r ˚ ´1 `2Z. Proof. In light of Lemma 10.1(2), we can assume that r‚ “ |π0 pΣσ0 0 q|. By the definitions of intrinsic and limiting orientation, we can also assume that kr “ 0 for all r ‰ r‚ . Let @ D 1 sZ2 , 1 “ w1 pV0ϕ0 q, rS‚1

@ D 1 sZ2 . 2 “ w1 pV0ϕ0 q, rS‚2

We denote by o2nd (resp. o:nd ) the orientation of V0ϕ0 |nd obtained by 1 back to nd. transferring the orientation ond (resp. o1nd ) around S‚2 Let r “ 1, 2. The decorated structure of C0 determines an orien1 . If k ‰ 0, this decorated structure thus determines an tation on S‚r ‚r 1 Ă S 1 from the node nd to the marked point x P S 1 oriented arc S‚r;1 ‚r ‚r ‚r 1 and an oriented arc S‚r;2 from this marked point back to nd. The first (resp. second) arc carries j11 p C0 q (resp. k‚r ´1´j11 p C0 q) marked R (resp. V R ) be the ordered points xi of C0 other than x‚r . Let Vr;1 r;2 direct sum of the jr1 p C0 q (resp. k‚r ´ jr1 p C0 q) fibers of V0ϕ0 at the 1 1 other than x‚r (resp. xi P S‚r;2 including x‚r ) marked points xi P S‚r;1 R and V R in the order they appear on the arc. If k‚r “ 0, we take Vr;1 r;2 : 1 2 to be the zero vector spaces. The orientation ond (resp. ond , ond , ond ) of V0ϕ0 |nd determines an orientation oxi (resp. o1xi , o2xi , o:xi ) of V0ϕ0 at 1 by the transfer in the positive direction each marked point xi P S‚r : 1 . If in addition s “ 1, 2, let o 1 2 of S‚r r;s (resp. or;s , or;s , or;s ) be the R. resulting orientation of Vr;s By definition and the assumptions above, `

V0ϕ0

˘ C0

` ˘ R R R R ” V0ϕ0 C0 ;r‚ “ V1;1 ‘V1;2 ‘V2;1 ‘V2;2 .

Orientations for Twisted Determinants

273

The two intrinsic orientations of the lemma are described by ` ˘ o C0 ;1 V0ϕ0 , ox “ o1;2 ‘o11;1 ‘o2;1 ‘o2;2 ðñ nk‚2 P 2Z, ` ˘ o C0 ;2 V0ϕ0 , ox “ o1;1 ‘o1;2 ‘o2;2 ‘o22;1 ðñ nk P 2Z.

(10.10) (10.11)

The limiting orientation of the lemma is described by o` C0

`

˘

#

ox “

o1;2 ‘o12;1 ‘o12;2 ‘o:1;1 ,

o12;2 ‘o:1;1 ‘o:1;2 ‘o22;1 ,

if rp C0 q “ 1; if rp C0 q “ 2.

(10.12)

Combining (10.10)–(10.12) with o:1;1

“

# 1 o1;1 o1;1

o:1;2 “ o1;2

iff j11 p C0 q2 “ 0; iff j11 p C0 qp1 `2 q “ 0; ` ˘ iff k‚1 ´j11 p C0 q p1 `2 q “ 0,

o22;1

“

# o2;1 o12;1

o12;2 “ o2;2

iff j21 p C0 q2 “ 0; iff j21 p C0 qp1 `2 q “ 0; ` ˘ iff k‚2 ´j21 p C0 q 1 “ 0,



we obtain the claim. 10.2

Orientations of Real CR-Operators

Suppose pΣ, σq is a smooth decorated symmetric surface and pV, ϕq is a real bundle over pΣ, σq. A tuple ox of orientations as in (9.1) determines an orientation opλpV, ϕq; ox q of every real CR-operator Dλ on the rank 1 real bundle pair λpV, ϕq as above Proposition 8.2. A relative Pin˘ -structure p on the real vector bundle V ϕ over Σσ Ă Σ is a relative OSpin-structure on the real vector bundle ϕ

V˘ ˘ “ V ϕ ‘p2˘1qλpV ϕ q compatible with the canonical orientation of this vector bundle. Via the construction at the beginning of Section 9.3, p thus determines an orientation op pV˘ , ϕ˘ q of every real CR-operator D˘ on the real bundle pair pV˘ , ϕ˘ q. Along with the canonical homotopy class of isomorphisms (7.7), the orientations opλpV, ϕq; ox q and op pV˘ , ϕ˘ q determine an orientation op pV, ϕ; ox q of every real CR-operator D on the real bundle pair pV, ϕq. Suppose pV, ϕq is a rank 1 odd-degree real bundle pair over pS 2 , τ q. In particular, the real line bundle V ϕ over S 1 Ă S 2 is not orientable.

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Let D be a real CR-operator on pV, ϕq. For an orientation ox1 of V ϕ at a point x1 in S 1 Ă S 2 , we denote by ` ˘ ` ˘ (10.13) o˘ 0 V, ϕ; ox1 ” oι 2 pp˘ pV ϕ qq V, ϕ; ox1 S

0

the orientation of D determined by the image (7.30) of the Pin˘ ϕ structure p˘ 0 pV q of Examples 1.23, 1.24, and 5.1 under the second map in (6.3) with X “ S 2 . Lemma 10.4. Suppose pΣ, σq is a smooth decorated symmetric surface, pV, ϕq is a rank n real bundle pair over pΣ, σq, p is a relative Pin˘ -structure on the real vector bundle V ϕ over Σσ Ă Σ, and D is a real CR-operator on pV, ϕq. Let ox be a tuple of orientations of V ϕ at points xr in Sr1 Ă Σσ as in (9.1). (1) The orientation op pV, ϕ; ox q of D does not depend on the choice of half-surface Σb˚ of an elemental component Σ˚ of pΣ, σq if and only if p pΣ˚ q “ 0. (2) The interchange in the ordering of two consecutive compo1 of Σσ preserves the orientation op pV, ϕ; ox q if nents Sr1 and Sr`1 and only if D ˘`@ D ˘ `@ 1 sZ2 `1 P Z2 . n`2Z ‰ w1 pV ϕ q, rSr1 sZ2 `1 w1 pV ϕ q, rSr`1 (3) The reversal of the component orientation oxr in (9.1) preserves op pV, ϕ; ox q if and only if w1 pV ϕ q|Sr1 ‰ 0. (4) If in addition η P H 2 pΣ, Σσ ; Z2 q, then the orientations op pV, ϕ; ox q and oη¨p pV, ϕ; ox q are the same if and only if xη, rΣb sZ2 y “ 0. Proof. The first statement follows from the CROrient 1os(1) property applied to pV˘ , ϕ˘ q and Proposition 9.12 applied to λpV, ϕq. The second claim follows from the CROrient 1os(2) property applied to pV˘ , ϕ˘ q and Proposition 9.13 applied to λpV, ϕq. The third claim follows from Proposition 9.14 applied to λpV, ϕq. The last statement follows from the RelSpinPin 6 property on page 82 and the first part of the CROrient 2(a) property applied  to pV˘ , ϕ˘ q. Lemma 10.5. Suppose pΣ, σq, pV, ϕq, p, and D are as in Lemma 10.4.

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275

(1) If o is an orientation on V ϕ and ox is a tuple of orientations of V ϕ at points xr in Sr1 Ă Σσ as in (9.1) obtained by restricting o, then the orientations op pV, ϕ; ox q and oR˘ pV, ϕq of D are the o ppq same if and only if (7.13) holds. (2) If pV, ϕq is a rank 1 degree 1 real bundle pair over pS 2 , τ q and ox1 is an orientation of V ϕ at a point x1 in S 1 Ă S 2 , then the orientation (10.13) is the intrinsic orientation opV, ϕ; ox1 q of Proposition 8.2. (3) If ox is a tuple of orientations of V ϕ at points xr in Sr1 Ă Σσ as in (9.1) and (7.15) is a decomposition of pΣ, σq into decorated symmetric surfaces of genera g1 and g2 , respectively, then the isomorphism (7.19) respects the orientations ` ˘ ` ˘ op ” op V, ϕ; ox , op;1 ” op|Σ1 V |Σ1 , ϕ|Σ1 ; ox |Σ1 ` ˘ (10.14) op;2 ” op|Σ2 V |Σ2 , ϕ|Σ2 ; ox |Σ2 , if and only if (7.20) with V1 ” V |Σ1 and V2 ” V |Σ2 holds. Proof. (1) By the RelSpinPin 7(ses6X ) and CROrient 5(a) properties (the latter applied three times if p P PΣ` pV ϕ q), the canonical homotopy class of isomorphisms (7.7) respects the orientations pV, ϕq of DpV,ϕq , and op pV˘ , ϕ˘ q of DpV,ϕq˘ , oR˘ o ppq ` ˘ ` ˘ o0 λpV, ϕq; λpoq ” oιΣ pos0 pλpV,ϕq,λpoqqq λpV, ϕq

(10.15)

of DλpV,ϕq , where ιΣ is the first map in (6.3) with X “ Σ, if and only if (7.13) holds. By Proposition 9.4(2), the orientation (10.15) equals opλpV, ϕq, ox q. These two statements imply the claim. (2) In this case, the real bundle pair pV, ϕq˘ is 3 ˘ 1 copies of pV, ϕq. By Proposition 8.9(2), the isomorphism (8.34) respects the orientation ` ˘ ` ˘ o˘ 0 p3˘1qpV, ϕq “ oι 2 pp˘ pV ϕ qq p3˘1qpV, ϕq S

0

ϕ of p3˘1qD induced by the relative OSpin-structure ιS 2 pp˘ 0 pV qq on ϕ˘ 1 2 the real vector bundle V˘ over S Ă S and the intrinsic orientation opV, ϕ; ox1 q on the factors λpDq on the right-hand side of (8.34). By definition, this means that the orientation (10.13) on the first factor of λpDq equals opV, ϕ; ox1 q.

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(3) For r “ 1, 2, we denote by V˘;r , ϕr , ϕ˘;r , pr , Dr , Dλ;r , D˘;r , prq and ox the restrictions of V˘ , ϕ, ϕ˘ , p, D, a real CR-operator Dλ on λpV, ϕq, the real CR-operator D˘ ” D‘Dλ on pV˘ , ϕ˘ q, and the tuple ox in (9.1), respectively, to Σr . The splitting in (7.6) induces the exact triples of Fredholm operators given by the rows in the commutative diagram of Figure 10.1. The splitting (7.18) induces the exact triples of Fredholm operators given by the left and middle columns in this diagram and by the individual summands in the right column. The isomorphism (7.19) is induced by the left column in this diagram. By definition, the middle (resp. top/bottom) row respects the orientations op , op pV˘ , ϕ˘ q, and opλpV, ϕq; ox q (resp. op;r , opr pV˘;r , prq ϕ˘;r q, and opλpVr , ϕr q; ox q). By the CROrient 4(a) property (resp. Proposition 9.15), the middle column (resp. the exact triple formed by the individual summands in the right column) respects the orientations op1 pV˘;1 , ϕ˘;1 q, op pV˘ , ϕ˘ q, and op2 pV˘;2 , ϕ˘;2 q p1q p2q (resp. opλpV1 , ϕ1 q; ox q, opλpV, ϕq; ox q, and opλpV2 , ϕ2 q; ox q). Since ˘` ˘ ` ˘` ˘ ` ind Dλ;1 ind D2 “ 1´g1 `deg V1 p1´g2 qn`deg V2 ,

(10.16)

Lemma 8.13 and the last two sentences imply that the left column in the diagram respects the orientations (10.14) in the Pin´ -case if and only if the number (10.16) is even. Since the exact triple formed by the individual summands in the right column in Figure 10.1 respects the orientations opλpV1 , p1q p2q ϕ1 q; ox q, opλpV, ϕq; ox q, and opλpV2 , ϕ2 q; ox q, Lemma 8.13 implies

Figure 10.1. Commutative square of exact rows and columns of Fredholm operators for the proof of Lemma 10.5(3).

Orientations for Twisted Determinants

277

that the right column in the Pin` -case respects the direct sum orientations if and only if the number ˘` ˘ ` ˘` ˘ ` ind Dλ;1 ind Dλ;2 “ 1´g1 `deg V1 1´g2 `deg V2 is even. Since ` ˘` ˘ ` ˘` ˘ ind 3Dλ;1 ind D2 – ind Dλ;1 ind D2 mod 2, Lemma 8.13 again, (10.16), and the two sentences above (10.16) imply that the left column respects the orientations (10.14) in the Pin` -case if and only if the number ` ˘` ˘ ` ˘` ˘ ind Dλ;1 ind Dλ;2 ` ind 3Dλ;1 ind D2 ˘ ` – pn`1q 1´g1 `deg V1 p1´g2 q mod 2 is even.

10.3



Degenerations and Exact Triples

We now describe the behavior of the orientations op pV, ϕ; ox q under flat degenerations of pΣ, σq to nodal symmetric surfaces as in the CROrient 7C and 7H3 properties of Section 7.3. We first suppose that C0 is a decorated symmetric surface with one conjugate pair of nodes nd˘ as in (7.36) and in the top left diagram of Figure 9.4 on page 172 so that xr P Sr1 for each r P rN s. We also suppose that r 0 , pΣ r 0, σ rb r0 q, and Σ pV0 , ϕ0 q is a real bundle pair over pΣ0 , σ0 q. Let C 0 be as below (7.36), pVr0 , ϕ r0 q be the lift of pV0 , ϕ0 q to a real bundle pair r 0, σ r0 q. A relative Pin˘ -structure p0 on the real vector bundle over pΣ ϕ0 p0 on the real V0 over Σσ0 0 Ă Σ0 lifts to a relative Pin˘ -structure r ϕ r0 σ r0 r r r vector bundle V0 over Σ0 Ă Σ0 . A tuple ox of orientations of the fibers of V0ϕ0 |xr as in (9.1) lifts to a tuple r ox of orientations of the ϕ r0 r corresponding fibers of V0 . Let D0 be a real CR-operator on pV0 , ϕ0 q. We denote its lift to a r 0 . The orientation r0 q by D real CR-operator on pVr0 , ϕ ` ˘ ` ˘ r (10.17) r0 ; r ox op0 ox ” orp0 Vr0 , ϕ r 0 and the complex orientation of V0 | ` determine an orientation of D nd ` ˘ ` ˘ (10.18) op0 ox ” op0 V0 , ϕ0 ; ox

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of D0 via the isomorphism (7.38). In an analogy with the intrinsic orientation of the CROrient 7C property, we call (10.18) the intrinsic orientation of D0 induced by p0 and ox . Suppose in addition that p U,rcq is a flat family of deformations of C0 as in (7.32), pV, ϕq is a real bundle pair over p U,rcq extendrc R R ing pV0 , ϕ0 q, sR 1 , . . . , sN are sections of U over ΔR with sr p0q “ xr σ0 1 for all Sr P π0 pΣ0 q, and D” tDt u is a family of real CR-operators on pVt , ϕt q as in (7.34) extending D0 . The decorated structure on C0 , the relative Pin˘ -structure p0 on V0ϕ0 , and the tuple of orientations ox of fibers of V0ϕ0 induce a decorated structure on the fiber pΣt , σt q of π for every t P ΔR , a relative Pin˘ -structure pt on Vtϕt , and a tuple ox;t of orientations of fibers of Vtϕt as above the CROrient 7C property and in (9.10). The latter in turn determine an orientation ` ˘ ot ” opt Vt , ϕt ; ox;t of Dt for each t P Δ˚R as above Lemma 10.4. These orientations vary continuously with t and extend to an orientation ` ˘ (10.19) o1p0 pox q ” o1p0 V0 , ϕ0 ; ox of D0 as above the CROrient 7C property on page 121. In an analogy with the limiting orientation of the CROrient 7C property, we call (10.19) the limiting orientation of D0 induced by ox . Lemma 10.6. Suppose C0 is a decorated marked symmetric surface as in (7.36) which contains precisely one conjugate pair pnd` , nd´ q of nodes and no other nodes and carries precisely one real marked point xr on each connected component Sr1 of Σσ0 0 , pV0 , ϕ0 q is a real bundle pair over C0 , and p is a relative Pin˘ -structure on the real vector bundle V0ϕ0 over Σσ0 0 Ă Σ0 . Let ox be a tuple of orientations of V0ϕ0 |xr as in (9.1) and D0 be a real CR-operator on pV0 , ϕ0 q. The intrinsic and limiting orientations, (10.18) and (10.19), of D0 are the same. r 0;λ be the lift of a real CR-operator D0;λ on λpV0 , ϕ0 q Proof. Let D r r0 q. We take to λpV0 , ϕ D0;˘ “ D0 ‘p2˘1qD0;λ ,

r 0;˘ “ D r 0 ‘p2˘1qD r 0;λ . D

(10.20)

These two decompositions and the one in (7.6) induce the exact triples of Fredholm operators given by the rows in the diagram of

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Figure 10.2. Commutative square of exact rows and columns of Fredholm operators for the proof of Lemma 10.6.

Figure 10.2. The exact triple (7.37) of Fredholm operators induces the exact triples of Fredholm operators given by the columns in this diagram. By definition, the columns in Figure 10.2 respect the complex orientations on the vector spaces in the bottom row, the orientations ` ˘ ` ˘ r r0 ; r ox , r r˘ q, and op0 ” orp0 pVr0;˘ , ϕ op0 ox ” orp0 Vr0 , ϕ ` ˘ r o0;λ pox q ” r o λpVr , ϕq; r r ox (10.21) r 0;˘ , and D r 0;λ constructed above Lemma 10.4 and Proposir0, D of D tions 9.4 and 9.1, respectively, and the associated intrinsic orientations op0 pox q, ` ˘ (10.22) op0 ” op0 pV0 q˘ , pϕ0 q˘ , and o0 pox q as in (10.18), the CROrient 7C property with os0 “ p0 , and (9.9). The bottom row respects the complex orientations of V0 |nd` , V0;˘ , and λC pV0 |nd` q. By definition, the middle row respects the three orientations in (10.21); the top row respects the limiting orientations o1p0 pox q, ` ˘ (10.23) o1p0 ” o1p0 pV0 q˘ , pϕ0 q˘ , and o10 pox q as in (10.19), the CROrient 7C(a) property with os0 “ p0 , and (9.11). By the CROrient 7C property and Proposition 9.2, op0 “ o1p0

and o0 pox q “ o10 pox q,

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respectively. Since the (real) dimension of V0 |nd` is even, Lemma 8.13 and the previous paragraph thus imply that the orientations (10.18)  and (10.19) are the same. We next suppose that C0 is a decorated symmetric surface with one H3 node nd as above (7.39) and in the top left diagram of Figure 9.5 on page 175 so that xr P Sr1 for each r P rN s different from the index r‚ of the singular topological component of Σσ0 0 and xr‚ “ nd (as in Proposition 9.3, we allow a “marked point” to be a node). We also suppose that pV0 , ϕ0 q is a real bundle pair r 0 , pΣ r 0, σ r b , nd1 P S 1 , and nd2 P S 1 be as in over pΣ0 , σ0 q. Let C r0 q, Σ 0 ‚1 ‚2 r0 q be the lift of pV0 , ϕ0 q to a real bundle and below (7.39) and pVr0 , ϕ r 0, σ r0 q. A relative Pin˘ -structure p0 on the real vector pair over pΣ ϕ0 p0 on the bundle V0 over Σσ0 0 Ă Σ0 lifts to a relative Pin˘ -structure r ϕ r0 σ r0 r r r real vector bundle V0 over Σ0 Ă Σ0 . A tuple ox of orientations of ox of orientations of the fibers of V0ϕ0 |xr as in (9.1) lifts to a tuple r ϕ r0 r the corresponding fibers of V0 as in (9.12). Let D0 be a real CR-operator on pV0 , ϕ0 q. We denote its lift to a r 0 . The orientation r0 q by D real CR-operator on pVr0 , ϕ ` ˘ ` ˘ r r0 ; r ox op0 ox ” orp0 Vr0 , ϕ

(10.24)

r 0 and the orientation ond ” oxr of V0 |nd determine an orientation of D ‚ ` ˘ ` ˘ op0 ox ” op0 V0 , ϕ0 ; ox

(10.25)

of D0 via the isomorphism (7.41). We call (10.25) the intrinsic orientation of D0 induced by p0 and ox . Suppose in addition that p U,rcq is a flat family of deformations of C0 as in (7.32), pV, ϕq is a real bundle pair over p U,rcq extendrc R ing pV0 , ϕ0 q, sR 1 , . . . , sN are sections of U over ΔR satisfying (9.15), and D” tDt u is a family of real CR-operators on pVt , ϕt q as in (7.34) extending D0 . The decorated structure on C0 determines an open subspace Δ` R Ă ΔR and a decorated structure on the fiber pΣt , σt q of π for every t P ΔR . The relative Pin˘ -structure p0 on V0ϕ0 and the tuple of orientations ox of fibers of V0ϕ0 induce a relative Pin˘ -structure pt on Vtϕt and a tuple ox;t of orientations of fibers of Vtϕt as above the CROrient 7H3 property and (9.16), respectively. For each t P Δ˚R , the

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decorated structure on pΣt , σt q, the relative Pin˘ -structure pt , and the tuple ox;t determine an orientation ` ˘ (10.26) ot ” opt Vt , ϕt ; ox;t of Dt as above Lemma 10.4. We denote by ` ˘ ` o` p0 pox q ” op0 V0 , ϕ0 ; ox

(10.27)

the orientation of D0 obtained as the continuous extension of the orientations (10.26) with t P Δ` R . We call (10.27) the limiting orientation of D0 . Let W1 pV0 , ϕ0 qr‚ be as in (9.18). Define ˇ (ˇ W1 pV0 , ϕ0 q ” ˇ Sr1 P π0 pΣσ0 0 q : r ‰ r‚ , w1 pV0ϕ0 q|Sr1 ‰ 0 ˇ ˇ (ˇ `ˇtr P r2s : w1 pV0ϕ0 q|S‚r 1 ‰ 0 ˇ. Lemma 10.7. Suppose C0 is a decorated marked symmetric surface as in (7.36) which contains precisely one H3 node nd and no other nodes and carries precisely one real marked point xr on each smooth connected component Sr1 of Σσ0 0 and a marked point xr‚ “ nd on the nodal connected component of Σσ0 0 . Let pV0 , ϕ0 q be a rank n real bundle pair over pΣ0 , σ0 q, p be a relative Pin˘ -structure on the real vector bundle V0ϕ0 over Σσ0 0 Ă Σ0 , ox be a tuple of orientations of V0ϕ0 |xr as in (9.1), and D0 be a real CR-operator on pV0 , ϕ0 q. The intrinsic and limiting orientations, (10.25) and (10.27), of D0 are the same if and only if ˘ ` ˘ ` 1 1 sZ2 y`1 xw1 pV0ϕ0 q, rS‚2 sZ2 y ` n W1 pV0 , ϕ0 q´r‚ xw1 pV0ϕ0 q, rS‚1

“

´ W1 pV0 , ϕ0 qr‚ # 2Z P Z2 , |π0 pΣσ0 0 q|`W1 pV0 , ϕ0 q`2Z P Z2 ,

if p P PΣ´0 pV0ϕ0 q;

if p P PΣ`0 pV0ϕ0 q.

r 0;λ , D0;˘ , and D r 0;˘ be as in the proof of Proof. Let D0;λ , D the canonical orientation on Lemma 10.6. We denote by o˘ nd ` ˘bp3˘1q ` ϕ ˘ . λ V0;˘0;˘ « λ V0ϕ0 The decompositions in (7.6) and (10.20) induce the exact triples of Fredholm operators given by the rows in the diagram of Figure 10.3.

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Figure 10.3. Commutative square of exact rows and columns of Fredholm operators for the proof of Lemma 10.7.

The exact triple (7.40) of Fredholm operators induces the exact triples of Fredholm operators given by the columns in this diagram. By definition, the left and middle columns in this figure and the exact triple formed by each summand in the right column respect the orientations ond , o˘ nd , and ond of the vector spaces in the bottom row, r 0;˘ , and D r 0;λ , respectively, and the r0, D the orientations (10.21) of D associated intrinsic orientations op0 pox q, op0 , and o0 pox q as in (10.25), (10.22), and (9.14). The bottom row respects the orientations ond , o˘ nd , and ond . By definition, the middle row respects the orientations in (10.21); the top row respects the limiting orientations o1p0 pox q, o1p0 , and o10 pox q as in (10.27), (10.23), and (9.17). By the CROrient 7H3(a) property applied to pV0 , ϕ0 q˘ , op0 “ o1p0 if and only if ˘ ` (10.28) pn`1q |π0 pΣσ0 0 q|´r‚ P 2Z. By Proposition 9.3 applied to λpV0 , ϕ0 q, o10 pox q “ o0 pox q if and only if (9.19) holds. By Lemma 8.13, the total number of rows and columns in Figure 10.3 which respect the orientations is of the same parity as ˘ ˘` ˘ `ˇ ` ind p2˘1qD0;λ dim V0ϕ0 |nd – n ˇπ0 pΣσ0 0 q|`W1 pV0 , ϕ0 q mod 2. (10.29) Combining the statements in this paragraph and the previous one, we obtain the claim in the Pin´ -case. Since the exact triple formed by the individual summands in the o0;λ pox q, right column in Figure 10.3 respects the orientations o0 pox q, r

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and ond , Lemma 8.13 implies that the right column in the Pin` -case respects the direct sum orientations if and only if the number ˇ ˘ ˇ ˘` ` ind D0;λ dim λpV0ϕ0 |nd q – ˇπ0 pΣσ0 0 qˇ `W1 pV0 , ϕ0 q mod 2 is even. Combining this with the statements in the previous two  paragraphs, we obtained the claim in the Pin` -case. We are now in a position to obtain an analog of the CROrient 5 property for the orientations constructed above Lemma 10.4. We first prove Proposition 10.8 under the assumption that the complex vector bundle V 1 in (7.21) is topologically trivial as a consequence of the already established CROrient 5(a) property on page 116. We then reduce the general case to this special case by bubbling off the non-trivial part of V 1 onto conjugate pairs of spheres and using the already established CROrient 7C(a) property on page 121 and Lemma 10.6. Proposition 10.8. Suppose pΣ, σq is a smooth decorated symmetric surface, e and o2x are as in Lemma 10.2, os1 is a relative OSpin1 structure on the real vector bundle V 1ϕ over Σσ Ă Σ, o1x is the tuple orientations for pV 1 , ϕ1 q as in (9.1) obtained by restricting the orien1 tation o1 of V 1ϕ determined by os1 , and p2 is a relative Pin˘ -structure 2 on V 2ϕ . The homotopy class of isomorphisms (7.24) respects the orientations ` ˘ ` ˘ op pox q ” oxxos1 ,p2 yyeR V, ϕ; xxo1x , o2x yy eR , oos1 ” oos1 V 1 , ϕ1 , ` ˘ (10.30) op2 po2x q ” op2 V 2 , ϕ2 ; o2x if and only if

ˆ ˙ |π0 pΣσ q| P 2Z. prk V qprk V `1q 2 1

2

(10.31)

Proof. Let p “ xxos1 , p2 yy eR and ox “ xxo1x , o2x yy eR . (1) We first assume that the restriction of V 1 to each connected component of Σ is of degree 0. Let D 1 , D 2 , and Dλ2 be real CRoperators on pV 1 , ϕ1 q, pV 2 , ϕ2 q, and λpV 2 , ϕ2 q, respectively, and set D “ D 1 ‘D 2 , 1

2 D˘ “ D 2 ‘p2˘1qDλ2 .

By the assumption that V 1ϕ is orientable and [17, Lemma 3.2], we can identify the rank 1 real bundle pair λpV 1 , ϕ1 q over pΣ, σq with the

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trivial rank 1 real bundle

` ˘ pV1 , ϕ1 q ” ΣˆC, σˆc 1

so that the induced identification of the real line bundles λpV 1ϕ q and Σσ ˆR respects the orientation o1 and the standard orientation on R. We denote by Dλ the real CR-operator on λpV, ϕq corresponding to Dλ2 under the induced identification of λpV, ϕq « λpV 1 , ϕ1 qbλpV 2 , ϕ2 q

with λpV 2 , ϕ2 q and take D˘ “ D‘p2˘1qDλ . 2 above induce the exact triples of The decompositions of D˘ and D˘ Fredholm operators given by the middle and bottom rows in the first diagram of Figure 10.4. The exact sequence e in (7.21) induces the exact triples given by the left and middle columns in this diagram.

Figure 10.4. Commutative squares of exact rows and columns of Fredholm operators for the proof of Proposition 10.8.

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By definition, the middle and bottom rows in the diagram respect the orientations op pox q of D and op2 pox q of D 2 , op pV˘ , ϕ˘ q of D˘ and op2 pV˘2 , ϕ2˘ q of D˘ , and ` ` ˘ ˘ oλ pox q ” o λpV, ϕq; ox and oλ po2x q ” o λpV 2 , ϕ2 q; o2x (10.32) of Dλ and Dλ2 , respectively. By the assumption on the identification of λpV 1 , ϕ1 q with pV1 , ϕ1 q, the identification in the right column respects the orientations (10.32). By the second identity in the RelSpinPin 7(ses3X ) property on page 82 and the CROrient 5(a) property, the middle column respects the orientations oos1 , op pV˘ , ϕ˘ q, and op2 pV˘2 , ϕ2˘ q if and only if (10.31) holds. Along with Lemma 8.13, the last three statements imply that the left column in the first diagram in Figure 10.4 respects the orientations (10.30) if and only if (10.31) holds, provided the restriction of V 1 to each connected component of Σ is of degree 0. (2) We now deduce the general case from the special case in (1) and Lemma 10.6. Let pΣ0 , σ0 q be a decorated marked symmetric surface consisting of pΣ, σq with conjugate pairs of copies of S 2 attached at one conjugate pair of points on each elemental component of pΣ, σq. We denote by Σ00 , S ` , Σ` 0 Ă Σ0 the copy of Σ, the set of the nodal points carried by the distinguished half Σb00 “ Σb , and the union of the additional copies of S 2 attached at S ` . Let p U,rcq be a family of deformations of C0 ” pΣ0 , σ0 q over ` Δ Ă C2|S | so that the fiber Ct1 of p U,rcq over some t1 P ΔR is pΣ, σq rc R R and sR 1 , . . . , sN be sections of U over ΔR with sr pt1 q “ xr for each 1 σ Sr P π0 pΣ q. Let r1 q ÝÑ pVr , ϕq r ÝÑ pVr 2 , ϕ r2 q ÝÑ 0 0 ÝÑ pVr 1 , ϕ

(10.33)

be a short exact sequence of real bundle pairs over p U,rcq restricting to e over t1 so that the restriction of Vr 1 to each topological component of Σ00 is of degree 0; such a sequence exists because the restriction of V 1 to each elemental component of pΣ, σq is of even degree. r and the restrictions of We denote the exact sequence (10.33) by e 1 1 1 1 2 2 1 ,ϕ r r r r q, pV , ϕ r q, and pV , ϕ r q to Σ00 by pVr00 r100 q, pVr00 , ϕ r00 q, and pV , ϕ 2 2 r r00 q, respectively. pV00 , ϕ

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For each t P ΔR , the tuples o1x , ox , and o2x of orientations of fibers of 2 V ϕ , and V 2ϕ induce tuples o1x;t , ox;t , and o2x;t of orientations of V 1ϕ r1 2ϕ r2 r 1ϕr1 , sR˚ Vr ϕr , fibers of Vrt t , Vrtϕrt , and Vrt t via the vector bundles sR˚ r V r 1 2ϕ r2 . Let os 2 be relative OSpin-structures on the real r r r V and p and sR˚ r 1 2 vector bundles Vr 1ϕr and Vr 2ϕr , respectively, over Urc Ă U restricting r 1 q and w2 pr p2 q vanish on each added to os1 and p2 on Σt1 so that w2 pos copy of S 2 . We define @@ 1 2 DD r r 1 |Σ00 , p00 “ r r ,r p er , os100 “ os p|Σ00 , p200 “ r p2 |Σ00 , p “ os 1ϕ1 ,

R

os1t

1

r |Σt , pt “ r “ os p|Σt , p2t “ r p2 |Σt @ t P ΔR , ` ˘ ` ˘ r1t , op;t pox q “ opt Vrt , ϕ rt ; ox;t , oos1 ;t “ oos1t Vrt1 , ϕ ` ˘ r2t ; o2x;t @ t P Δ˚R . op2t po2x q “ op2t Vrt2 , ϕ

(10.34)

(10.35)

Let D1 ” tDt1 u and D2 ” tDt2 u be families of real CR-operators on and pVt2 , ϕ2t q, respectively, as in (7.34), and

pVt1 , ϕ1t q

Dt “ Dt1 ‘Dt2 @t P ΔR .

(10.36)

1 , D , and D 2 the real CR-operators on pV 1 r1 ,ϕ We denote by D00 00 00 00 r00 q, 1 1 2 2 1 r00 q, and pVr00 , ϕ r00 q, respectively, induced by D0 , D0 , and D02 pVr00 , ϕ ` ` ` and by D00 , D00 , and D00 the real CR-operators on Vr 1 |Σ` , Vr |Σ` , 0 0 and Vr 1 | ` , respectively, induced by D 1 , D0 , and D 2 . The orientaΣ0

0

0

tions (10.3) of Dt1 , Dt , and Dt2 extend continuously to orientations ` ˘ ` ˘ r10 , o1p;0 pox q ” op0 Vr0 , ϕ r0 ; ox;0 , and o1os1 ;0 ” oos10 Vr01 , ϕ ` ˘ r20 ; o2x;0 (10.37) o1p2 ;0 po2x q ” op20 Vr02 , ϕ

of D01 , D0 , and D02 , respectively; these are the analogs of the limiting orientations of the CROrient 7C(a) property. We denote by ` ˘ ` ˘ r10 , op;0 pox q ” op0 Vr0 , ϕ r0 ; ox;0 , and oos1 ;0 ” oos10 Vr01 , ϕ ` ˘ r20 ; o2x;0 op2 ;0 po2x q ” op20 Vr02 , ϕ (10.38) the corresponding analogs of the intrinsic orientation of the CROrient 7C(a) property. By Lemma 10.6 applied |S ` | times, these orientations are the same as the limiting orientations (10.37).

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The exact triple (7.37), the decomposition (7.18), and the isomorphism (7.8) induce the exact triples of Fredholm operators given by the rows in the second diagram of Figure 10.4. The restriction of the exact sequence (10.33) to Σ0 induces the exact triples given by the columns in this diagram. Since the decompositions (10.36) induce an isomorphism λp Dq « λp D1 qbλp D2 q of line bundle over ΔR and the orientations (10.37) restrict to the orientations (10.30) for t “ t1 , it is sufficient to show that the left column respects the limiting orientations (10.38) if and only if (10.31) holds. The right column and the exact triple formed by the second summands in the middle column respect the complex orientations of all 1 to each topological component terms. Since the restriction of Vr00 of Σ00 is of degree 0, (1) above implies that the exact triple formed by the first summands in the middle column respects the orientations ` 1 1˘ ` ˘ ,ϕ r0 , op;00 pox q ” op00 Vr0 , ϕ r00 ; ox;0 , and oos1 ;00 ” oos100 Vr00 ` 2 2 2 ˘ ,ϕ r00 ; ox;0 op2 ;00 po2x q ” op200 Vr00 1 , D , and D 2 , respectively, if and only if (10.31) holds. Since of D00 00 00 the (real) index of D01` is even, Lemma 8.13 then implies that the middle column respect the direct sum orientations ` ˘ ` ˘ r oos10 Vr01 , ϕ r10 , r op0 Vr0 , ϕ r0 ; ox;0 , and op;0 pox q ” r oos1 ;0 ” r ` ˘ r op20 Vr02 , ϕ r20 ; o2x;0 op2 ;0 po2x q ” r (10.39)

if and only if (10.31) holds. The orientations ` 1 1˘ ` ˘ C r r , oC pox q ” op Vr0 , ϕ r0 ; ox;0 , and oC 0 0 p;0 os1 ;0 ” oos10 V0 , ϕ ` 2 2 2 ˘ 2 r r ;o (10.40) oC 0 x;0 p2 ;0 pox q ” op20 V0 , ϕ of D01 , D0 , and D02 , respectively, determined by the orientations (10.39) and the complex orientations of the vector spaces in the right column via the rows in the second diagram of Figure 10.4 are the analogs of the C-split orientations of Corollary 7.3. By the proof of Lemma 10.5(3), these orientations are the same as

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the orientations (10.38). Along with the sentence after (10.38), this implies that the rows in this diagram respect the orientations (10.37), the orientations (10.39), and the complex orientations of Vr01 |S ` , Vr0 |S ` , and Vr02 |S ` . Since the right column respects these complex orientations, the middle column respects the orientations (10.39) if and only if (10.31) holds, and the (real) dimension of Vr 1 |S ` is even, Lemma 8.13 implies that the left column respects the limiting orien tations (10.37) if and only if (10.31) holds. 10.4

Properties of Twisted Orientations

Suppose C is a smooth decorated marked symmetric surface as in (7.3), pV, ϕq is a real bundle pair over C, p is a relative Pin˘ structure on the real vector bundle V ϕ over Σσ Ă Σ, and ox is a tuple of orientations of fibers of V ϕ as in (10.3). For any real CRoperator D on the real bundle pair pV, ϕq, we denote by o C;p pV, ϕ; ox q the orientation of the twisted determinant (10.2) of D induced by the orientations o CpV ϕ , ox q of λRCpV, ϕq and op pV, ϕ; ox q of D defined in Section 10.1 and 10.2, respectively. Suppose C is a marked symmetric surface with the underlying symmetric surface pS 2 , τ q and k “ 2 real marked points, pV, ϕq is a rank 1 odd-degree real bundle pair over pS 2 , τ q, and ox1 is an orientation of Vxϕ1 . We then denote by ` ˘ o˘ C;0 V, ϕ; ox1 ” o C;ι

˘ ϕ S 2 pp0 pV qq

`

V, ϕ; ox1

˘

(10.41)

r CpDq determined by the image (7.30) of the Pin˘ the orientation on λ ˘ ϕ structure p0 pV q of Examples 1.23, 1.24, and 5.1 under the second map in (6.3) with X “ S 2 . The first three statements of Corollary 10.9 extend the three statements of the CROrient 1p property on page 112 to real bundle pairs pV, ϕq over smooth marked symmetric surfaces C without the balancing restriction. The last statement of this corollary, the first and last statements of Corollary 10.10, and Corollary 10.11 extend the CROrient 2(b), 3, 4(b), and 5(b) properties in the same way. Corollaries 10.12 and 10.13 extend the CROrient 7C(b) and 7H3(b) properties of Section 7.3 to real bundle pairs pV0 , ϕ0 q over marked

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symmetric surfaces C0 without the balancing restriction. According to the middle statement of Corollary 10.10, the orientations o C;p pV, ϕ; ox q satisfy the conclusion of the CROrient 6(b) property on page 117. Corollary 10.9. Suppose C is a smooth decorated marked symmetric surface, pV, ϕq is a rank n real bundle pair over C, p is a relative Pin˘ -structure on the real vector bundle V ϕ over Σσ Ă Σ, and D is a real CR-operator on pV, ϕq. Let ox be a tuple of orientations of V ϕ at points x˚r in Sr1 Ă Σσ as in (10.3). r CpDq does not depend on the (1) The orientation o C;p pV, ϕ; ox q on λ b choice of half-surface Σ˚ of an elemental component Σ˚ of pΣ, σq if and only if ˙ ˆ ˙ ˙ ˆ ÿ ˆ@ D kr ´1 kr ´1 ϕ 1 `n “ p pΣ˚ q P Z2 . w1 pV q, rSr sZ2 1 2 σ 1 Sr Pπ0 pΣ˚ q

(2) The interchange in the ordering of two consecutive compo1 of Σσ preserves the orientation o C;p pV, ϕ; ox q nents Sr1 and Sr`1 if and only if ` ˘ `@ D ˘`@ D ˘ 1 n kr kr`1 `1 `2Z ‰ w1 pV ϕ q, rSr1 sZ2 `1 w1 pV ϕ q, rSr`1 sZ2 `1 P Z2 .

(3) The interchange of two real marked points xjir p Cq and xj r1 p Cq on i the same connected component Sr1 of Σσ with 2 ď i, i1 ď kr preserves o C;p pV ϕ , ox q. The combination of the interchange of the real points xj1r p Cq and xjir p Cq with 2 ď i ď kr and the replacement of the component ox˚r ” ox˚r in (10.3) by ox˚r preserves j p Cq 1

j p Cq i

o C;p pV, ϕ; ox q if and only if (10.5) holds. (4) The reversal of the component orientation ox˚r in (10.3) preserves o C;p pV, ϕ; ox q if and only if @ D kr `2Z “ w1 pV ϕ q, rSr1 sZ2 `1 P 2Z. (5) If in addition η P H 2 pΣ, Σσ ; Z2 q, then the orientations o C;p pV, ϕ; ox q and o C;η¨p pV, ϕ; ox q are the same if and only if xη, rΣb sZ2 y “ 0. Proof. The five statements of this corollary follow from the four statements of Lemma 10.1 and the four statements of  Lemma 10.4.

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Corollary 10.10. Suppose C, pV, ϕq, p, and D are as in Corollary 10.9. (1) If o is an orientation on V ϕ and ox is a tuple of orientations of V ϕ at points x˚r in Sr1 Ă Σσ as in (10.3) obtained by restrictr CpDq corresponds ing o, then the orientation o C;p pV, ϕ; ox q on λ to the homotopy class of isomorphisms of λpDq and λRCpV, ϕq determined by the orientations oR˘ pV, ϕq and λRCpoq if and only o ppq if (7.13) holds. (2) If pΣ, σq “ pS 2 , τ q, k “ 2, rk V “ 1, deg V “ 1, and ox1 is an r CpDq cororientation of Vxϕ1 , then (10.41) is the orientation of λ responding to the homotopy class of the isomorphism (7.28) with a “ 1. (3) If ox is a tuple of orientations of V ϕ as in (10.3) and C “ C1 \ C2 is a decomposition of C into decorated marked symmetric surfaces as above (7.17) of genera g1 and g2 , respectively, then ` ˘ ` ˘ o C;p pV, ϕq “ o C1 ;p| C1 V | C1 , ϕ| C1 ; ox | C1 \o C2 ;p| C2 V | C2 , ϕ| C2 ; ox | C2 if and only if (7.20) with V1 ” V | C1 and V2 ” V | C2 holds. Proof. (1) By the assumption on ox , the orientations λRCpoq and o CpV ϕ , ox q of λRCpV, ϕq are the same. Along with the definition of r CpDq above Corollary 10.9 and the orientation o C;p pV, ϕ; ox q of λ Lemma 10.5(1), this implies the claim. (2) By Lemma 10.5(2), the orientation (10.13) of D is the same as the intrinsic orientation opV, ϕ; ox1 q defined above Proposition 8.2. By Proposition 8.2(2), ˘ ` ˘ ` ˘ ` o V, ϕ; ox1 ” o2,0 V, ϕ; ox1 “ o2,0 ox1 . The claim thus follows from the definitions of the twisted orientation (10.41) and the evaluation orientation o2,0 pox q. (3) Since the decomposition C“ C1 \ C2 respects the orderings of the connected components of the fixed loci, the isomorphism (7.17) respects the orientations ` ˘ ` ˘ ` ˘ o C V ϕ , ox , o C1 V1ϕ1 , ox | C1 , and o C2 V2ϕ2 , ox | C2 . The claim thus follows from Lemma 10.5(3).



Orientations for Twisted Determinants

291

Corollary 10.11. Suppose C, e, and o2x are as in Lemma 10.2, 1 os1 is a relative OSpin-structure on the real vector bundle V 1ϕ over Σσ Ă Σ, and o1x is the tuple orientations for pV 1 , ϕ1 q as in (10.4) 1 obtained by restricting the orientation o1 of V 1ϕ determined by os1 . 2 If p2 P PΣ pV 2ϕ q, then ˘ ` ˘ ` ` ˘ o C;xxos1 ,p2 yyeR V, ϕ; xxo1x , o2x yy eR “ oos1 pV 1 , ϕ1 qλRCpos1 q eo C;p2 V 2 , ϕ2 ; o2x

if and only if (7.26) holds. Proof. This claim follows immediately from Proposition 10.8 and  Lemma 10.2. It remains to describe the behavior of the orientations o C;p pV, ϕ; ox q under the flat degenerations of the CROrient 7C and 7H3 properties in Section 7.3. We first suppose that C0 is a decorated symmetric surface with one conjugate pair of nodes nd˘ as in (7.36), pV0 , ϕ0 q is a real bundle pair over pΣ0 , σ0 q, p0 is a relative Pin˘ -structure on the real vector bundle V0ϕ0 over Σσ0 0 Ă Σ0 , and ox is a tuple of orientations of the fibers of V0ϕ0 at points x˚r P Sr1 as in (10.3). Let D0 be a real CR-operator on pV0 , ϕ0 q. We define the associated intrinsic and limiting orientations, ` ˘ ` ˘ ` ˘ ` ˘ o C0 ;p0 ox ” o C0 ;p0 V0 , ϕ0 ; ox and o1C0 ;p0 ox ” o1C0 ;p0 V0 , ϕ0 ; ox , (10.42) of the twisted determinant r C pD0 q ” λR pV0 , ϕ0 q˚ bλpD0 q λ C0 0

(10.43)

of D0 to be the orientations induced by the intrinsic and limiting orientations of λRC0 pV0 , ϕ0 q defined after Lemma 10.2 and the intrinsic and limiting orientations of D0 in (10.18) and (10.19), respectively. The observation below (10.6) and Lemma 10.6 immediately give the following conclusion. Corollary 10.12. The intrinsic and limiting orientations (10.42) on r C pD0 q are the same. λ 0

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292

Suppose finally that C0 is a decorated marked symmetric surface as in (7.36) which contains precisely one H3 node nd and no other nodes, pV0 , ϕ0 q and p0 are as above Corollary 10.12, ox is a tuple of orientations of the fibers of V0ϕ0 at points x˚r P Sr1 for r ‰ r‚ and x˚r‚ “ nd as above (10.7). Let D0 be a real CR-operator on pV0 , ϕ0 q. We define the associated intrinsic and limiting orientations, ` ˘ ` ˘ ` ˘ ` ˘ o C0 ;p0 ;1 ox ” o C0 ;p0 ;1 V0 , ϕ0 ; ox , o C0 ;p0 ;2 ox ” o C0 ;p0 ;2 V0 , ϕ0 ; ox , ` ˘ ` ˘ ` (10.44) and o` C0 ;p0 ox ” o C0 ;p0 V0 , ϕ0 ; ox , of the twisted determinant (10.43) of D0 to be the orientations induced by the intrinsic and limiting orientations of λRC0 pV0 , ϕ0 q as below (10.8) and in (10.9) and the intrinsic and limiting orientations of D0 in (10.25) and (10.27). Let R p C0 ; V0ϕ0 q, δr˚ pV0 , ϕ0 q, and r˚ p C0 ; V0ϕ0 q be as above Lemma 10.3 and W1 pV0 , ϕ0 qr‚ and W1 pV0 , ϕ0 q be as above Lemma 10.7. These two lemmas immediately give the following conclusion. Corollary 10.13. Let r ˚ “ 1, 2. The intrinsic and limiting orientar tions o C0 ;p0 ;r˚ pox q and o` C0 ;p0 pox q in (10.42) on λ C0 pD0 q are the same if and only if ˛ ¨ ˚` ˘ ˚ pn`1q ˚ kr‚ ´1 δR p C0 q`δr˚ p C0 q ` ˝ ÿ

`

ÿ σ

‹ ‹ pkr ´1q‹ ‚

Sr1 Pπ0 pΣ0 0 q rąr‚

pkr ´1q ´W1 pV0 , ϕ0 qr‚

σ Sr1 Pπ0 pΣ0 0 q

rąr‚

D˘ @ 1 1 ` kr‚ ´1` w1 pV0ϕ0 q, rS‚1 sZ2 `rS‚2 sZ2 δR p C0 q ˘ 1 ` 1 ` k‚r˚ ´1`xw1 pV0ϕ0 q, rS‚r ˚ sZ2 y jr ˚ p C0 q ` ˘ ` ˘ 1 1 sZ2 y k‚2 ´1`xw1 pV0ϕ0 q, rS‚2 sZ2 y ` k‚1 ´1 r ˚ ` xw1 pV0ϕ0 q, rS‚1 ` ˘ “ n r ˚ ´1 `2Z # if p P PΣ´0 pV0ϕ0 q; np|π0 pΣσ0 0 q|`W1 pV0 , ϕ0 qq P Z2 , ` pn`1qp|π0 pΣσ0 0 q|`W1 pV0 , ϕ0 qq P Z2 , if p P PΣ`0 pV0ϕ0 q. `

Part III

Real Enumerative Geometry

Enumerative geometry of algebraic varieties is a field of mathematics that dates back to the nineteenth century. The general goal of this subject is to count geometric objects that satisfy pre-specified geometric conditions. The objects are typically complex curves in smooth algebraic manifolds that are usually required to be of a specified genus, to represent a given homology class, and to meet a given collection of geometric constraints. The prototypical example of such a count is the number Nd of degree d rational curves that pass through 3d ´ 1 points in general position in the complex projective plane P2 . As the space of p3d´1q-tuples of points in P2 in general position is path-connected, Nd does not depend on the choice of such a tuple. It is fairly straightforward to find that N1 , N2 “ 1 and N3 “ 12; see Section 12.1. The result N4 “ 620, first obtained in [48] in the middle of the nineteenth century, is significantly trickier. The higher-degree numbers Nd remained unknown until the early 1990s, when a recursive formula for these numbers, which became known as Kontsevich’s recursion, was deduced in [27] from the WDVV relation of string theory; see (12.5). It was mathematically confirmed shortly after in [31, 37] using the interpretation of the numbers Nd in terms of moduli spaces of J-holomorphic maps from the Riemann sphere P1 as in [23].

293

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While a degree d polynomial over C has precisely d roots (counted with multiplicity), a degree d polynomial over R can have any number d1 of roots so that 0 ď d1 ď d and d1 is of the same parity as d. The number of real curves in a real algebraic variety that are of a specified genus, represent a given homology class, and meet a given collection of geometric constraints likewise generally depends on the position of the constraints and not just on their type. It may even come as a surprise that there can be non-trivial lower bounds on counts of real curves and other geometric objects. For example, the number of real cubics passing through 8 general points in RP2 can be only 8, 10, or 12 (the complex count); see [10]. Invariant signed counts of real rational J-holomorphic curves in compact real symplectic fourfolds and sixfolds pX, ω, φq, now known as Welschinger’s invariants, were first defined in [44–46] and interpreted in terms of moduli spaces of J-holomorphic maps from the disk D2 in [39]. An adaptation of the interpretation of [39] in terms of real J-holomorphic maps from the Riemann sphere P1 later appeared in [14]. The two moduli-theoretic perspectives on Welschinger’s invariants lead to the WDVV-type relations for these curve counts announced in [40] and established 12 years later in [6,8]. In contrast to the intrinsically defined curve signs of [44] for real symplectic fourfolds, the curve signs of [14, 39] depend on the choice of a relative Pin-structure on the fixed locus X φ of the involution φ on X. The curve signs of [45, 46] for real symplectic sixfolds depend on the choice of an OSpin-structure on X φ but in a more intrinsic way than in [14,39]. The main theorems of Part , Theorems 14.1 and 14.2, provide precise relations between the signs of [44–46] and [14,39]. While the systematic approach behind the proofs of these theorems is motivated by the approach outlined in [39, Section 8], the statements of these theorems are different from the sign comparison predictions in [39, 40]. Chapter 11 obtains the key topological results concerning Pinstructures on a surface Y used in the proof of Theorem 14.1. Theorems 11.1 and 11.2 in this chapter relate the restriction of a Pin-structure p on Y along an immersed loop α in Y and the number of nodes of α. The proof of these theorems uses the classical perspective of Definition 1.1 on Spin- and Pin-structures.

Real Enumerative Geometry

295

Chapter 12 begins by recalling classical perspectives on counting curves of low degree in the complex projective plane P2 and motivating Welschinger’s definition of invariant signed counts of real rational J-holomorphic curves in real symplectic fourfolds in [44]. After introducing a general real symplectic setting, we recall this definition and the proof from [44] that the resulting counts of real rational curves are indeed independent of the auxiliary choices involved. Figure 12.2 on page 236 summarizes this proof. Along the way, we provide an explicit formula for the Hessian of the projection πM in this figure and point out the fundamental reason why this proof does not extend to counting real curves in positive genera. In Chapter 13, we describe invariant counts of real rational curves in real symplectic manifolds as the degrees of relatively orientable pseudocycles from moduli spaces of real pseudoholomorphic maps with signed marked points. This further re-interpretation of the aforementioned moduli-theoretic perspectives on Welschinger’s invariants originates in [6] in the case of real symplectic fourfolds and was extended in [8] to real symplectic sixfolds. We explain why these degrees give well-defined counts of real rational curves, which depend only on the homology classes of the constraints, and gather key properties of the resulting invariants in Theorems 13.1 and 13.2. Just as in [14,39], our description of these invariants involves orienting determinants of real Cauchy-Riemann operators on the pullbacks of the real bundle pT X, dφq by parametrizations of the real curves being counted. These parametrizations are immersions if the constraints are chosen generically. Proposition 13.8 re-interprets the already given definition of real curve counts in terms of orienting determinants of real Cauchy–Riemann operators on the normal bundles pNu, ϕq to the relevant immersions. Both definitions of the real curve counts in Chapter 13 and their comparison use the properties of orientations of determinants of real Cauchy–Riemann operators provided by Theorem 7.1, which in turn uses the trivializations perspective of Definition 1.3 on Spin- and Pin-structures and the trivializations perspective of Definition 6.3 on relative Spin- and Pin-structures. Theorems 14.1 and 14.2 provide a comparison of the counts of real rational curves defined in Chapter 13 with Welschinger’s versions of these counts in real symplectic fourfolds and sixfolds defined

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in [44–46], respectively. We deduce the two comparison theorems from Proposition 13.8; the proof of the first theorem also uses Theorem 11.1. Thus, the proof of Theorem 14.1 depends on the perspectives of Definitions 1.1 and 1.3 on Spin- and Pin-structures being equivalent in a way respecting the properties of these structures listed in Section 1.2; such an equivalence is provided by Theorem 1.4. Chapter 14 also contains examples of real curve counting invariants in basic real symplectic fourfolds and sixfolds. These examples illustrate some of the implications of the properties provided by Theorems 13.1 and 13.2 and of the comparisons of Theorems 14.1 and 14.2.

Chapter 11

Pin-Structures and Immersions

Let Y be a smooth manifold. For α P LpY q, define ´α P LpY q by ´αpzq “ αpzq @ z P S 1 Ă C. A smooth map α : S 1 ÝÑ Y is an immersion if dz α ‰ 0 for all z P S 1 . For such a map, let

Nα ”

α˚ T Y ÝÑ S 1 Im dα

denote its normal bundle. Thus, an immersion α determines an exact sequence 0 ÝÑ T S 1 ÝÑ α˚ T Y ÝÑ Nα ÝÑ 0

(11.1)

of real vector bundles over S 1 , which we denote by epαq. A node of such a map is a pair tz1 , z2 u of distinct points of S 1 such that αpz1 q “ αpz2 q; a triple self-intersection point of α corresponds to 3 nodes in this definition. We call an immersion α admissible if Im dz1 α ‰ Im dz2 α for every node tz1 , z2 u of α. The number of nodes of an admissible immersion α is finite; we denote it by δpαq. Let L˚ pY q Ă LpY q denote the set of admissible immersions into Y .

297

298

11.1

Spin/Pin-Structures & Real Enumerative Geometry

Main Statements and Examples

Suppose α : S 1 ÝÑ Y is an immersion. By Example 1.8, the standard orientation on S 1 Ă C determines an OSpin-structure os0 pT S 1 q on the first non-trivial vector bundle in the short exact sequence (11.1). By the SpinPin 1 property in Section 1.2, each of the other two bundles in (11.1) admits a Pin˘ -structure. Along with the H 1 pY ; Z2 qequivariance of the second map in (1.24) in the second input and the SpinPin 2 property, this implies that the map

P˘ pNαq ÝÑ P˘ pα˚ T Y q,

p2 ÝÑ xxos0 pT S 1 q, p2 yy epαq ,

(11.2)

is an H 1 pY ; Z2 q-equivariant bijection. By the SpinPin 2 property, P˘ pNαq consists of two elements. Let Y be a smooth surface (manifold of real dimension 2), possibly with boundary, and α : S 1 ÝÑ Y be an immersion. The last nontrivial vector bundle in (11.1) is then a line bundle. The paragraph above Lemma 1.18 and the statement of Lemma 1.19 identify the lefthand side of (11.2) with Z2 by specifying a Pin˘ -structure p˘ 0 pNαq corresponding to 0 P Z2 . For a Pin˘ -structure p on T Y , we define tp pαq P Z2 by DD @@ ˚ (11.3) os0 pT S 1 q, p˘ tp pαq pNαq epαq “ α p. If Nα is orientable, the identification of P˘ pNαq with Z2 does not depend on the orientation of α. Along with the first statement in the SpinPin 7(ses3) property, this implies that tp p´αq “ tp pαq @ p P P˘ pT Y q

if w1 pα˚ T Y q “ 0.

If Nα is not orientable, the identification of P` pNαq with Z2 does not depend on the orientation of α and the identification of P´ pNαq with Z2 does. Along with the first statement in the SpinPin 7(ses3) property, this implies that tp p´αq “ tp pαq @ p P P´ pT Y q, @ D tp p´αq “ tp pαq` w1 pY q, rαsZ2 @ p P P` pT Y q.

(11.4)

For os P OSppT Y q, let tos pαq ” tp pαq be the element of Z2 determined by the preimage p of os under (1.15). By the SpinPin 4 property

Pin-Structures and Immersions

299

and (1.26), this element is the same for the two possibilities for the domain of (1.15). We call a topological surface Y closed if it is compact and without boundary. For such a surface and α, β P H1 pY ; Z2 q, let @ D α¨β ” pPDY αqpPDY βq, rY sZ2 P Z2 denote the homology intersection number of α and β. Theorem 11.1. Let Y be a smooth closed surface. For every Pin´ structure p on T Y, the map μp : H1 pY ; Z2 q ÝÑ Z2 ,

μp prαsZ2 q “ 1`tp pαq`δpαq @ α P L˚ pY q,

is well-defined and satisfies ` ˘ μp α`β “ μp pαq`μp pβq ` α¨β ` xw1 pY q, αyxw1 pY q, βy

@ α, β P H1 pY ; Z2 q.

(11.5)

Furthermore, μη¨p pαq “ μp pαq ` xη, αy

@ η P H 1 pY ; Z2 q, α P H1 pY ; Z2 q.

(11.6)

Theorem 11.2. Let Y be a smooth closed surface. For every Pin` structure p on T Y, the map μp : H1 pY ; Zq ÝÑ Z2 ,

μp prαsZ q “ 1`tp pαq`δpαq @ α P L˚ pY q,

is well defined and satisfies ` ˘ μp p0q “ 0, μp α`β “ μp pαq`μp pβq`αZ2 ¨βZ2 where αZ2 , βZ2 Furthermore,

P

@ α, β P H1 pY ; Zq, (11.7) H1 pY ; Z2 q are the Z2 -reductions of α, β.

μη¨p pαq “ μp pαq ` xη, αZ2 y

@ η P H 1 pY ; Z2 q, α P H1 pY ; Zq.

(11.8)

Example 11.3. Let α : S 1 ÝÑ R2 be the standard inclusion of the unit circle and α´ : S 1 ÝÑ SOp2q be as in the n “ 2 case of the proof of Lemma 2.2. The standard orientations on S 1 and R2 determine an orientation on Nα via the exact

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sequence (11.1); it is given by the inward pointing radial direction. The Spin-structure on ` ˘ˇ T R2 |S 1 “ R2 ˆR2 ˇS 1 “ S 1 ˆR2 induced by the canonical Spin-structures on the oriented line bundles T S 1 and Nα via the exact sequence (11.1) is described by the section ` ` ˘ ˘ e2πit ÝÑ e2πit , α´ ptq . S 1 ÝÑ SO T R2 |S 1 “ S 1 ˆSOp2q, Since this loop generates π1 pSOp2qq, this OSpin-structure differs from the restriction of the unique OSpin-structure os on the oriented vector bundle T R2 over R2 . Thus, tos pαq “ 1. This implies that μp prαsq “ 0 for any Pin-structure p on a smooth closed surface Y and a homotopically trivial admissible loop α in Y . Example 11.4. Let os0 be the OSpin-structure on the 2-torus pS 1 q2 induced by the standard homotopy class of trivializations ˘ ` T pS 1 q2 “ T S 1 ˆT S 1 “ pS 1 q2 ˆR2 of T Y . For any parametrization α of a circle zˆS 1 or S 1 ˆz in pS 1 q2 , @@ DD α˚ os0 “ os0 pT S 1 q, os0 pNαq epαq and so tos0 pαq “ 0. Thus, μos0 pαq “ 1

` ˘ @ α P H1 pS 1 q2 ; Z2 ´t0u.

(11.9)

Example 11.5. Let Y be the infinite Mobius band, SγR;1 Ă Y be ` its unit circle bundle, and p` 0 be the Pin -structure on Y defined in Example 1.25. By the SpinPin 2(a) property, Y admits two Pin` structures. By (1.43) and the second statement in (11.4), tp` pαq “ 0 0 for every smooth parametrization α : S 1 ÝÑ Y of SγR;1 . Since the restriction homomorphism ` ˘ H 1 pY ; Z2 q ÝÑ H 1 SγR;1 ; Z2 is trivial, the SpinPin 2(a) property and the H 1 pY ; Z2 q-equivariance of (1.30) in the first input then imply that tp` pαq “ 0 for the other Pin` -structure p` 1 on Y as well.

1

Pin-Structures and Immersions

301

´ 2 2 Example 11.6. Let p0 ” p´ 0 pRP q and p1 ” p1 pRP q be the two ´ Pin -structures on RP2 as in Example 1.16. By (4.40) and the first statement in (11.4), tp´ pαq “ 0 for every smooth homotopically non1 trivial embedding α of S 1 into RP2 . Along with the conclusion of Example 11.3 and (11.6), this implies that

μp´ prαsq 1 # 0, if rαs “ 0 P H1 pRP2 ; Z2 q; “ 1, if rαs ‰ 0 P H1 pRP2 ; Z2 q;

` ˘ μp´ prαsq “ 0 @ rαs P H1 RP2 ; Z2 . 0

(11.10) Remark 11.7. Let Y be as in Theorems 11.1 and 11.2. Denote by Q´ pY q and Q` pY q the spaces of maps μ : H1 pY ; Z2 q ÝÑ Z2

and

μ : H1 pY ; Zq ÝÑ Z2

satisfying (11.5) and (11.7), respectively, with μp replaced by μ. The group H 1 pY ; Z2 q acts on Q´ pY q and Q` pY q freely and transitively by tη¨μupαq “ μpαq`xη, αy

@ η P H 1 pY ; Z2 q, α P H1 pY ; Z2 q

tη¨μupαq “ μpαq`xη, αZ2 y

@ η P H 1 pY ; Z2 q, α P H1 pY ; Zq,

and

respectively. By the SpinPin 1(a) property in Section 1.2 and Wu’s relations (11.12), T Y admits a Pin´ -structure. For the same reasons, T Y admits a Pin` -structure if Q` pY q ‰ H. Thus, the assignments

P˘ pY q ÝÑ Q˘ pY q,

p ÝÑ μp ,

(11.11)

of Theorems 11.1 and 11.2 are natural H 1 pY ; Z2 q-equivariant bijections. A quadratic enhancement (of the intersection form) on Y is a map q : H1 pY ; Z2 q ÝÑ Z4

` ˘ ` ˘ s.t. q α`β “ qpαq`qpβq`ι α¨β @ α, β P H1 pY ; Z2 q,

where ι : Z2 ÝÑ Z4 is the inclusion. A natural H 1 pY ; Z2 q-equivariant bijection between P´ pY q and the space of quadratic enhancements on Y of the same flavor as (11.11) is provided by [28, Theorem 3.2], based on the perspective on the Kervaire invariant in [5].

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302

11.2

Admissible Immersions into Surfaces

We deduce Theorems 11.1 and 11.2 from Proposition 11.8. The latter is in turn a consequence of the four lemmas stated in this section and proved in the following. We recall that Wu’s relations [34, p. 132] imply that w2 pY q “ w12 pY q P H 2 pY ; Z2 q and xw1 pY q, αy “ α2 P Z2 @ α P H1 pY ; Z2 q

(11.12)

for a smooth closed surface Y . Proposition 11.8. Let Y be a smooth closed surface and α1 , . . . , αm : S 1 ÝÑ Y be admissible loops. If m ÿ

rαi sZ2 “ 0 P H1 pY ; Z2 q

(11.13)

i“1

and p P P´ pT Y q, then m ÿ `

˘ tp pαi q`δpαi q

i“1

“m`

ÿ´ D@ D¯ @ P Z2 . αi ¨αj ` w1 pY q, rαi sZ2 w1 pY q, rαj sZ2 iăj

(11.14) If p11.13q holds with Z-coefficients and p P P` pT Y q, then m ÿ ÿ ˘ ` αi ¨αj P Z2 . tp pαi q`δpαi q “ m ` i“1

(11.15)

iăj

Proof of Theorem 11.1. Let p be a Pin´ -structure on T Y . If admissible loops α1 , α2 : S 1 ÝÑ Y agree in H1 pY ; Z2 q, then (11.14) and (11.12) give ˘ ` ˘ @ D ` tp pα1 q`δpα1 q ` tp pα2 q`δpα2 q “ α1 ¨α1 ` w1 pY q, rα1 sZ2 “ 0. Thus, the map μp : H1 pY ; Z2 q ÝÑ Z2 ,

` ˘ μp rαsZ2 “ tp pαq`δpαq`1,

Pin-Structures and Immersions

303

is well defined (α : S 1 ÝÑ Y is an admissible loop above). By the m “ 3 case of (11.14) and (11.12), ` ˘ μp α`β ` μp pαq`μp pβq “ α¨β ` xw1 pY q, αyxw1 pY q, βy ` pα`βq¨pα`βq`xw1 pY q, α`βy “ α¨β ` xw1 pY q, αyxw1 pY q, βy. This establishes (11.5). Since the H 1 pY ; Z2 q-action of the SpinPin 2 property is natural and free, @ @ D D tη¨p pαq “ tp pαq` α˚ η, rS 1 sZ2 “ tp pαq ` η, rαsZ2 (11.16) for every η P H 1 pY ; Z2 q and every admissible loop α : S 1 ÝÑ Y . This implies (11.6).  Proof of Corollary 11.2. Let p be a Pin` -structure on T Y . If admissible loops α1 , α2 : S 1 ÝÑ Y agree in H1 pY ; Zq, then (11.15) and (11.12) give ˘ ` ˘ @ D ` tp pα1 q`δpα1 q ` tp p´α2 q`δpα2 q “ α1 ¨α1 “ w1 pY q, rα1 sZ2 . Along with the second statement in (11.4), this implies that the map ` ˘ μp rαsZ2 “ tp pαq`δpαq`1, μp : H1 pY ; Zq ÝÑ Z2 , is well defined and satisfies D @ μp pαq`μp p´αq “ w1 pY q, αZ2

@ α P H1 pY ; Zq.

(11.17)

By the m “ 3 case of (11.15) and (11.12), ` ˘ μp α`β `μp p´αq`μp p´βq “ α¨β `pα`βq¨pα`βq “ α¨β `xw1 pY q, αy ` xw1 pY q, βy. Along with (11.17), this implies the second claim in (11.7). The first claim in (11.7) follows immediately from Example 11.3. The rela tion (11.8) follows from (11.16). We will deduce Proposition 11.8 from the four lemmas stated in the following; they are proved in Section 11.3. Lemma 11.9 obtains an analog of this proposition for the boundary components of a compact bordered surface. Lemma 11.10 is used in the proof of

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Proposition 11.8 to deduce its case for disjoint embedded loops from Lemma 11.9. Lemma 11.11 and the crucial Lemma 11.12 reduce the general case of this proposition to this special case. Lemma 11.9. Let Y be a compact surface with boundary components α1 , . . . , αm . If p is a Pin` -structure on T Y, then m ÿ

tp pαi q “ χpY q`2Z P Z2 .

(11.18)

i“1

If p is a Pin´ -structure on T Y, then m ÿ

tp pαi q “ m`2Z P Z2 .

(11.19)

i“1

Lemma 11.10. Let Y be a smooth surface and C1 , . . . , Cm Ă Y be a collection of disjoint simple circles such that m ÿ “ ‰ Ci Z2 “ 0 P H1 pY ; Z2 q.

(11.20)

i“1

(1) There exists an immersion f : Σ ÝÑ Y from a compact surface with boundary components pBΣq1 , . . . , pBΣqm so that the restriction of f to each pBΣqi is a homeomorphism onto Ci . (2) If in addition the circles Ci are oriented and p11.20q holds with Z coefficients, then there exists an immersion f : Σ ÝÑ Y from a compact oriented surface with boundary components pBΣq1 , . . . , pBΣqm so that the restriction of f to each pBΣqi is an orientation-preserving homeomorphism onto Ci . Let Y be a smooth surface and α1 , . . . , αm : S 1 ÝÑ Y be admissible loops. By applying a small deformation, we can deform these loops so that all their intersection and self-intersection points are simple nodes, i.e. ˇ (ˇ ˇ pi, zq P t1, . . . , muˆS 1 : αi pzq “ p ˇ ď 2 @ p P Y,

Im dz1 αi ‰ Im dz2 αj @ pi, z1 q ‰ pj, z2 q.

We can then smooth out all intersection and self-intersection points in accordance with the orientations of the loops inside of small

Pin-Structures and Immersions

Figure 11.1.

305

Smoothing a node of a collection of oriented curves.

coordinate charts as in Figure 11.1. We call the new collection α11 , . . . , α1m1 : S 1 ÝÑ Y of oriented loops a smoothing of the collection α1 , . . . , αm . Lemma 11.11. Let Y be a smooth surface and α1 , . . . , αm : S 1 ÝÑ Y be admissible loops. If α11 , . . . , α1m1 is a smoothing of α1 , . . . , αm , then 1

m ÿ

rα1i sZ

“

i“1

m ÿ

rαi sZ P H1 pY ; Zq.

(11.21)

i“1

If in addition Y is closed, then m ÿ

m1 ” m `

δpαi q `

i“1

ÿ

αi ¨αj mod 2.

(11.22)

iăj

Lemma 11.12. Let Y, α1 , . . . , αm , and α11 , . . . , α1m1 be as in Lemma 11.11. (a) If p is a Pin` -structure on T Y, then 1

m ÿ

m ÿ

tp pα1i q “

i“1

tp pαi q.

(11.23)

i“1

(b) If p is a Pin´ -structure on T Y, then 1

m ÿ

tp pα1i q `

i“1

ÿ@

w1 pY q, rα1i sZ2

D@

w1 pY q, rα1j sZ2

D

iăj

“

m ÿ i“1

tp pαi q `

ÿ@

w1 pY q, rαi sZ2

iăj

D@

D w1 pY q, rαj sZ2 .

(11.24)

Spin/Pin-Structures & Real Enumerative Geometry

306

Proof of Proposition 11.8. By Lemmas 11.10 and 11.12, we can replace the collection α1 , . . . , αm by its smoothing. Thus, we can assume that the images C1 , . . . , Cm of the loops α1 , . . . , αm are disjoint circles. Along with (11.13) and (11.12), this implies that @ D w1 pY q, rCi sZ2 “ 0

@ i “ 1, . . . , m.

(11.25)

Thus, (11.14) and (11.15) in this situation reduce to m ÿ

tp pαi q ” m

mod 2.

(11.26)

i“1

Since the normal bundle of each Ci is orientable by (11.25), tp pαi q does not depend on the orientation of αi even for p P P` pT Y q in this case. Let f : Σ ÝÑ Y be as in Lemma 11.10. Thus, m ÿ i“1

1

tp pαi q ”

m ÿ

` ˘ tf ˚ p f |pBΣqi mod 2

@ p P P˘ pT Y q.

(11.27)

i“1

Along with (11.19) with pY, pq replaced by pΣ, f ˚ pq, (11.26) for all p P P´ pT Y q. Suppose (11.13) holds with Z-coefficients. Let f : as in Lemma 11.1011.10. Since Σ is oriented, χpΣq is parity as m1 . Along with (11.18) with pY, pq replaced this implies (11.26) for all p P P` pT Y q.

this implies Σ ÝÑ Y be of the same by pΣ, f ˚ pq, 

Example 11.13. Let Y be a smooth closed surface. By (11.5) and (11.12), μp p0q “ 0 for every p P P´ pT Y q; this is consistent with the conclusion of Example 11.3. By (11.7) and (11.12), @ D μp pαq`μp p´αq “ w1 pY q, αZ2

@ α P H1 pY ; Zq, p P P` pT Y q ;

this is consistent with the second statement in (11.4). This relation in particular implies that the normal bundle to an (embedded) circle C in a smooth closed connected unorientable surface Y representing the unique two-torsion element α of H1 pY ; Zq is trivializable if T Y admits a Pin` -structure. This can be obtained directly from the evenness

Pin-Structures and Immersions

307

of χpY q, as such a class α is the sum of the basis elements α1 , . . . , αm in the proof of Corollary 11.14. Corollary 11.14. Let Y be a closed connected unorientable surface. If the dimension of H1 pY ; Z2 q is at most 3, then T Y admits a canonical Pin´ -structure p´ 0 pY q; it is preserved by every homeomorphism of Y . Proof. We first recall an observation from the proof of [15, Lemma 2.2]. By [36, Theorem 77.5], Y is the connected sum of m copies of RP2 for some m P Z` ; the dimension of H1 pY ; Z2 q then is m. By [36, Theorem 77.5], Y can be represented by the labeling scheme α1 α1 . . .αm αm ; see Figure 11.2. From the labeling scheme, it is immediate that αi ¨αi “ 1 and αi ¨αj “ 0 if i ‰ j. Thus, there exists a basis α1 , . . . , αm for H1 pY ; Z2 q diagonalizing the Z2 -intersection form. By the Universal Coefficient Theorem for Cohomology [35, Theorems 53.5], the homomorphism ` ˘  ( κpηq pαq “ xη, αy, κ : H 1 pY ; Z2 q ÝÑ HomZ2 H1 pY ; Z2 q, Z2 , is an isomorphism. By the SpinPin 1 property in Section 1.2 and (11.12), T Y admits a Pin´ -structure p. Along with (11.6), these two statements imply that T Y admits a unique Pin´ -structure p´ 0 pY q so that μp´ pY q pαi q “ 0 0

@ i “ 1, . . . , m.

(11.28)

If m ď 3, a basis α1 , . . . , αm diagonalizing the Z2 -intersection form is unique up to the permutations of its elements. The Pin´ -structure determined by (11.28) then does not depend on the choice of such a  basis and is preserved by every homeomorphism of Y .

Figure 11.2. Labeling scheme for Y “ RP2#RP2#RP2 and a deformation α11 of the loop α1 used to compute the intersection product on H1 pY ; Z2 q.

308

Spin/Pin-Structures & Real Enumerative Geometry

Remark 11.15. If m “ 4 and α1 , . . . , α4 are as in the proof of Corollary 11.14, then α11 ” α2`α3`α4 ,

α12 ” α1`α3`α4 ,

α13 ” α1`α2`α4 ,

α14 ” α1`α2`α3

are another basis for H1 pY ; Z2 q diagonalizing the Z2 -intersection form and μp´ pY q pα1i q “ 1 for all i. Every disjoint collection C1 , . . . 0 Cm Ă Y of circles representing a diagonalizing basis for H1 pY ; Z2 q presents Y as the real blowup of S 2 at m distinct points (with S 2 obtained from Y by contracting the circles). For any two collections x1 , . . . , xm and x11 , . . . , x1m of distinct points on S 2 , there exists a homeomorphism from the blowup Y at the first set to the blowup Y 1 at the second set so that the i-th “exceptional divisor” Ci for the first blowup is taken to Ci1 for all i. For any two diagonalizing bases α1 , . . . , αm and α11 , . . . , α1m for H1 pY ; Z2 q, there thus exists a homeomorphism f of Y so that f˚ αi “ α1i for all i. This implies that Corollary 11.14 does not extend to m ě 4. 11.3

Proofs of Lemmas 11.9–11.12

We now establish the four lemmas stated in Section 11.2. Proof of Lemma 11.9. We can assume that Y is connected and m ě 1. Let Σ be the closed surface obtained from Y by attaching disks along the boundary components of Y . By the SpinPin 1 property in Section 1.2 and (11.12), T Σ admits a Pin´ -structure; if the Euler characteristic χpΣq “ χpY q`m is even, then T Σ also admits a Pin` -structure. By Example 11.3, (11.18) and (11.19) hold if p is the restriction p0 of a Pin˘ -structure on T Σ. By Corollary A.14, any other Pin˘ -structure p on T Y differs from p0 on an even number of boundary components. This establishes (11.18) if χpY q and m are of the same parity and (11.19) in all cases. Suppose χpY q and m are not of the same parity, i.e. χpΣq is odd, and thus Σ does not admit a Pin` -structure by the SpinPin 1 property. Let Σ1 Ă Σ be the surface obtained from Y by attaching disks along the boundary components α2 , . . . , αm of Y . By the SpinPin 1

Pin-Structures and Immersions

309

property, T Σ1 admits a Pin` -structure p1 . If tp1 pα1 q “ 1, then Example 11.3 implies that p1 extends to a Pin` -structure on T Σ. Thus, tp1 pα1 q “ 0 ” χpΣ1 q “ χpY q`pm´1q mod 2. Along with Example 11.3, this implies that (11.18) holds if p is the restriction p0 of p1 . By Corollary A.14, any other Pin` -structure p on T Y differs from p0 on an even number of boundary components. This establishes (11.18) in the remaining case.  Proof of Lemma 11.10. Let Σ1 , . . . , Σn be the topological components of the surface obtained by cutting Y along the circles C1 , . . . , Cm . Triangulate Y so that each Ci is a subcomplex (and no edges of the same 2-simplex are identified). Then, m ÿ

Ci “ B

i“1

k ÿ

Δj P C1 pY ; Z2 q

(11.29)

j“1

for some 2-simplices Δj Ă Y . Since the right-hand side of (11.29) contains no 1-simplices in the interior of any Σj , Σ ” Δ1 Y¨ ¨ ¨YΔk is the union of some of the bordered surfaces Σ1 , . . . , Σn ; the boundary of Σ is C1 Y¨ ¨ ¨YCm . By the above, the boundary of each surface Σj above is a union of some of the circles C1 , . . . , Cm . Suppose the circle Ci are oriented and (11.20) holds with Z coefficients. Then, m ÿ

Ci “ B

i“1

k ÿ

aj Δj P C1 pY ; Zq

(11.30)

j“1

P Z`

and oriented simplices Δj Ă Y so that each unorifor some aj ented simplex appears at most once. For each a P Z` , let Σ1a be the union of the 2-simplices Δj with aj “ a. Since the right-hand side of (11.30) contains no 1-simplices in the interior of any Σj , each Σ1a is the union of some of the bordered surfaces Σ1 , . . . , Σn . The orientations of the simplices Δj orient Σ1a . Define Σ“

8 ğ

t1, . . . , auˆΣ1a ,

f : Σ ÝÑ Y, f pb, xq “ x @ x P Σ1a ,

a“1

b “ 1, . . . , a, a P Z` . By (11.30), the initial orientation of each Ci agrees with its boundary  orientation as a component of BΣ.

310

Spin/Pin-Structures & Real Enumerative Geometry

Proof of Lemma 11.11. The right-hand sides in (11.21) and (11.22) are invariant under small deformations of the loops αi . Thus, we can assume that all intersection and self-intersection points of α1 , . . . , αm are simple nodes. Smoothing each node then changes the number of loops by one. This implies (11.22); (11.21) is immediate from the construction of the smoothing.  Proof of Lemma 11.12. The right-hand sides in (11.23) and (11.24) are invariant under small deformations of the loops αi . Thus, we can assume that all intersection and self-intersection points of α1 , . . . , αm are simple nodes. We show in the following that (11.23) and (11.24) hold if α11 , . . . , α1m1 are obtained from α1 , . . . , αm by smoothing one node or intersection point p of α1 , . . . , αm . This suffices to establish the two identities. If U0 , U1 Ă S 1 are open (proper, connected) arcs covering S 1 , we denote by U01 Ă U0 XU1 the connected component that follows U0 ´U1 and precedes U1 ´U0 with respect to the standard orientation of S 1 and by U10 Ă U0 XU1 the other connected component. Let I2 , I2;1 P Op2q, rI2 P Spinp1q Ă Pin˘ p1q, and rI2 , pI2 P Spinp2q Ă Pin˘ p2q

be as in Sections 2.1 and 2.2. Fix a metric on T Y . We establish (11.14) and (11.15) using the perspective of Definition 1.1(a). Let α : S 1 ÝÑ Y be an immersion. We trivialize α˚ T Y with its metric over U0 and U1 by taking the first vector field to be the oriented vector field on S 1 of unit length with respect to the pullback metric and choosing the second vector field so that the transition α from U to U equals I on U . This implies that function g01 1 0 2 01 # ˇ if w1 pα˚ T Y q “ 0; I2 , α ˇ “ g01 U10 I2;1 , if w1 pα˚ T Y q ‰ 0. 1 for os pT S 1 q on T S 1 equals r The transition function gr01 I1 . 0 If Nα (or equivalently α˚ T Y ) is orientable, the canonical Pin˘ 2 “ r01 structure p˘ 0 pNιq on Nα is specified by the transition function g rI1 . By (3.35), the transition function ` 1 ˘ 2 (11.31) g01 , gr01 : U0 XU1 ÝÑ Pin˘ p2q g01 ” ι2;1 r r

r specifying the Pin˘ -structure xxos0 pT S 1 q, p˘ 0 pNιqyy epαq then equals I2 . ˚ The above trivializations for α T Y can thus be lifted to trivializations

Pin-Structures and Immersions

311

for the Pin˘ -structure α˚ p on α˚ T Y so that the associated transition p function gr01 satisfies t pαq p ˇˇ p ˇˇ “ rI2 , gr01 “ pI2p , (11.32) gr01 U01 U10 if α˚ T Y is orientable. If the vector bundle α˚ T Y is not orientable, the transition function (11.31) specifying the Pin˘ -structure xxos0 pT S 1 q, p˘ 0 pNιqyy epαq is described by ˇ ˇ “ rI2 , gr01 ˇ “ rI‹ , g01 ˇ r U01

U10

2;1

for a certain element rI‹2;1 P Pin˘ p2q in the preimage of I2;1 under (2.26). The above trivializations for α˚ T Y can thus be lifted to trivializations for the Pin˘ -structure α˚ p on α˚ T Y so that the p associated transition function gr01 satisfies ˇ t pαq p ˇ p ˇˇ rI2 , “ g r “ rI‹2;1pI2p , (11.33) gr01 01 U01 U10 if α˚ T Y is not orientable. Let p P Y be either a simple node of a loop α so that smoothing p breaks α into loops αa and αb , as in Figure 11.3, or a transverse intersection point of distinct loops αa and αb so that smoothing p combines αa and αb into a single loop α. Since α is homologous to the sum of αa and αb , D @ D @ D @ (11.34) w1 pY q, rαa sZ2 ` w1 pY q, rαb sZ2 “ w1 pY q, rαsZ2 . Let U Ă Y be a square coordinate neighborhood of p intersecting the collection α1 , . . . , αm along the two diagonals and U 2 Ă U 1 Ă U be

Figure 11.3.

The space S and the loops α, αa , αb Ă S.

Spin/Pin-Structures & Real Enumerative Geometry

312

neighborhoods of p so that U 2 Ă U 1 and U 1 Ă U . We smooth out p inside of U 2 and define ` ˘L S “ U 1 \α´1 pY ´U 2 q „, U 1 Q αpzq „ z P S 1 @ z P α´1 pU 1 ´U 2 q, # x, if x P U 1 ; f : S ÝÑ Y, f pxq “ U0 “ S ´α´1 pY ´U q. ` ˘ αpzq, if z P α´1 Y ´U 2 ;

Let Ua , Ub Ă α´1 pY ´U 1 q be the connected components so that Ua is contained in the domain of the loop αa and Ub is contained in the domain of the loop αb . The collection tUa , Ub , U0 u is then an open cover of S. For ‚ “ ta, bu, we denote by U0‚ Ă U0 XU‚ the connected component that follows U0´UaYUb and precedes U‚´U0 with respect to the standard orientation of S 1 and by U‚0 Ă U0 XU‚ the other connected component. Let ξ P ΓpS; f ˚ T Y q be such that ξ|Ua is the standard oriented unit vector field on the domain S 1 of αa , ξ|Ub is its analog for αb , and ξ|U 1 is as depicted in Figure 11.3. We trivialize f ˚ T Y over Ua , Ub , and U0 by taking the first vector field to be ξ and choosing the f second vector field so that the transition functions g0a from Ua to U0 f and g0b from Ub to U0 satisfy f ˇˇ f ˇˇ “ I2 and g0b “ I2 . g0a U0a U 0b

This implies that f ˇˇ g0‚ U‚0

# “

if w1 pα˚‚ T Y q “ 0, ‚ “ a, b;

I2 ,

I2;1 , if w1 pα˚‚ T Y q ‰ 0, ‚ “ a, b.

In light of (11.32) and (11.33), these trivializations for f ˚ T Y can be lifted to trivializations for the Pin˘ -structure f ˚ p on f ˚ T Y so that p p and r g0b satisfy the associated transition functions r g0a ˇ

ˇ

p ˇ g0a r , grp ˇ U0a 0b U0b

“ rI2 ,

ˇ

p ˇ g0‚ r U‚0

“

# tp pα‚ q pI , 2

if w1 pα˚‚ T Y q “ 0, ‚ “ a, b;

rI‹ pI tp pα‚ q , if w1 pα˚ T Y q ‰ 0, ‚ “ a, b. 2;1 2 ‚

(11.35) Let ι : S 1 ÝÑ S be the continuous map so that α “ f ˝ι. This map identifies the open subsets Ua , Ub , U0a , Ua0 , U0b , Ub0 Ă S with their preimages. We denote by U0$ Ă ι´1 pU0 q the connected component containing Ua0 and U0b and by U0% Ă ι´1 pU0 q

Pin-Structures and Immersions

Figure 11.4. for α˚ p.

313

Open cover of the domain of α and transition functions (11.36)

the other connected component; see Figure 11.4. The collection tUa , Ub , U0$ , U0% u is then an open cover of S 1 and Ua XU0% “ U0a ,

Ua XU0$ “ Ua0 ,

Ub XU0$ “ U0b ,

Ub XU0% “ Ub0 ;

all other intersections are empty. The pullbacks by ι of the above trivializations of f ˚ T Y % of ι˚ f ˚ T Y “ α˚ T Y over induce trivializations Φa , Φb , Φ$ 0 , Φ0 $ % Ua , Ub , U0 , U0 , respectively, so that the first vector field in each trivialization is the standard oriented unit vector field on the domain S 1 of α. The pullbacks by ι of the above trivializations of f ˚ p induce trivr b, Φ r $, Φ r % of ι˚ f ˚ p “ α˚ p over Ua , Ub , U $ , U % , respecr a, Φ ializations Φ 0 0 0 0 p tively. By (11.35), the associated transition functions gra% from U0% p p p to Ua , gr$a from Ua to U0$ , grb$ from U0$ to Ub , and gr%b from Ub to U0% are given by p “ rI2 , gra% # tp pαa q pI , if w1 pα˚a T Y q “ 0; 2 p gr$a “ rI‹ pI tp pαa q , if w1 pα˚ T Y q ‰ 0; a 2;1 2 p “ rI2 , grb$ # tp pαb q pI , if w1 pα˚b T Y q “ 0; 2 p “ gr%b rI‹ pI tp pαb q , if w1 pα˚ T Y q ‰ 0. 2;1 2 b

(11.36)

r b and Φ r$ Suppose w1 pα˚a T Y q “ 0. Multiplying the trivializations Φ 0 t pα q p a ´1 p by pI2 q turns the last transition function in (11.36) into p p tp pαa q I2 g%b r

“

# tp pαa q`tp pαb q pI , 2

if w1 pα˚b T Y q “ 0;

rI‹ pI tp pαa q`tp pαb q , if w1 pα˚ T Y q ‰ 0; 2;1 2 b

(11.37)

Spin/Pin-Structures & Real Enumerative Geometry

314

and makes the remaining three transition functions rI2 . This modification does not effect the associated trivializations of α˚ T Y . Comparing (11.37) with (11.32) and (11.33), we conclude that tp pαq “ tp pαa q ` tp pαb q if w1 pα˚a T Y q “ 0 or w1 pα˚b T Y q “ 0. Suppose w1 pα˚a T Y q ‰ 0 and w1 pα˚b T Y q ‰ 0. Multiplying the trivir b and Φ r $ by prI‹ pI tp pαa q q´1 turns the last transition funcalizations Φ 2;1 2 0 tion in (11.36) into p r‹ p tp pαa q I2;1 I2 g%b r

# tp pαa q`tp pαb q pI , if p P P`pT Y q; ` ‹ ˘2 tp pαa q`tp pαb q 2 “ rI2;1 pI2 “ pI tp pαa q`tp pαb q`1 , if p P P´pT Y q; 2

(11.38) and makes the remaining three transition functions rI2 . This modifi˚ cation multiplies the associated trivializations Φb and Φ$ 0 of α T Y by I2;1 and thus leaves the first vector field unchanged. Comparing (11.38) with (11.33), we conclude that # tp pαa q`tp pαb q, if p P P` pT Y q; tp pαq “ tp pαa q`tp pαb q`1, if p P P´ pT Y q; if w1 pα˚a T Y q ‰ 0 and w1 pα˚b T Y q ‰ 0. By the last two paragraphs, # tp pαq “

if p P P` pT Y q; @ D@ D tp pαa q`tp pαb q` w1 pY q, rαa sZ2 w1 pY q, rαb sZ2 , if p P P´ pT Y q.

tp pαa q`tp pαb q,

Combining this with (11.34), we obtain (11.23) and (11.24), with α11 , . . . , α1m1 being loops obtained from α1 , . . . , αm by smoothing one  node or intersection point p of α1 , . . . , αm . Remark 11.16. Let α be as in the paragraph containing (11.33). By Examples 1.23 and 1.24, the canonical Pin˘ -structure p˘ 0 pNιq 2 | r on Nα is specified by the transition function satisfying gr01 U01 “ I1 2 ‹ r r r and gr01 |U10 “ I1;1 . Along with (2.36), this implies that I2;1 “ I2;1 .

Chapter 12

Counts of Rational Curves on Surfaces

Our next aim is to recall Welschinger’s definition of invariant counts of real rational curves in real symplectic fourfolds in [44]. This is done in Section 12.3, after introducing a general setting for counts of real curves. Welschinger’s geometric definition is motivated by a classical computation of counts of cubic curves in the complex projective plane P2 , which we describe in Sections 12.1 and 12.2 following the proof of [10, Proposition 4.7.3] and [41, Section 3.iv]. The invariance of Welschinger’s counts is deduced in Section 12.5 from Proposition 12.5; the essence of this argument (which follows [44]) is indicated by Figure 12.2 on page 336. Proposition 12.5 is established in Section 12.6; in the process, we give an explicit formula for the Hessian of the projection πM in (12.37) at a critical point. 12.1

Complex Low-Degree Curves in P2

For m P Z` and d P Zě0 , we denote by HPdm the complex vector space of all homogeneous polynomials of degree d P Zě0 on Cm and by ˘L ˘ ` ` P HPdm ” HPdm ´t0u C˚ its complex projectivization. The dimension of this space is ˘ `d`m´1 ´1. Let m´1  ( γC;2 ” p, vq P P2 ˆC3 : v P  Ă C3 ÝÑ P2 315

316

Spin/Pin-Structures & Real Enumerative Geometry

be the complex tautological line bundle. Each P P HPd3 determines a bd ˚ ˚bd q « γC;2 holomorphic section sP of the holomorphic line bundle pγC;2 2 over P by  (` ˘ sP pq p, vqbd “ P pvq . Furthermore, every holomorphic section of this line bundle equals sP for some P P HPd3 . The section sP is transverse to the zero set if and only if 0 P C is a regular value of P |C3 ´t0u . For d P Z` , a subset C Ă P2 is a reduced (resp. irreducible, smooth) degree d curve if C “ sP´1 p0q for some polynomial P P HPd3 so that P 1 is not divisible by Q2 for any polynomial Q P HPd3 with d1 P Z` (resp. P is irreducible, 0 P C is a regular value of P |Cm ´t0u ). A reduced degree d curve C Ă P2 thus corresponds to an element rP s P PpHPd3 q with sP´1 p0q “ C. A point p P C is then called smooth (resp. singular) if dp sP ‰ 0 (resp. dp sP “ 0). In particular, a reduced curve C Ă P2 is smooth if and only if all its points are smooth. For any rP s P PpHPd3 q, there exist integers m1 , . . . , mr P Z` and distinct irreducible curves C1 , . . . , Cr Ă P2 of degrees d1 , . . . , dr P Z` , respectively, so that s´1 P p0q “

r ď

Cs

s“1

and d “

r ÿ

ms ds .

(12.1)

s“1

2 The subset s´1 P p0q Ă P is a reduced curve if and only if m1 , . . . , mr “ 1 in (12.1). Every line C Ă P2 , i.e. a degree 1 curve, is smooth. Its homology class L generates H2 pP2 ; Zq. A smooth degree d curve C Ă P2 is a compact connected Riemann surface of genus ˆ ˙ d´1 . gd ” 2

In particular, all lines and smooth conics C Ă P2 , i.e. degree 2 curves, are of genus 0; all smooth cubics C Ă P2 , i.e. degree 3 curves, are of genus 1. For every reduced degree d curve C Ă P2 , there exists a holomorphic map u : Σ ÝÑ P2 from a compact Riemann surface so that u is an embedding outside of finitely many points and upΣq “ C; see [22, p. 498]. This map, called the normalization of C, is unique up to the

Counts of Rational Curves on Surfaces

317

obvious equivalence. If C is irreducible, Σ is connected of genus at most gd . In such a case, the (geometric) genus gpCq of C is defined to be the genus of Σ. An irreducible curve C Ă P2 is called rational if gpCq “ 0, i.e. it is the image of a holomorphic map u : P1 ÝÑ P2 . If C Ă P2 is any reduced curve, a point p P C is called a simple node if à ˇ ´1 ˇ ˇu ppqˇ “ 2 and Im dz u “ Tp P2 , zPu´1 ppq

where u : Σ ÝÑ P2 is the normalization of C. If C Ă P2 is an irreducible degree d curve and all its singular points are simple nodes, then the number of these nodes is gd ´gpCq. If d P Z` and polynomials P0 , P1 , P2 P HPd2 have no common factor, the map ` ˘ uP0 P1 P2 : P1 ÝÑ P2 , uP0 P1 P 22 rZ0 , Z1 s ‰ “ “ P0 pZ0 , Z1 q, P1 pZ0 , Z1 q, P2 pZ0 , Z1 q , (12.2) is well-defined, holomorphic, and of degree d, i.e. (` ˘  uP0 P1 P 22 ˚ rP1 sZ “ dL P H2 pP2 ; Zq. Furthermore, every holomorphic degree d map u : P1 ÝÑ P2 equals uP0 P1 P2 for some polynomials P0 , P1 , P2 P HPd2 with no common factor. Thus, the space P0 pP2 , dq of (parametrized) holomorphic degree d maps u : P1 ÝÑ P2 is canonically identified with a dense open subspace of ˘ ` P HPd2 ˆHPd2 ˆHPd2 « P3pd`1q´1 . We denote by P˚0 pP2 , dq Ă P0 pP2 , dq the dense open subspace of maps u so that upP1 q Ă P2 is a reduced degree d curve. The group PSL2 C of holomorphic automorphisms of P1 acts on P0 pP2 , dq by composition on the right, preserving the image of each map in P2 . The restriction of this action to P˚0 pP2 , dq is free. The quotient L M˚0 pP2 , dq ” P˚0 pP2 , dq PSL2 C parametrizing irreducible degree C rational curves C Ă P2 is a complex manifold of dimension 3d ´ 1. For a generic element

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rus P M˚0 pP2 , dq, the holomorphic map u is an immersion so that the only singular points of its image upP1 q Ă P2 are simple nodes. Since the genus of a smooth degree d curve in P2 is gd , the number of such nodes is gd . For a point p P P2 , the requirement that p P s´1 P p0q is a linear condition on the coefficients of a polynomial P P HPd3 . Thus,  ` ( ` d˘ ˘ ˘ ` P HPd3 p ” rP s P P HPd3 : p P s´1 P p0q Ă P HP3 is a hyperplane. For a generic k-tuple p ” pp1 , . . . , pk q of points in P2 , the corresponding hyperplanes are transverse and ˘ ˘  ` ( ` P HPd3 p ” rP s P P HPd3 : p1 , . . . , pk P s´1 P p0q ”

k č ˘ ˘ ` ` P HPd3 p Ă P HPd3 i“1

i

is a linear subspace of (complex) codimension k. Similarly,  ( M˚0 pP2 , dqp ” rus P M˚0 pP2 , dq : p1 , . . . , pk P upP1 q Ă M˚0 pP2 , dq is a complex submanifold of (complex) dimension 3d ´ 1 ´ k for a generic k-tuple p as above. For d P Z` , we denote by Nd the number of degree d rational curves C Ă P2 that pass through 3d´1 points in general position in P2 . This is the cardinality of the zero-dimensional manifold M˚0 pP2 , dqp for a generic p3d ´ 1q-tuple p of points in P2 . If P P HPd3 is the limit of a sequence of polynomials Pi P HPd3 cutting out such curves and m1 , . . . , mr , C1 , . . . , Cr , and d1 , . . . , dr are as in (12.1), then every curve Cs is rational. Since r ÿ s“1

dimC M˚0 pP2 , ds q “

r ÿ

p3ds ´1q ď 3d´1

s“1

2 and the equality above holds if and only of s´1 P p0q Ă P is an irreducible degree d curve, the number Nd is finite. Since two distinct points p1 and p2 in P2 determine a line in P2 , N1 “ 1. This is also the intersection number of two distinct hyperplanes, PpHP13 qp1 and PpHP13 qp2 , in PpHP13 q « P2 . Since five points

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p1 , . . . , p5 in general position in P2 (no three are on a line) determine a conic in P2 and this conic is smooth (not a union of two lines), N2 “ 1. This is also the intersection number of five distinct hyperplanes, PpHP23 qp1 , . . . , PpHP23 qp5 , in PpHP23 q « P5 . Eight points p1 , . . . , p8 in general position in P2 determine a family of irreducible cubics parametrized by the intersection PpHP33 qp1 ...p8 “ PpHP33 qp1 X¨ ¨ ¨XPpHP33 qp8 « P1

(12.3)

of eight hyperplanes in PpHP33 q « P9 . This pencil of cubics consists of the irreducible curves p0q Ă P2 Crλ0 ,λ1 s ” sλ´1 0 P0 `λ1 P1 with rλ0 , λ1 s P P1 for some fixed non-zero polynomials P0 , P1 P HP33 so that p0q, sP´1 p0q Ă P2 sP´1 0 1 are smooth cubics intersecting transversely at 9 points, including p1 , . . . , p8 . Every pair of distinct curves Crλ0 ,λ1 s in this pencil intersects at these 9 points transversely as well. Every point of p0qXsP´1 p0q lies on a unique cubic Crλ0 ,λ1 s . A generic eleP2 ´sP´1 0 1 ment Crλ0 ,λ1 s is a smooth cubic and thus of genus 1. The remaining elements have exactly one singular point, a simple node, at a point p0qXsP´1 p0q. Thus, each of the singular curves in the pencil not in sP´1 0 1 is a rational cubic. By definition, the number of these curves is N3 . p0qXsP´1 p0q. For each Let S be the complex blowup of P2 at sP´1 0 1 rλ0 , λ1 s P P1 , let C rλ0 ,λ1 s Ă S be the proper transform of Crλ0 ,λ1 s , i.e. the closure of p0qXsP´1 p0q Ă P2´sP´1 p0qXsP´1 p0q “ S´sP´1 p0qXsP´1 p0q. Crλ0 ,λ1 s´sP´1 0 1 0 1 0 1 p0qXsP´1 p0q in P2 is replaced by a Since each of the 9 points of sP´1 0 1 copy of P1 in S, the Euler characteristic of S is given by ` ˘ (12.4) χpSq “ χpP2 q`9 χpP1 q´χpptq “ 12 . Since every pair of distinct curves Crλ0 ,λ1 s intersects only at p0qXsP´1 p0q and transversely, their proper transforms C rλ0 ,λ1 s Ă S sP´1 0 1

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are disjoint. Thus, the map π : S ÝÑ P1 ,

πppq “ rλ0 , λ1 s @ p P C rλ0 ,λ1 s , rλ0 , λ1 s P P1 ,

is well defined and holomorphic. A generic fiber of this map is a 2-torus pS 1 q2 . The N3 remaining fibers are obtained from a generic fiber by collapsing a circle S 1 to a point. Thus, ˘ ˘ ` χpSq “ χppS 1 q2 χpP1 q`Nd χpptq´χpS 1 q “ Nd . Comparing this with (12.4), we conclude that Nd “ 12. An alternative classical-style computation of the number N3 appears in [50, Section 2.3]. A generic rational degree 4 curve C Ă P2 has g4 “ 3 nodes. This makes a direct computation of the number N4 much harder, but this was accomplished in the middle of the nineteenth century; see [48, p. 378]. A classical-style modern-terminology computation of N4 appears in [50, Section 3]. The higher-degree numbers Nd were first obtained from Kontsevich’s recursion, ¸ˆ ˜ ˙ ÿ 1 pd1 ´d2 q2 3d´2 d1 d2 Nd1 Nd2 d1 d2 ´ 2 Nd “ 6pd´1q d `d “d 3d ´ 2 3d1 ´1 1

@ d ě 2,

2

N1 “ 1,

(12.5)

announced in [27] and established in [31, 37]; see also [50, Section 4]. 12.2

Real Low-Degree Curves in P2

For n P Z` , let τn : Pn ÝÑ Pn ,

˘ “ ` ‰ τn rZ0 , Z1 , . . . , Zn s “ Z0 , Z1 , . . . , Zn , (12.6)

be the standard conjugation on the complex projective space Pn . The fixed locus of τn is the real projective space RPn Ă Pn . The conjugation τ1 is equivalent to the conjugation τ in (7.27) via a linear change of coordinates on P1 . We call a subset C Ă Pn real if τn pCq “ C. We call a tuple pC1 , . . . , Ck q of subsets of Pn real if

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the tuple pτn pC1 q, . . . , τn pCk qq is a permutation of pC1 , . . . , Ck q. For m P Z` and d P Zě0 , we denote by RHPdm Ă HPdm

and

` ˘ ` ˘ RP RHPdm Ă P HPdm

the real vector subspace of polynomials with real coefficients and its real projectivization, respectively. If P P HPdm , the subspace s´1 P p0q Ă ´1 ´1 d m´1 P is real if and only if sP p0q “ sQ p0q for some Q P RHPm . Let C Ă P2 be a real reduced curve. A (simple) node p P C can be of 3 types: (E) p P RP2 is an isolated point of RC ” C XRP2 , (H) p P RP2 is a non-isolated point of RC, (C) p P P2 ´RP2 is a non-real point of C. The nodes of type C come in pairs tp, τ2 ppqu. A pair pp1 , p2 q of distinct points p1 and p2 of P2 is real if either p1 , p2 P RP2 or τ2 pp1 q “ p2 . In either case, the unique real line passing through these two points is real. A tuple pp1 , . . . , p5 q of five distinct points of P2 is real if it consists k points of RP2 and l conjugate pairs of points of P2 ´ RP2 with k ` 2l “ 5. If these points are in general position, the unique conic passing through these five points is real in all three cases. Since the degree 1 and 2 curves being counted τ2 of real rational are rational and smooth, Welschinger’s counts Nd,l 2 degree 1 and 2 curves in P , as defined in Section 12.3, are thus τ2 τ2 , N1,1 “ 1, N1,0

and

τ2 τ2 τ2 N2,0 , N2,1 , N2,2 “ 1.

(12.7)

A tuple pp1 , . . . , p8 q of eight distinct points of P2 is real if it consists k points of RP2 and l conjugate pairs of points of P2 ´RP2 with k ` 2l “ 8. If these points are in general position, they determine a family of real irreducible cubics parametrized by RPpRHP33 qp1 ...p8 “ RPpRHP33 qXPpHP33 qp1 ...p8 « RP1 , with PpHP33 qp1 ...p8 Ă PpHP33 q as in (12.3). The 9-point subset p0q X sP´1 p0q Ă P2 Bp ” sP´1 0 1

(12.8)

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shared by every pair of distinct cubics in the pencil PpHP33 qp1 ...p8 is real in this case. Its intersection with RP2 thus consists of the k points pi P RP2 and another point p9 . Since there is a unique cubic passing through 9 general points and the tuple p is real, for every p P RP2 there exists rP s P RPpRHP33 q such that p P sP´1 p0q. Let S be the complex blowup of P2 at Bp . For each P P PpHP33 qp1 ...p8 , let sP´1 p0q Ă S be the proper transform of sP´1 p0q. The proper transform RSk Ă S of RP2 is the real blowup of RP2 at the k`1 points of RP2XBp . Since each of these points in RP2 is replaced by a copy of RP1 in RSk , ` ˘ χpRSk q “ χpRP2 q`pk`1q χpRP1 q´χpptq “ ´k . (12.9) By the last sentence of the previous paragraph and the same reasoning as below (12.4), the map π : RSk ÝÑ RPpRHP33 qp1 ...p8 ,

πppq “ rP s @ p P sP´1 p0q,

rP s P RPpRHP33 qp1 ...p8 ,

(12.10)

is well defined and smooth. All but finitely many (at most N3 “ 12) fibers of this map are intersections of smooth real cubics with RP2 , i.e. either a single copy of S 1 or two copies of S 1 ; these fibers do not contribute to χpRSk q. The remaining fibers are the intersections of real cubics with a single node, which must be of type E or H, with RP2 , i.e. either a single copy of S 1 and an isolated point or a wedge of two copies of S 1 . These fibers have Euler characteristics `1 and ´1, respectively. Thus,  (  ( χpRSk q “ # E-nodal cubics ´ # H-nodal cubics . Comparing this with (12.9), we conclude that  (  ( # H-nodal cubics ´ # E-nodal cubics “ k . All of these singular cubics are rational. By the previous paragraph, the number τ2 “ 8 ´ 2l N3,l

(12.11)

of real rational cubics passing through k real points and l conjugate pairs of points in general position in P2 with k`2l “ 8 and counted

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323

with sign based on the parity of the number of E-type nodes does not depend on the position of the points. As shown in [44], this invariance property extends to counts of real rational curves in other real symplectic fourfolds; see Theorem 12.1. On the other hand, as noted in [44, Section 2.1], the analogous signed count in higher genera can depend on the choice of the points. For example, if P0 , P1 P HP43 are polynomials generating the pencil of degree 4 curves passing through a generic tuple p of dim PpHP43 q ´ 1 “ 13 points, then the intersection Bp in (12.8) contains 42´13 other points. If p is a real tuple, then either 1 or 3 of these points lie in RP2 and (12.9) gives ´χpRSk q P tk, k`2u. The projection (12.10) is still well defined; the Euler characteristics of its smooth, E-nodal, and H-nodal fibers are still 0, 1, and ´1, respectively (but these fibers may contain one or two more copies of S 1 ). Thus,  (  ( # H-nodal degree 4 curves ´ # E-nodal degree 4 curves  ( P k, k`2 . Both values for this count of real genus 2 degree 4 curves in P2 are possible; see the end of [44, Section 2.1]. 12.3

Welschinger’s Invariants in Dimension 4

A real symplectic manifold is a triple pX, ω, φq consisting of a symplectic manifold pX, ωq and a diffeomorphism φ of X with itself so that φ2 “ idX and φ˚ ω “ ´ω. By [33, Lemma 3], the fixed locus X φ of φ is then a Lagrangian submanifold of pX, ωq. For a real symplectic manifold pX, ω, φq, define  ( H2φ pX; Zq “ B P H2 pX; Zq : φ˚ pBq “ ´B . For a connected component Y of X φ , let ` ˘ BY ;Z2 : H2 X, Y ; Z ÝÑ H1 pY ; Zq ÝÑ H1 pY ; Z2 q be the composition of the boundary homomorphism of the homology relative exact sequence for the pair pX, Y q with the mod 2 reduction

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324

of the coefficients. We denote by dX;Y : H2 pX, Y ; Zq ÝÑ H2φ pX; Zq the natural homomorphism which glues each map f : pΣ, BΣq ÝÑ pX, Y q from an oriented bordered surface with the map φ ˝ f from pΣ, BΣq with the opposite orientation; see [8, Section 1.1]. Define r φ pX, Y q “ H 2

` ˘ ( dX;Y pB 1 q, BY ;Z2 pB 1 q : B 1 P H2 pX, Y ; Zq

Ă H2φ pX; Zq‘H1 pY ; Z2 q.

(12.12)

By [4, Proposition 4.1], D @ D @ w2 pXq, B “ w1 pY q, b P Z2

r φ pX, Y q. @ pB, bq P H 2

(12.13)

The homomorphism pdX;Y , BY ;Z2 q on H2 pX, Y ; Zq descends to a surjective homomorphism L ( φ ΦX,Y : H 2 pX, Y ; Zq ” H2 pX, Y ; Zq B 1 `φ˚ B 1 : B 1 P H2 pX, Y ; Zq r φ pX, Y q. ÝÑ H 2

(12.14)

If pΣ, σq is a symmetric surface as in Section 7.1 with separating fixed locus Σσ , Σb is a half-surface of pΣ, σq, and u : Σ ÝÑ X is a continuous map such that u˝σ “ φ˝u and upΣσ q Ă Y , then ΦX,Y

`“

‰˘ ` ˘ r φ pX, Y q. u˚ prΣb , Σσ sZ q “ u˚ prΣsZ q, u˚ prΣσ sZ2 q P H 2 (12.15) φ

Furthermore, the element ru˚ prΣb , Σσ sZ qs P H 2 pX, Y ; Zq is independent of the choice of the half-surface Σb . We denote by Jω the space of ω-compatible (or -tamed) almost complex structures on X and by Jωφ Ă Jω the subspace of almost complex structures J such that φ˚ J “ ´J. Let c1 pX, ωq ” c1 pT X, Jq P H 2 pX; Zq

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325

be the first Chern class of T X with respect to some J P Jω ; it is independent of such a choice. Since φ˚ ω “ ´ω, φ˚ c1 pX, ωq “ ´c1 pX, ωq.

(12.16)

Define ω : H2 pX; Zq ÝÑ Z,

@ D 1 ω pBq “ c1 pX, ωq, B ` dimR X ´3. 2 (12.17)

r “ ω pBq. r ” pB, bq P H r φ pX, Y q, we define ω pBq For B 2 Let J P Jω . A smooth map u : Σ ÝÑ X from a Riemann surface pΣ, jΣ q is J-holomorphic if du˝jΣ “ J ˝du : T Σ ÝÑ u˚ T X . Such a map is called simple if u´1 pu´1 pzqq “ tzu for all but countably many points z P Σ. For B P H2 pX; Zq, a subset C Ă X is a reduced J-holomorphic degree B curve if there exist a compact Riemann surface Σ and a simple J-holomorphic map ` ˘ (12.18) u : Σ ÝÑ X s.t. C “ upΣq, u˚ rΣsZ “ B. As in the setting of Section 12.1, the map u, called the normalization of C, is unique up to the obvious equivalence; see the proof of [32, Proposition 2.5.1]. If Σ is connected, then C is called irreducible and the (geometric) genus gpCq of C is again defined to be the genus of Σ. Such a curve is called rational if it is of genus 0, i.e. if Σ “ P1 in (12.18). If J P Jωφ , a degree B curve C Ă X is real if φpCq “ C. If so, then B P H2φ pX; Zq and there exist r ” pB, bq P H r φ pX, Y q B 2

(12.19)

and a unique involution σ on the Riemann surface Σ in (12.18) so that ` ˘ ` ˘ (12.20) u˝σ “ φ˝u and u˚ rΣσ sZ2 “ b P H1 Y ; Z2 . r In such a case, we say that the curve C is of degree B.

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Let J P Jω and C Ă X be a J-holomorphic curve. A point p P C is a simple node if there exists a simple map u as in (12.18) such that u´1 ppq consists of two distinct points z1 , z2 P Σ with dz1 u, dz2 u ‰ 0,

Im dz1 u ‰ Im dz2 u Ă Tp X .

If J P Jωφ and C Ă X is real, a (simple) node p of C can be of three types: E, H, and C as in Section 12.2 with pP2 , RP2 q replaced by pX, X φ q. We denote by δE pCq and δH pCq the numbers of simple nodes of C of types E and H, respectively. The nodes of type C come in pairs tp, φppqu; we denote the number of such pairs by δC pCq. Suppose pX, ω, φq is a compact connected real symplectic fourfold and Y Ă X φ is a connected component. For B1 , B2 P H2 pX; Zq, we denote by @ D B1 ¨B2 ” PDX pB1 q, B2 P Z the homology intersection number of B1 and B2 . For B P H2 pX; Zq, let B 2 ” B ¨B P Z be the homology self-intersection number of B. By the definition of dX;Y , B1 ¨B2 `2Z “ b1 ¨b2 P Z2

r φ pX, Y q. @pB1 , b1 q, pB2 , b2 q P H 2 (12.21)

If J P Jωφ , C Ă X is a real irreducible J-holomorphic curve, and all singular points of C are (simple) nodes, then gpCq ` δE pCq ` δH pCq ` 2δC pCq “ 1`

˘ 1` 2 B ´xc1 pX, ωq, By . 2 (12.22)

Suppose in addition that B P H2φ pX; Zq and l P Zě0 are such that k ” ω pBq´2l P Zě0 .

(12.23)

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327

For a generic J P Jωφ , the set Mφp pB; Jq of real irreducible rational J-holomorphic degree B curves C Ă X passing through a tuple ˘ ` R ` ` k φ l (12.24) p ” ppR 1 , . . . , pk q, pp1 , . . . , pl q P Y ˆpX ´X q of k points in Y and l points in X ´ X φ in general position is then finite. Furthermore, every such curve is the image of an immersion u as in (12.18) with Σ “ P1 and transverse intersection points. By (12.23), either k ą 0 or xc1 pX, ωq, By is odd. Along with (12.15) and (12.13), this implies that for every C P Mφp pB; Jq there exists a J-holomorphic immersion u as in (12.18) with Σ “ P1 satisfying the first condition in (12.20) with σ “ τ . If k ą 0, the set r Jq of real rational Mφp pB; Jq thus decomposes into the subsets Mp pB; r irreducible J-holomorphic degree B curves with r “ pB, bq P H r φ pX, Y q Ă H φ pX; Zq‘H1 pY ; Z2 q ; B 2 2 if k “ 0, the above decomposition needs to be taken over all connected components of X φ . Let ÿ p´1qδE pCq . (12.25) N φr pY q ” B,l

r CPMp pB;Jq

Theorem 12.1 ([44, Theorem 0.1]). Suppose pX, ω, φq is a compact connected real symplectic fourfold and Y Ă X φ is a connected rPH r φ pX, Y q is as in (12.19) component of the fixed locus of φ. If B 2 and l, k P Zě0 are as in (12.23), the number N φr pY q in (12.25) is independent of generic choices of

J P Jωφ

and

B,l p P Y k ˆX l .

r Jq splits into subsets As indicated by (12.15), each set Mp pB; ´1 r The reasoning for Theorem 12.1 Mp prB 1 s; Jq with rB 1 s P ΦX,Y pBq. implies that each number ÿ φ p´1qδE pCq NrB 1 s,l pY q ” CPMp prB 1 s;Jq

is also independent of generic choices of J P Jωφ and p P Y k ˆX l . Any two tuples p of distinct points as in (12.24) differ by a symplectomorphism ψ of pX, ωq such that ψ˝φ “ φ˝ψ. For the purposes

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of Theorem 12.1, it thus is sufficient to establish the independence of the number N φr pY q in (12.25) of a generic choice of J P Jωφ . B,l

Let τ, η : P1 ÝÑ P1 be the involutions on P1 as in (7.27). If pX, ω, φq is a real symplectic manifold, let  H2φ pX; Zqσω ” u˚ prP1 sZ q : u P CpP1 ; Xq, u˝σ D ( @ with σ “ τ, η, “ φ˝u, ω, u˚ prP1 sZ q ą 0 φ φ φ τ η H2φ pX; ZqRS ω ” H2 pX; Zqω YH2 pX; Zqω Ă H2 pX; Zq

be the subsets of ω-effective σ-spherical and real spherical classes. For the purposes of establishing Theorem 12.1, it can be assumed that r Jq is B P H2φ pX; Zqτω and xc1 pX, ωq, By ě 1; otherwise, the set Mp pB; φ empty for a generic choice of J P Jω . The geometric proof of this theorem in [44] and in Section 12.5 is carried out under the assumption that B ‰ B1 `mB2 @ m P Z` , B1 , B2 P H2φ pX; ZqRS ω with D @ (12.26) m ě 2, c1 pX, ωq, B2 “ 0. As pointed out in [47, Remark 2.12], this condition had been overlooked in [44]; see Remark 12.6 for more details. The condition (12.26) is automatically satisfied if D @ @ B P H2φ pX; ZqRS c1 pX, ωq, B ‰ 0 ω . 12.4

Moduli Spaces of Real Maps

Suppose pX, ω, φq is a real symplectic manifold, Y Ă X φ is a conrPH r φ pX, Y q is as in (12.19), and J P Jωφ . Let nected component, B 2 pΣ, σ, jq be a (possibly nodal) symmetric Riemann surface with norr σ r map from pΣ, σ, jq malization pΣ, rq. A real J-holomorphic degree B is a continuous map u : Σ ÝÑ X such that ` ˘ ` ˘ u˝σ “ φ˝u, u˚ rΣsZ “ B, u˚ rΣσ sZ2 “ b, r ÝÑ X induced by u is J-holomorphic. Such a and the map u r: Σ map u is called simple if u r is simple.

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329

r Jq the moduli space of equivLet k, l P Zě0 . We denote by Mk,l pB; r J-holomorphic maps from pP1 , τ q with alence classes of real degree B k real marked points and l conjugate pairs of marked points modulo reparametrizations by the subgroup PSLτ2 C Ă PSL2 C of automorphisms of pP1 , τ q. Let r Jq Ą Mk,l pB; r Jq Mk,l pB; r Jq are be its stable-map compactification. The elements of Mk,l pB; the equivalence classes of tuples ˘ ` u “ u : Σ ÝÑ X, σ, pxi qiPrks , pzi` , zi´ qiPrls

(12.27)

such that ` ˘ C ” Σ, σ, pxi qiPrks , pzi` , zi´ qiPrls

(12.28)

is a marked symmetric Riemann surface, which is a tree of spheres, with k real marked points and l conjugate pairs of marked points r map from pΣ, σq. A marked and u is a real J-holomorphic degree B map u as in (12.27) is called simple if u is simple. We denote by r Jq Ă Mk,l pB; r Jq M˚k,l pB;

and

˚ r r Jq (12.29) Mk,l pB; Jq Ă Mk,l pB;

the subspaces of the equivalence classes of simple maps. For a real J-holomorphic map u from pΣ, σ, jq, u˚ pT X, dφq is a real bundle pair over pΣ, σq. With the notation as in (7.4), let φ : Γpuq ” ΓpΣ; u˚ T Xqu DJ;u

˚ dφ

˘u˚ dφ 0,1 ` ÝÑ Γ0,1 Σ; u˚ pT X, Jq J puq ” Γj (12.30)

be the linearization of the B¯J -operator on the space of real maps from pΣ, σq. This is a real CR-operator on the real bundle pair u˚ pT X, dφq over pΣ, σq in the terminology of Section 7.1 and is the restriction of the linearization DJ;u of the B¯J -operator, ˘ ` ˘ 1` Σ; f ˚ pT X, Jq , B¯J f “ df `J ˝df ˝j P Γ0,1 j 2

(12.31)

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on the space of all smooth maps from Σ to X. If ∇ is a connection in pT X, Jq compatible with a J-invariant Riemannian metric on X, ˘ 1` ˘ 1` u ∇ ξ `J∇uj ξ ´ T∇ pdu, ξq`JT∇ pdu˝j, ξq 2 2 ˘ 0,1 ` ˚ (12.32) P Γj Σ; u pT X, Jq @ ξ P ΓpΣ; u˚ T Xq,

DJ;u ξ “

where ∇u is the pullback connection in u˚ pT X, Jq and T∇ is the torsion of ∇; see [52, Proposition 3.13]. For a real marked J-holomorphic map u as in (12.27), let  φ : Γpuq ” ξ P Γpuq : ξpxi q P Im dxi u @i P rks, DJ;u ( 0,1 ξpzi` q P Im dz ` u @i P rls ÝÑ Γ0,1 J puq ” ΓJ puq i

(12.33)

φ in (12.30). be the restriction of DJ;u Suppose dimR X “ 4, J P Jω , and C Ă X is a J-holomorphic curve. Let u : Σ ÝÑ C be the normalization of C. A point p P C is a cusp of C if u´1 ppq consists of a single point z0 P Σ with

D ˇˇ duˇ ‰ 0 P Tp X, dz z0 " * " 2 ˇ * D ˇˇ D ˇ duˇ duˇ Ć Im Ă Tp X; Im dz 2 dz z0 z0

dz0 u “ 0,

by the first condition above, the second and third conditions are independent of the choice of the connection in pT X, Jq used to define the covariant derivative. A point p P C is a tacnode of C if u´1 ppq consists of two distinct points z1 , z2 P Σ with Im dz1 u “ Im dz2 u Ă Tp X, dz1 u, dz2 u ‰ 0, ˇ D ˇˇ D ˇ Im duˇ `Im dz1 u ‰ Im duˇ `Im dz2 u Ă Tp X ; dz z1 dz z2 the last condition above is again independent of the choice of the connection in pT X, Jq used to define the covariant derivative due to the preceding conditions. A point p P C is a (simple) triple point of C

Counts of Rational Curves on Surfaces

331

if u´1 ppq consists of three distinct points z1 , z2 , z3 P Σ with dz1 u, dz2 u, dz3 u ‰ 0,

Im dzi u ‰ Im dzj u Ă Tp X @ i, j P r3s, i ‰ j.

Example 12.2. Let φ be the standard component-wise conjugation on C2 . Its fixed locus is R2 Ă C2 . For each t P R, the holomorphic map ut : C ÝÑ C2 ,

` ˘ ut pzq “ z 2 ´2t, z 3 ´3tz ,

is real with respect to the standard conjugation on C. The only singular point of the real J-holomorphic curve u0 pCq Ă C2 is a cusp at p0, 0q P R2 ; it is a non-isolated point of u0 pCqXR2 . The image of du0 is contained in the complex line subbundle  ( Tu0 ” pz, p2, 3zqcq : z, c P C Ă CˆC2 “ u˚0 T C2 ÝÑ C ; the homomorphism du0 maps T C isomorphically onto this subbundle outside of z “ 0 (where du0 vanishes). Furthermore, ˇ ` ˘ d ˇˇ ut ˇ P Γ C; Tu0 . dt t“0 For t P R˚ , the only singular point ? of the real J-holomorphic curve ut pCq Ă C2 is the node at ut p˘ 3tq “ pt, 0q. This node is of type E if t P R´ and of type H if t P R` . This family of curves is illustrated by the top row in Figure 12.1.

Figure 12.1. The real loci of a real curve with a real cusp, tacnode, or triple point and of their real deformations; the dashed lines indicate pairs of conjugate branches of the curve or of its deformations at the singular point.

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Example 12.3. Let φ be the standard component-wise conjugation on C2 , ` ˘ σE : ΣE ÝÑ ΣE , σE pr, zq “ 3´r, z , ` ˘ ΣE , ΣH “ t1, 2uˆC, σH : ΣH ÝÑ ΣH , σH pr, zq “ r, z . For each t P R, the holomorphic maps ` ˘ uE;t pr, zq “ z, p´1qr ipz 2 ´tq , ` ˘ ÝÑ C2 , uH;t pr, zq “ z, p´1qr pz 2 ´tq ,

uE;t : ΣE ÝÑ C2 , uH;t : ΣH

are real with respect to the involutions σE on ΣE and σH on ΣH , respectively. The only singular point of the real J-holomorphic curves uE;0 pΣE q, uH;0 pΣH q Ă C2 is a tacnode at p0, 0q P R2 ; it is an isolated point of uE;0 pΣE qXR2 and a non-isolated point of uH;0 pΣH qXR2 . For t P R˚ , the only singular points of the real J-holomorphic curves uE;tpΣE q, uH;t pΣH q Ă C2 ? ? are the two nodes at uE;t pr, ˘ tq and uH;t pr, ˘ tq, respectively. If t P R´ , the two nodes form a conjugate pair of C-nodes. If t P R` , both nodes of uE;t pΣE q (resp. uH;t pΣH q) are of type E (resp. H). These two families of curves are illustrated by the middle row in Figure 12.1. Example 12.4. Let φ be the standard component-wise conjugation on C2 , ΣE , ΣH “ t1, 2, 3uˆC, σE : ΣE ÝÑ ΣE , σH : ΣH ÝÑ ΣH , # ` ˘ pr, zq, if r “ 1; ˘ σH pr, zq “ r, z . σE pr, zq “ ` 5´r, z , if r “ 2, 3; For each t P R, the holomorphic maps uE;t : ΣE ÝÑ C2 ,

uH;t : ΣH ÝÑ C2 ,

Counts of Rational Curves on Surfaces

# uE;tpr, zq “ # uH;t pr, zq “

333

pz, 0q, if r “ 1; ` ˘ r z, p´1q iz´t , if r “ 2, 3; pz, 0q, if r “ 1; ` ˘ r z, p´1q z´t , if r “ 2, 3;

are real with respect to the involutions σE on ΣE and σH on ΣH , respectively. The only singular point of the real J-holomorphic curves uE;0 pΣE q, uH;0 pΣH q Ă C2 is a triple point at p0, 0q P R2 ; it is an isolated point of the intersections of two of the branches of uE;0 pΣE q with R2 and a non-isolated point of the intersection of each of the three branches of uH;0 pΣH q with R2 . For t P R˚ , the only singular points of the real J-holomorphic curve uE;tpΣE q Ă C2 (resp. uH;t pΣH q Ă C2 ) are the conjugate pair of C-nodes at uE;tp1, ˘itq and an E-node at uE;tp2, 0q (resp. three H-nodes at uH;t p1, ˘tq, uH;t p2, 0q). These two families of curves are illustrated by the bottom row in Figure 12.1. 12.5

Proof of Theorem 12.1

Suppose pX, ω, φq is a real symplectic manifold, Y Ă X φ is a conr P H r φ pX, Y q is as in (12.19), J P Jωφ , and nected component, B 2 k, l P Zě0 . For a tuple p of points as in (12.24), define ‰ “ r Jq : Σ, σ, pxi qiPrks , pzi` , zi´ qiPrls P Mk,l pB; ( ` ` upxi q “ pR i @i P rks, upzi q “ pi @i P rls , r Jqp “ M˚ pB; r JqXMk,l pB; r Jqp , M˚k,l pB; k,l r Jqp “ Mk,l pB;

r Jqp “ Mk,l pB; r JqXMk,l pB; r Jqp , Mk,l pB; ˚ r ˚ r r Jqp . Mk,l pB; Jqp “ Mk,l pB; JqXMk,l pB; r Jqp pp, Jq-regular if the operator D φ We call rus P Mk,l pB; J;u in (12.33) is onto.

Spin/Pin-Structures & Real Enumerative Geometry

334

If J ” pJt qtPr0,1s is a path in Jωφ , let ď r Jqp “ r Jt qp , ttuˆM˚k,l pB; M˚k,l pB; tPr0,1s

r Jqp “ Mk,l pB;

ď

r Jt qp , ttuˆMk,l pB;

tPr0,1s ˚ r Mk,l pB; Jqp “

ď

˚ r ttuˆMk,l pB; Jt qp ,

tPr0,1s

r Jqp “ Mk,l pB;

ď

r Jt qp . ttuˆMk,l pB;

tPr0,1s

If the path J is C 1 -smooth with respect to the C m -topology on Jωφ for some m P Z` , let ` ˘ d J9t ” Jt P Γ X; HomC ppT X, Jq, pT X, ´Jqq dt be the derivative of J with respect to t. For t P r0, 1s and a real marked Jt -holomorphic map u as in (12.27), define φ : RˆΓpuq ÝÑ Γ0,1 DJ;t,u Jt puq,

τ φ pτ, ξq “ DJφt ;u pξq` J9t ˝du˝j. (12.34) DJ;t,u 2 r Jqp pp, Jq-regular if this operator is onto. We call pt, rusq P Mk,l pB; r PH r φ pX, Y q, and l, k P Zě0 Suppose now that pX, ω, φq, Y Ă X φ , B 2 are as in Theorem 12.1 and p P Y k ˆX l is a tuple of distinct points. If J0 , J1 P Jωφ are generic, then r Jt qp “ M˚ pB; r Jt qp Mk,l pB; k,l

(12.35)

r Jt qp with t “ 0, 1 are pp, Jt qfor t “ 0, 1, all elements of M˚k,l pB; regular, and the only singularities of the image upP1 q Ă X of each r Jt qp are simple nodes. In particular, the counts element rus P M˚k,l pB; ÿ ÿ 1 p´1qδE pCq “ p´1qδE pupP qq N φr pY ; Jt q ” B,l

r tq CPMp pB;J

r t qp rusPMk,l pB;J

(12.36) with t “ 0, 1 are well defined. We show in the following that they are the same, thus establishing Theorem 12.1.

Counts of Rational Curves on Surfaces

335

Let J ” pJt qtPr0,1s be a generic path in Jωφ . By standard transversality arguments, such as those in [32, Section 3.2], every element pt, rusq ˚ r ˚ r of Mk,l pB; Jqp is pp, Jq-regular. Thus, the space Mk,l pB; Jqp is a onedimensional manifold with ˚

r Jqp “ M˚ pB; r J1 qp \M˚ pB; r J0 qp B Mk,l pB; k,l k,l and the projection ˚ r πM : Mk,l pB; Jqp ÝÑ r0, 1s,

` ˘ πM t, rus “ t,

(12.37)

is smooth. Under the standard identification Tt R « R, the homomorphism ‰ τ“ 9 Jt ˝du˝j ÝÑ rτ s, (12.38) cok DJφt ;u ÝÑ cok dt,u πM , 2 ˚ r Jqp with u as is well defined for every element pt, rusq of Mk,l pB; 1 in (12.27); it is an isomorphism if Σ “ P . By transversality arguments as in [32, Section 3.2] and [44, Section 2.2], there exists a finite subset I0 Ă p0, 1q so that (12.35) holds r Jt qp are pp, Jt q-regular, for all t P r0, 1s´ I0 , all elements of M˚k,l pB; and the only singularities of the image upP1 q Ă X of each element rus of this space are simple nodes. In particular, the counts (12.36) with r Jt qp are t P r0, 1s´I0 are well defined. Since all elements rus of Mk,l pB; pp, Jt q-regular if t R I0 , these counts are constant on the connected components of r0, 1s´I0 . Under the assumption (12.26), ˚ r r Jqp . Mk,l pB; Jqp “ Mk,l pB;

(12.39)

The space on the right-hand side above is compact (for any choice of a C 1 -path J). In this case, it thus remains to verify that the counts (12.36) are the same on the two sides of each point of I0 . The crucial proposition in the following is specific to the genus 0 setting of Theorem 12.1; see Remark 12.8. Proposition 12.5 ([44, Lemma 2.13]). Suppose pX, ω, φq, Y Ă r P H r φ pX, Y q, and l, k P Zě0 are as in Theorem 12.1 and X φ, B 2 p P Y k ˆX l is a tuple of distinct points. r Jqp . With the notation as in (12.27), p1q Let J P Jωφ and rus P Mk,l pB; rus is not pp, Jq-regular if and only if u is not an immersion.

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Spin/Pin-Structures & Real Enumerative Geometry

p2q Let J0 , J1 P Jωφ be generic and J ” pJt qtPr0,1s be a generic path ˚ r in Jωφ . With the notation as in (12.27), pt, rusq P Mk,l pB; Jqp is a critical point of the projection πM in (12.37) if and only if Σ “ P1 and u is not an immersion. Furthermore, all critical points of πM are non-degenerate. For the dimensional reasons detailed in [44, Section 2.2], an ele˚ r ment pt, rusq P Mk,l pB; Jqp with t P I0 and u as in (12.27) can be only of the following types: (W0) Σ “ P1 and the only singularities of upΣq are simple nodes. (W1) Σ “ P1 , one of the singularities of upP1 q is a tacnode lying in X φ , and the remaining singularities of upP1 q are simple nodes. (W2) Σ “ P1 , one of the singularities of upP1 q is a triple point lying in X φ , and the remaining singularities of upP1 q are simple nodes. (W3) Σ is a wedge of two copies of P1 , with the node mapped by u to X φ , and the only singularities of upΣq are simple nodes. (W4) Σ “ P1 , one of the singularities of upP1 q is a cusp lying in 1 X φ ´tpR i uiPrks , and the remaining singularities of upP q are simple nodes. ˚ r Jqp of types (W1)–(W4) are indicated The special points of Mk,l pB; in Figure 12.2. The Hessian of πM in (12.37) at a critical point pt, rusq as in (W4) is explicitly given by (12.60).

Figure 12.2. The special points of the projection (12.37): t P r0, 1s and pt, rusq P ˚ r Jqp with t P I0 and u as in (W1)–(W4). The signs indicate the value of Mk,l pB; ˚ δE pCq r Jt qp with rus as p´1q for the curve C Ă X determined by each rus P Mk,l pB; in (W0).

Counts of Rational Curves on Surfaces

337

By Proposition 12.5(2), the restriction of πM to a neighborhood ˚ r Jqp is a diffeomorphism onto a neighborhood of pt, rusq in Mk,l pB; of t in r0, 1s in the cases (W0)–(W3), i.e. for every τ P p0, 1q´I0 sufr Jτ qp close ficiently close to t, there exists a unique ruτ s P M˚k,l pB; to rut s in the Gromov topology. In the case (W0), the nodes of the image uτ pP1 q of the map component uτ of uτ correspond to the nodes of ut pP1 q and are of the same type as the nodes of ut pP1 q ” upP1 q. In the case (W3), the simple nodes of ut pP1 q other than the image of the node of Σ correspond to nearby nodes of uτ pP1 q of the same type. In the cases (W1) and (W2), the simple nodes of ut pP1 q correspond to nearby nodes of uτ pP1 q of the same type. In addition, uτ pP1 q has two (resp. three) simple nodes near the tacnode (resp. triple point) of ut pP1 q. As illustrated by Examples 12.3 and 12.4, the parity of the number of E-nodes among these additional nodes does not depend on whether τ ă t or τ ą t. Thus, in the cases (W0)–(W3), there exists ˚ r ´1 Jqp so that W XπM pτ q cona neighborhood W of pt, rusq in Mk,l pB; sists of a single element ruτ s for every τ P r0, 1s´I0 sufficiently close 1 to t and p´1qδC puτ pP qq does not depend on the choice of such τ . ˚ r Jqp is as in (W4). By ProposiSuppose pt, rusq P Mk,l pB; tion 12.5(2) again, there exist δ P R` and a neighborhood W of pt, rusq ˚ r in Mk,l pB; Jqp so that pt´δ, tq, pt, t`δq Ă p0, 1q´I0 and either ˇ ˇ ˇW Xπ ´1 pτ qˇ “ 2 @ τ P pt´δ, tq, M

´1 pτ q “ H @ τ P pt, t`δq, or W XπM ˇ ˇ ˇW Xπ ´1 pτ qˇ “ 2 @ τ P pt, t`δq, M ´1 pτ q “ H W XπM

@ τ P pt´δ, tq.

Along with the description of ker DJφt ;u in the proof of Proposition 12.5(1) in Section 12.6 and Example 12.2, this gives ÿ

p´1qδE pupP

1 qq

“0

@ τ P pt´δ, tq, pt, t`δq.

´1 rusPW XπM pτ q

Combining this with the previous paragraph, we conclude that the counts (12.36) are the same on the two sides of each point of I0 .

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Figure 12.3. Real multiply covered Jt -holomorphic maps potentially appearing on the right-hand side of (12.39) in the absence of the restriction (12.26). The degrees of the maps on the irreducible components of the domains are shown next to the corresponding components. The double-headed arrows labeled by σ indicate the involutions on the entire domains of the maps. The smaller doubleheaded arrows indicate the involutions on the real images of the corresponding irreducible components of the domain.

Remark 12.6. The restriction (12.26) necessary for the geometric arguments in [44] behind Theorem 12.1 is overlooked in [44]. Without this restriction, the right-hand side in (12.39) could contain multiply covered Jt -holomorphic maps represented by the two diagrams in Figure 12.3. In light of the situation in the projective setting of [1], the end of the proof of Theorem 0.1 and Remark 2.12 in [47] suggest a potential geometric approach to showing that multiply covered Jt -holomorphic maps as in the first diagram of Figure 12.3 do not appear as limits of simple maps in a one-parameter family J of almost complex structures as in Theorem 12.1. The potential appearance of multiply covered Jt -holomorphic maps as in the second diagram of Figure 12.3 is not mentioned in [47]. On the other hand, these maps create no difficulties in the virtual class approach of [39, Section 7].

12.6

Proof of Proposition 12.5

For a real bundle pair pV, ϕq over a topological space X with an involution φ, we define the dual real bundle pair pV ˚ , ϕ˚ q over pX, φq by ϕ˚ : V ˚ ÝÑ V ˚ ,



( ` ˘ ϕ˚ pαq pvq “ α ϕpvq @ v P V.

If in addition X is a smooth manifold, σ is a smooth involution, J is an almost complex structure on X such that φ˚ J “ ´J, and p, q P Zě0 , we define the real bundle pair pT ˚ X p,q bC V, ϕp,q q of V -valued

Counts of Rational Curves on Surfaces

339

pp, qq-forms over pX, φq by ϕp,q : T ˚ X p,q bC V ÝÑ T ˚ X p,q bC V,  p,q ( ` ` ˘˘ ϕ pαq pxq 9 “ ϕ α dφpxq 9 @ x9 P T X p,q . 1

1

For x P X, αbv P Tx˚ X p,q bC Vx , and β bu P Tx˚ X p ,q bC Vx˚ , let ` ˘ ` ˘ 1 1 αbv ^ β bu “ upvqα^β P Tx˚ X p`p ,q`q . For a real CR-operator D on a real bundle pair pV, ϕq over a smooth symmetric Riemann surface pΣ, σ, jq as in (7.5), we denote by ` ˘pϕ˚ q1,0 D ˚ : Γ Σ; T ˚ Σ1,0 bC V ˚ ´  ( ” ψ P ΓpΣ; T ˚ Σ1,0 bC V ˚ q : ψ˝σ “ ´pϕ˚ q1,0 pψq ˘pϕ˚ q1,1 ` ÝÑ Γ Σ; T ˚ Σ1,1 bC V ˚ ´  ` ( ˘ ” ω P Γ Σ; T ˚ Σ1,1 bC V ˚ : ω˝σ “ ´pϕ˚ q1,1 pωq the negative of the formal adjoint of D with respect to the Serre pairings ż ` ˘ ˚ 1,1 ϕ ˚ 1,1 ˚ pϕ q ÝÑ C, ξ bω ÝÑ ξ ^ω , ΓpΣ; V q bR Γ Σ; T Σ bC V ´ żΣ ` ˘ ˚ 1,0 ϕ ˚ 1,0 ˚ pϕ q ÝÑ C, ζ bψ ÝÑ ζ ^ψ . Γ0,1 j pΣ; V q bR Γ Σ; T Σ bC V ´ Σ

This operator is the restriction of a real CR-operator on the complex vector bundle T ˚ Σ1,0 bC V ˚ over Σ to the subspace of sections that are anti-invariant with respect to ϕ1,0 . It satisfies dpξ ^ψq “ pDξq^ψ`ξ^pD ˚ ψq @ ξ P ΓpΣ; V qϕ , ˘pϕ˚ q1,0 ` . ψ P Γ Σ; T ˚ Σ1,0 bC V ˚ ´

(12.40)

Since σ reverses the orientation of Σ, the above pairings take values in R. The following statement thus follows from [25, Lemma 2.3.2]. Lemma 12.7 (Serre duality). Let D be a real CR-operator on a real bundle pair pV, ϕq over a smooth symmetric Riemann surface pΣ, σ, jq. The pairing ż ˘ ` ˘ ` ˚ cok D b ker D ÝÑ R, rζsbψ ÝÑ ζ ^ψ , Σ

is well defined and non-degenerate.

340

Spin/Pin-Structures & Real Enumerative Geometry

Proof of Proposition 12.5(1). The approach here is based on the proof of [26, Theorem 2]. By Carleman Similarity Principle [32, Theorem 2.3.5], dz u is injective for all but finitely many points z P P1 and the image of du is contained in a complex line subbundle Tu Ă u˚ T X over P1 . Since u is a real map, the conjugation u˚ dφ preserves this subbundle. Let

Nu ”

u˚ T X ÝÑ P1 Tu

be the normal bundle of u and ϕ be the conjugation on Nu so that 0 ÝÑ p Tu, u˚ dφq ÝÑ u˚ pT X, dφq ÝÑ pNu, ϕq ÝÑ 0 q

(12.41)

is an exact sequence of real bundle pairs over pP1 , τ q. By Carleman Similarity Principle again, every zero of ˘ ` du P Γ P1 ; T ˚ P1 bC Tu contributes positively to the Euler class of the complex line bundle T ˚ P1 bC Tu over P1 . Thus, deg Tu ě deg T P1 “ 2

and

˚

deg Nu “ deg u T X ´deg Tu ď k`2l´1;

(12.42)

the last inequality above follows from (12.23) and the first inequality. Each of the inequalities in (12.42) is an equality if and only if u is an immersion. For any real almost complex submanifold C of pX, φ, Jq, the linearization of the B¯J -operator on the space of real maps from pP1 , τ q to pC, φ|C q is a restriction of the linearization of the B¯J -operator on the space of real maps from pP1 , τ q to pX, φq. In particular, ˘ ˚ φ ` u˚ dφ , ΓpP1 ; Tuqu dφ Ă Γ0,1 DJ;u j pΣ; Tuq φ descends where j is the standard complex structure on P1 . Thus, DJ;u to a real CR-operator Du2 on the real bundle pair pNu, ϕq over pP1 , τ q. Let

Du1 : ΓpP1 ; Tuqu

˚ dφ

u ÝÑ Γ0,1 j pΣ; Tuq

˚ dφ

and

Counts of Rational Curves on Surfaces

341

 Du2 : ξ P ΓpP1 ; Nuqϕ : ξpxi q “ 0 @i P rks, ξpzi` q ( 1 ϕ “ 0 @i P rls ÝÑ Γ0,1 j pP ; Nuq φ be the restrictions of DJ;u and Du2 , respectively. The operator Du2 is also induced by Du . We denote by B¯ the real CR-operator on the rank 1 real bun¯ dle pT P1 , dτ q over pP1 , τ q determined by the holomorphic B-operator 1 on T P . The differential du induces an injective homomorphism

du : ker B¯ ÝÑ ker Du1 , which is an isomorphism if and only if u is an immersion. By Lemma 12.7, ` ˘˚ cok Du1 « ker Du1˚ . By Carleman Similarity Principle, every zero of a non-zero element η P ker Du1˚ contributes positively to the Euler class of the complex line bundle T ˚ P1 bC Tu˚ . Since ˘ ` deg T ˚ P1 bC Tu˚ ď ´4 by (12.42), we conclude that cok Du1 “ t0u.

(12.43)

The conjugation ϕ on Nu induces a conjugation on the complex line bundle

Nup´uq ” Nu b

k â i“1

OP1 p´xi q b

l â

OP1 p´zi` ´zi´ q ,

i“1

which we denote in the same way. By (12.42), deg Nup´uq ď ´1

(12.44)

and the equality holds if and only if u is an immersion. By the twisting construction of [38, Lemma 2.4.1], Du2 induces a real

342

Spin/Pin-Structures & Real Enumerative Geometry

CR-operator pDu2 q´u on the real bundle pair pNup´uq, ϕq so that there are isomorphisms «

kerpDu2 q´u ÝÑ ker Du2

«

and cok pDu2 q´u ÝÑ cok Du2 .

By Carleman Similarity Principle, every zero of a non-zero element ξ P kerpDu2 q´u contributes positively to the Euler class of the complex line bundle Nup´uq. Along with (12.44), this gives ker Du2 “ t0u,

dim cok Du2 “ ´ind pDu2 q´u

“ ´1 ´ deg Nup´uq ě 0,

(12.45)

and the inequality above is an equality if and only if u is an immersion. Suppose dz u ‰ 0 whenever z P txi uiPrks , tzi` uiPrls . The real φ CR-operator Du1 is then the restriction of the operator Du ” DJ;u in (12.33) and 0 ÝÑ Du1 ÝÑ Du ÝÑ Du2 ÝÑ 0

(12.46)

is an exact triple of real CR-operators. Thus, there is an exact sequence 0 ÝÑ ker Du1 ÝÑ ker Du ÝÑ ker Du2 ÝÑ cok Du1 ÝÑ cok Du ÝÑ cok Du2 ÝÑ 0

(12.47)

of vector spaces. Along with (12.43) and (12.45), this implies that Du is surjective if and only if u is an immersion. If dz u “ 0 for some z P txi uiPrks , tzi` uiPrls , the operator Du1 in (12.46) should be replaced by a real CR-operator Du1 on a real bundle pair obtained by twisting down p Tu, u˚ dφq at the marked points of u where du vanishes. Along with (12.45), the exactness of the sequence (12.47) with Du1 replaced by Du1 then implies that Du is not surjective. This establishes the first claim of the  proposition. Remark 12.8. Let g P Z` . The genus g analog of (12.44) is the bound deg Nup´uq ď g´1 with the equality if and only if u is an immersion. While a generic real CR-operator on a degree g´1 complex line bundle over a genus g

Counts of Rational Curves on Surfaces

343

Riemann surface is an isomorphism, the subspace of such operators that are not isomorphisms is of codimension 1. The critical points of the analog of the projection (12.37) in positive genera can thus include, at least a priori, elements pt, rusq so that the domain of u is smooth and u is an immersion. The number δE pus pP1 qq of E-nodes does not change along a parametrization pτs , rus sqsPp´δ,δq ˚ r of Mk,l pB; Jqp around such pt, rusq. By the last paragraph of Section 12.2, the genus g analog of the counts (12.36) can change around at least some such values of t. The critical point pt, rusq must thus be a saddle point (possibly degenerate) in such cases, similar to the point labeled (W4) in Figure 12.2. In contrast to the situation with (W4), the curves us pP1 q with s ‰ 0 contribute to the genus g analog of (12.36) with the same sign. Proof of Proposition 12.5(2). For dimensional reasons, the domain Σ of u is either P1 or a wedge of two copies of P1 . In the former case, u is as in (W0)–(W2) or in (W4) in Section 12.5. Since J is φ in (12.34) is onto in these cases and thus generic, the operator DJ;t,u ` ˘ ` ˘ ( ˚ r ¯ : D φ pτ, ξq “ 0 . Jqp “ τ, rξs P Rˆ Γpuq{dupker Bq Tpt,rusq Mk,l pB; J;t,u By Proposition 12.5(1), the operator DJφt ;u is (resp. is not) onto in the cases (W0)–(W2) (resp. the case (W4)). Since the homomorphism (12.38) is an isomorphism in these cases, it follows that pt, rusq is not (resp. is) a critical point of πM in the cases (W0)–(W2) (resp. the case (W4)). In the case (W4), the non-degeneracy of the critical point pt, rusq of πM is implied by [38, Theorem 2]. In the following, we re-establish this non-degeneracy by a less computationally heavy argument, without use of Bianchi’s first identity. Suppose u is as in (W4) and x0 P S 1 ´txi uiPrks is the unique point so that dx0 u “ 0. Since dx0 u “ 0, the second and third derivatives of u at x0 , D2x0 u and q˝D3x0 u, are well defined. Since J is generic, D2x0 u ‰ 0 P pTx˚0 P1 qb2 bC Tu and q˝D3x0 u ‰ 0 P pTx˚0 P1 qb3 bC Nu. (12.48) Since the section du of the complex line bundle T ˚ P1 bC Tu over P1 has a unique zero at x0 and it is transverse, deg Tu “ 3 and

deg Nup´uq “ ´2 .

(12.49)

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Along with the exactness of (12.47), (12.43), and (12.45), this implies that the homomorphisms L L ¯ ÝÑ ker Du dupker Bq ¯ and cok Du ÝÑ cok D 2 ker Du1 dupker Bq u (12.50) are isomorphisms and the dimensions of all vector spaces above are 1. L ¯ Thus, Let ξ P ker Du1 Ă Γpuq be so that rξs generates ker Du1 dupker Bq. ` ˘( ˚ r Tpt,rusq Mk,l pB; Jqp “ SpanR 0, rξs . ξpx0 q ‰ 0, (12.51) By Lemma 12.7 and the second homomorphism in (12.50) being an isomorphism, the pairing ż ˘ ` ˘ ` 2˚ ζ ^ψ, cok Du bR ker Du ÝÑ R, ζ bψ ÝÑ P1

is well defined and non-degenerate. Thus, ker Du2˚ is one-dimensional. Let ψ be a generator for ker Du2˚ . Since the homomorphism (12.38) is an isomorphism, ż ˘ ` 1 (12.52) J9t ˝du˝j ^ψ P R˚ . ct ” 2 P1 By (12.49), deg T ˚ P1 bC Nup´uq˚ “ 0. By an exact sequence as in (7.49) and the reasoning as for (12.43), the homomorphism  ( ker Du2˚ ÝÑ α P Tx˚0 P1 bC Nup´uq˚ |x0 : pϕ˚ q1,0 pαq “ ´α , ψ 1 ÝÑ ψ 1 px0 q, is onto. Along with (12.48) and (12.51), this implies that ξpx0 qb2 bC pq˝D3x0 uq ^ψpx0 q P i R˚ . pD2x0 uqb2

(12.53)

We call a smooth map exp : T X ÝÑ X exponential-like if exp|X “ idX and ` ˘ dx exp “ idTx X , idTx X : Tx pT Xq “ Tx X ‘Tx X ÝÑ Tx X @ x P X, where the second equality is the canonical splitting of Tx pT Xq into the horizontal and vertical tangent spaces along the zero section

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345

X Ă T X. Given such a map, a connection ∇ in T X, x P X, and v P Tx X, we denote by Π∇ v : Tx X ÝÑ Texppvq X the parallel transport with respect to ∇ along the “geodesic” γv : r0, 1s ÝÑ X,

γv psq “ exppsvq.

Suppose exp is a φ-equivariant exponential-like map on X, ∇ is a φ-equivariant connection in pT X, Jt q compatible with a φ- and Jt invariant metric gt on X, and p P R` with p ą 2. We denote by } ¨ }p,1 and } ¨ }p the Sobolev Lp1 and Lp -norms on Γpuq and Γ0,1 Jt puq, respectively, determined by the metric gt and connection ∇. Define 0,1 ∇ ∇ , Nexp;J : Γpuq ÝÑ Γ0,1 B¯exp;J Jt puq “ ΓJt puq, t ;u t ;u  ( ˘ ` ∇ 1 q “ Π∇ ´1 ˝ B ¯J exppξ 1 q , pξ B¯exp;J 1 t ;u ξ t ∇ ∇ pξ 1 q “ B¯exp;J pξ 1 q ´ DJφt ;u pξ 1 q, Nexp;J t ;u t ;u

(12.54)

∇ p0q “ 0. By [52, Propowith B¯Jt as in (12.31). By definition, Nexp;J t ;u ` sition 3.13], there exist C, δ P R so that › › ∇ ` ˘ ∇ ›Nexp;J ;u pξ 1 q´Nexp;J pξ 2 q›p ď C }ξ 1 }p,1 `}ξ 2 }p,1 }ξ 1 ´ξ 2 }p,1 t t ;u

@ ξ 1 , ξ 2 P Γpuq with }ξ 1 }p,1 , }ξ 2 }p,1 ď δ.

(12.55)

˚ r Jqp Let pt ` τs , rus sqsPp´δ1 ,δ1 q be a parametrization of Mk,l pB; around pt, rusq so that ˇ ˇ ˘ ` ˘ ` d ˇˇ d ˇˇ τs ˇ us ˇ “ 0, “ ξ. (12.56) τ0 , ru0 s “ 0, rus , ds s“0 ds s“0

For all s P p´δ1 , δ1 q sufficiently close to 0, there exists a unique ξs P Γpuq such that us “ exppξs q : P1 ÝÑ X

and

}ξs }p,1 ă δ .

By the last condition in (12.56) and (12.55), there exists C P R` so that › ∇ › ∇ › ď Cs3 pξ q´N psξq }ξs ´sξ}p,1 ď Cs2 and ›Nexp;J s ;u ;u exp;J t t p

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for all s close to 0. Since DJφt ;u pξ 1 q^ψ integrates to 0 for all ξ 1 P Γpuq, it follows that ˇ ˇż ż ˇ ˇ ` ∇ ˘ ˘ ` ∇ ¯ ¯ ˇ (12.57) Bexp;Jt ;u pξs q ^ψ´ Bexp;Jt ;u psξq ^ψ ˇˇ ď Cs3 . ˇ P1

P1

Since B¯Jt`τs us “ 0, (12.31) implies that ˇ ˇ ż ˇ ˇ ˘ ` ∇ ¯ ˇ ď C|τs |2 ˇct τs ` pξ q ^ψ B s ;u exp;J t ˇ ˇ

(12.58)

P1

for some C P R` fixed and for all s P R close to 0. We show at the end of this section that d2 ds2

ˆż

`

P1

˙ˇ ˇ ˘ ∇ ¯ Bexp;Jt ;u psξq ^ψ ˇˇ

s“0

˜ “ iπ

¸ ξpx0 qb2 bC pq˝D3x0 uq ^ψpx0 q pD2x0 uqb2 (12.59)

if exp is the exponential map determined by the connection ∇. Along with (12.58), (12.57), and (12.52), (12.59) implies that the Hessian of the projection πM in (12.37) at pt, rusq is given by ¸ ˜ b2 b pq˝D3 uq 2 ˇˇ q ξpx d π τ 0 C sˇ x0 2 πM pξ, ξq ” “ ^ψpx0 q . Dpt,rusq ds2 ˇs“0 ict pD2x0 uqb2 (12.60) Combining this with (12.53), we conclude that the critical point pt, rusq of πM is non-degenerate. Suppose now that the domain Σ of u is a wedge of two copies of P1 , i.e. u is as in (W3) on page 336. By (12.23) and (12.13), either k ‰ 0 or b ‰ 0. Thus, the node of the domain Σ of u is of type H and u corresponds to a pair of real Jt -holomorphic maps ` ˘ and u1 ” u1 : P1 ÝÑ X, τ, pxi qiPt0u\S R , pzi` , zi´ qiPt0u\S C 1 1 ` ˘ (12.61) u2 ” u2 : P1 ÝÑ X, τ, pxi qiPt0u\S R , pzi` , zi´ qiPt0u\S C 2

2

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347

r1 ” pB1 , b1 q and B r2 ” pB2 , b2 q with S1R\S2R “ rks and S1C\S2C “ rls. Let B be the degrees of u1 and u2 , respectively, as in (12.19). Since J is generic, ˇ Rˇ ˇ Cˇ ˇ ˇ ˇ ˇ ˇS1 ˇ `2ˇS1 ˇ “ ω pB1 q and ˇS2R ˇ `2ˇS2C ˇ “ ω pB2 q`1 , (12.62) after possibly re-ordering the components. For the same reason, both components of u are immersions and the only singular points of upΣq ` are simple nodes at points of X distinct from all points pR i , pi in (12.24). ˚ r Jqp The manifold structure of the moduli space Mk,l pB; around pt, rusq and the non-degeneracy of this critical point of the projection πM in (12.37) are implied by gluing techniques for pseudoholomorphic curves in the presence of obstructions, such as in [49]. For the purposes of Theorem 12.1, the weaker description of the structure of πM provided by the proof of [44, Proposition 2.1.4] suffices. In the following, we give a version of the argument in [44] with a simpler isotopy part. We can assume that there is a path r Jqp with pt`s, rus sqsPp´δ,0q in M˚k,l pB; ` ˘ lim rus s “ rus . t´δ, t`δ XI0 “ ttu and sÝÑ0´

˚

We denote by UM the space of pairs pJ, ru1 sq so that J P Jω and ru1 s is a simple marked J-holomorphic degree B map as in (12.27) satisfying u1 pxi q “ pR i @ i P rks,

u1 pzi` q “ p` i @ i P rls,

u1 pzi´ q “ φpp` i q @ i P rls.

(12.63)

We first show for any J P Jω (not necessarily real) close to Jt there exists at most one marked degree B J-holomorphic map ru1 s (not necessarily real) as in (12.27) satisfying (12.63) which is close to rus. Let ru2 s be another marked degree B J-holomorphic map satisfying (12.63) which is close to rus. Suppose first that the domain Σ1 of u1 is P1 . By [32, Proposition E.2.2], u1 and u2 intersect at finitely many points and each intersection point contributes positively to the homology intersection number ` ˘ ` ˘ u1˚ rΣ1 sZ ¨u2˚ rΣ2 sZ “ B 2 . By (12.22), the curve u1 pP1 q has 1 ` pB 2 ´ xc1 pX, ωq, Byq{2 nodes; they lie near the nodes of upΣq other than the image of the node

348

Spin/Pin-Structures & Real Enumerative Geometry

of Σ and thus near the corresponding nodes of u2 pΣ2 q. There are two intersection points of u1 and u2 near each node of u1 pP1 q. In addition, ` ` u1 and u2 intersect at the points pR i , pi , φppi q. Along with (12.23), this implies that ` ˘ B 2 ě 2 1`pB 2 ´xc1 pX, ωq, Byq{2 `pk`2lq “ B 2 `1 ; (12.64) this is a contradiction. If the domains Σ1 , Σ2 of u1 , u2 are not P1 , the marked maps u1 and u2 split into components u11 , u12 and u21 , u22 as in (12.61) so that u11 , u21 (resp. u12 , u22 ) are close to u1 (resp. u2 ). If u11 pP1 q ‰ u21 pP1 q (resp. u12 pP1 q ‰ u22 pP1 q), (12.64) with pB, k, lq replaced by pB1 , |S1R |, |S1C |q (resp. pB2 , |S2R |, |S2C |q) and (12.62) yield a con˚ tradiction. Thus, there exists a neighborhood W of pJt , rusq in UM so that the restriction of the projection ` ˘ ˚ π : UM ÝÑ Jω , π J, ru1 s “ J, ˚

to W is injective. Let W1 Ă UM be a neighborhood of pJt , rusq so 1 that W Ă W. We can assume that pJt`s , rus sq P W1 for all s P p´δ, 0q. 1 ˚ We denote by UM Ă UM the subspace of pairs pJ, ru1 sq so that either the domain of u1 is nodal or u1 is not an immersion. Let 1 ˚ 1 Jω1 “ πpUM q. Similar to [32, Proposition 3.4.2], UM ´ UM is a separable Banach manifold (after an appropriate completion) and the restriction ˚

1

π : UM ´ UM ÝÑ Jω is a local diffeomorphism. Similar to [32, Proposition 6.2.7], 1

Jω1 ” πpUM q Ă Jω is a countable union of Banach submanifolds of codimension at least 2. Thus, for there exists a continuous map p´δ, δqˆp0, δq ÝÑ Jω ,

ps, τ q ÝÑ Js,τ ,

such that Js,τ ÝÑ Jt`s as τ ÝÑ 0 and Js,τ P Jω1 for at most countably many pairs ps, tq.

Counts of Rational Curves on Surfaces

For each s P p0, δq, define γs : p0, πq ÝÑ p´δ, δqˆp0, δq,

349

` ˘ γs pθq “ ´ s cos θ, s sin θ .

Let I10 Ă p0, δq be the subset of points s such that Jγs pθq P Jω1 for some θ P p0, πq; this subset is at most countable. Since us is pp, Jt`s qregular for every s P p´δ, 0q, the restriction of π to a neighborhood of pJt`s , rus sq is a diffeomorphism onto an open neighborhood of Jt`s in Jω . For every s P p0, δq´I10 , the path Jγs in Jω thus lifts to a ˚ path pJγs , ruγs sq in UM with ˘ ` ˘ ` lim Jγs pθq , ruγs pθq s “ Jt´s , ru´s s . θÝÑ0

Suppose there exists a sequence psr , θr q in pp0, δq ´ I10 q ˆ p0, πq such that ` ˘ lim sr “ 0 and Jγsr pθr q , ruγsr pθr q s R W1 @ r P Z` . rÝÑ8

For each r P Z` , let  ` ˘ ( θr˚ “ min θ P p0, πq : Jγsr pθq , ruγsr pθq s R W1 P p0, θr s . By Gromov’s Compactness, a subsequence of pJγsr pθr˚ q , ruγsr pθr˚ q sq converges to some ˘ ` ˚ 1 Jt , ru1 s P W ´ W1 Ă W Ă UM . However, this contradicts the injectivity of the restriction of π to W. By the above conclusion and the injectivity of π| W, ˘ ˘ ` ` 1 Jt`s , rus s ” lim Jγs pθq , ruγs pθq s P W Ă W θÝÑπ

for all s P p0, δq sufficiently small and ˘ ` ˘ ` lim Jt`s , rus s “ Jt , rus . sÝÑ0

Since Jt`s P Jωφ , the injectivity of π| W also implies that rus s is the equivalence class of a real J-holomorphic map. Thus, the restriction of the projection πM in (12.37) to a neighborhood of pJt , rusq r Jqp is a homeomorphism onto a neighborhood of t in in M˚k,l pB;  r0, 1s.

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Proof of (12.59). Since the section du of the complex line bundle T ˚ P1 bC Tu over P1 has a unique zero at x0 and it is transverse, ξ “ r for some meromorphic vector field ξr on P1 which is holomorphic dupξq on P1´tx0 u and has a simple pole at x0 . We show in the following that ˇ ˇ ˘ˇ ˘ˇ D2 ¯ ` d2 `¯∇ φ ` u ˘ ˇ ˇ B B exppsξq ∇ξrξ ; psξq “ “ ´D J t ;u exp;J J t ˇ ˇ t ;u ds2 Ds2 s“0 s“0 (12.65) . Along the first equality above holds by the definition of Π∇ 1 ξ with (12.40), this implies that d2 ds2

ˆż

`

P1

˙ˇ ˇ ˘ ∇ ¯ Bexp;Jt ;u psξq ^ψ ˇˇ

s“0

¿ “ lim

`

˘ ∇uξrξ ^ψ ,

ÝÑ0 BB px q 0

(12.66)

where B px0 q Ă P1 is the ball of radius  around x0 in some coordinate chart. By Carleman’s Similarity Principle and (12.48), we can choose a real holomorphic coordinate z on a neighborhood U of x0 P P1 , real C2 -valued coordinates pw1 , w2 q on a neighborhood of upx0 q P X, a2 , a3 P R˚ , and a C2 -valued Lp1 -function g on U so that ˇ B ˇˇ “ , gpx0 q “ 0, Bz ˇx0 x0 ` ˘ pw1 , w2 q˝u “ a2 z 2 , a3 z 3 `z 3 gpzq : U ÝÑ C2 , ˇ ˇ r z ξ ˇˇ

(12.67)

and the coordinates pw1 , w2 q identify Jt |upx0 q with the standard complex structure on T0 C2 ; see also [38, Lemma 4.3.4]. By (12.67), ξpx0 q “ p2a2 , 0q with respect to the coordinates pw1 , w2 q. On a neighborhood of x0 P P1 , 1`f pzq B ξr “ z Bz

(12.68)

for some holomorphic function f with f p0q “ 0. Since dx0 u “ 0, the residue on the right-hand side of (12.66) and the derivatives on the right-hand side of (12.59) do not depend on the choice of the connection ∇ in T X. We can thus compute them using the standard connection on C2 . By (12.67), (12.68), and the condition that ξ is a

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351

section of Tu, D2x0 u “ pdzqb2 bp2a2 , 0q, ξpx0 qb2 bC pq˝D3x0 uq qp0, 1q^ψpx0 q ^ψpx0 q “ 6a3 , 2 b2 dz pDx0 uq ¿ lim

q˝D3x0 u “ pdzqb3 bqp0, 6a3 q, `

˘ qp0, 1q^ψpx0 q . ∇uξrξ ^ψ “ 6πia3 dz

ÝÑ0 BB px0 q

Combining the two equalities in the right column above with (12.66), we obtain (12.59). It remains to verify the second equality in (12.65). It is sufficient to do so at the points of P1´tx0 u where ξr is non-zero. We can choose a holomorphic coordinate z ” τ `iτ 1 around any such point so that B{Bτ “ ξr and thus ˇ ˇ d exppsξqˇˇ du du d exppsξqˇˇ “ ξ, “ uτ ” “ uτ 1 ” 1 “ Jt ξ, dτ ˇs“0 dτ dτ 1 ˇs“0 dτ (12.69) ˙ ˆ 2 ˘( `  r “ D ξ `Jt D D ξ ´T∇ uτ , D ξ 2 DJφt ;u ∇uξrξ pξq dτ 2 dτ 1 dτ dτ ˙ ˆ D (12.70) ´Jt T∇ uτ 1 , ξ , dτ ˘(  ` r “ d exppsξq `Jt d exppsξq 2 B¯Jt exppsξq pξq dτ dτ 1

(12.71)

on this coordinate chart. Since DJφt ;u ξ “ 0, the torsion T∇ of ∇ is antisymmetric in the two inputs, and ∇Jt “ 0, (12.32), the definition of the curvature R∇ of ∇, and (12.69) give ˆ ˙ D D D2 ξ “ ´Jt ξ ´T∇ puτ 1 , ξq dτ 2 dτ dτ 1 ˙ ˆ D D ξ ´ R∇ puτ 1 , uτ qξ “ ´Jt dτ 1 dτ ¸ ˜ ˙ ˆ ˆ ˙ (  D D uτ 1 , ξ `T∇ uτ 1 , ξ . ` Jt ∇ξ T∇ puτ 1 , ξq`T∇ dτ dτ

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352

Combining this with (12.70), (12.69), and ∇Jt “ 0, we obtain ` ˘(  r 2 DJφt ;u ∇uξrξ pξq ˜ ¸ ˆ ˆ ˙ ˙  ( D D “ ´T∇ ξ, ξ `Jt R∇ pJt ξ, ξqξ ` ∇ξ T∇ pJt ξ, ξq`T∇ Jt ξ, ξ . dτ dτ (12.72) Since the exponential map exp is determined by the connection ∇, ˇ D d exppsξq d exppsξq ˇˇ “ ξ and “ 0. (12.73) ˇ ds ds ds s“0 By the definition of the torsion T∇ of ∇, ` ˙ ` ˆ d exp sξq d exp sξq D d exppsξq D d exppsξq “ ´T∇ , . ds dτ dτ ds dτ ds Along with the definition of the curvature R∇ of ∇ and the first equations in (12.69) and (12.73), this gives ˇ ˇ ˘ ` D D d exppsξq ˇˇ D2 d exppsξq ˇˇ “ ´ R∇ uτ , ξ ξ ˇ ˇ 2 ds dτ dτ ds ds s“0 s“0 ˜ ˙ ˆ (  D ξ ´T∇ puτ , ξq, ξ ´ ∇ξ T∇ puτ , ξq`T∇ dτ ¸ ˇ ˙ ˆ D d exppsξq ˇˇ . `T∇ uτ , ˇ ds ds s“0 Combining this with the second equation in (12.73) gives ˇ ˘  ` ( D2 d exppsξq ˇˇ “ ´ R , ξ ξ ´ ∇ T u puτ , ξq ∇ τ ∇ ξ ˇ ds2 dτ s“0 ˙ ˆ D ξ ´T∇ puτ , ξq, ξ . ´ T∇ dτ Similarly,

ˇ ˘  ` ( D2 d exppsξq ˇˇ “ ´R∇ uτ 1 , ξ ξ ´ ∇ξ T∇ puτ 1 , ξq ˇ 2 1 ds dτ s“0 ˙ ˆ D 1 ξ ´T∇ puτ , ξq, ξ . ´ T∇ dτ 1

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353

By (12.69), the definition of T∇ again, and ∇Jt “ 0, D D D D uτ 1 “ Jt ξ . ξ ´T∇ puτ 1 , ξq “ 1 uτ ´T∇ puτ 1 , uτ q “ 1 dτ dτ dτ dτ Since T∇ and R∇ are anti-symmetric in the first two inputs and ∇Jt “ 0, (12.69), (12.71), and the last three equations yield ˇ ˘( ¯ˇ D2 ´¯ ` r BJt exppsξq pξq ˇˇ 2 Ds2 s“0 ¸ ˜ ˙ ˙ ˆ ˆ ˘  ` ( D D ξ, ξ ´Jt R∇ Jt ξ, ξ ξ ` ∇ξ T∇ pJt ξ, ξq`T∇ Jt ξ, ξ . “ ´T∇ dτ dτ (12.74) Since T∇ is anti-symmetric in the two inputs, (12.72) and (12.74)  confirm (12.65).

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Chapter 13

Counts of Stable Real Rational Maps

We next describe invariant counts of real rational curves in real symplectic manifolds and properties of these counts based on a perspective originating in [39] and developed further in [6, 8, 14]. We recall cases in which such invariants can be defined in Section 13.1 and summarize their properties in Theorems 13.1 and 13.2; some implications of the two theorems are illustrated in Section 14.3. The relevant notation and orientations for the Deligne–Mumford τ spaces Mk,l of stable real curves and for the moduli spaces of stable real maps are specified in Sections 13.2 and 13.3, respectively. In Section 13.4, we define the signs of the constrained maps appearing in the invariant counts of Theorems 13.1 and 13.2, indicate why these counts are indeed invariant in the cases specified in Section 13.1, and complete the proof of Theorem 13.1. For generic choices of the constraints, these maps are immersions. Proposition 13.8, established in Section 13.6, interprets the sign definition of Section 13.4 in terms of the normal bundles to these maps. This proposition is used in Section 13.6 to complete the proof of Theorem 13.2 and in Section 14.4 to relate the moduli-theoretic definitions of the real curve signs arising from [39] with the intrinsic ones introduced in [44–46]. Proposition 13.8 is a consequence of Lemma 13.10, proved in Section 13.7, which reformulates the definitions of the orientations at stable immersive maps in terms of the normal bundles to the immersions.

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13.1

Invariance and Properties

Let pX, ω, φq be a real symplectic manifold of real dimension 2n, as defined at the beginning of Section 12.3, Y Ă X φ be a connected r φ pX, Y q be as in (12.12). component of the fixed locus of φ, and H 2 We also define à p ˚ pX, Y q “ Hp pX; Zq‘Hn´1 pX ´Y ; Zq, H p‰n´1

p ˚ pX, Y q “ H

à

H p pX; Zq ‘H n`1 pX, Y ; Zq.

p‰n`1

p ˚ pX, Y q be the subspace of even-degree elements. p 2˚ pX, Y q Ă H Let H ě0 and h ” phi : Hi ÝÑ XqiPrls be a tuple of cycles Let k P Z in X, i.e. smooth maps from compact oriented manifolds, or more generally of pseudocycles as in [51]. With the notation as in (12.17) r ” pB, bq be an element of H r φ pX, Y q and just as in the following, let B 2 such that l ÿ ` ˘ r “ pn´1qk` 2n´dim Hi ´2 . (13.1) ω pBq i“1

If

J P Jωφ

is generic, the elements of h are in general position, and ˘ ` R k (13.2) p ” pR 1 , . . . , pk P Y

is a tuple of points in Y in general position, then the space r Jq of real rational irreducible J-holomorphic degree B r Mp;h pB; R R curves C Ă X passing through the points p1 , . . . , pk and meeting the images of h1 , . . . , hl is a zero-dimensional manifold. Suppose in addition l˚ P rls. As described in Section 13.4, a relative Pin˘ -structure p on Y Ă X determines a sign sp;l˚ ;h pCq for each r Jq if element C of Mp;h pB; D @ (13.3) k`2Z ‰ w2 pXq, B P Z2 . A relative OSpin-structure p on Y determines a sign sp;l˚ ;h pCq without the assumption (13.3). For a Pin˘ - or OSpin-structure p on Y , we set sp;l˚ ;h pCq “ sιX ppq;l˚ ;h pCq, with ιX as in (6.3).

Counts of Stable Real Rational Maps

357

If in addition pX, ω, Bq satisfies certain positivity conditions, the r Jq is finite. The sum discrete set Mp;h pB; ÿ pY q ” sp;l˚ ;h pCq (13.4) N φ;p ˚ r B;k,l ;h

r CPMp;h pB;Jq

is well defined and independent of the choices of J, cycles hi in their p ˚ pX, Y q, and the points pR in Y if respective homology classes in H i # ˘ n, if i P rls´rl˚ s; 1` mod 2, dim hi – dim hi P 2Z, 2 n´1, if i P rl˚ s´t1u; (13.5) and either (C1) n “ 2, p is a relative Pin´ -structure on Y Ă X with xw2 ppq, B 1 y “ r φ pX, Y q, or 0 for every pB 1 , b1 q P H 2 (C2) n “ 2, p is a relative Pin` -structure on Y Ă X with xw2 ppq, B 1 y “ r φ pX, Y q, or xw1 pY q, b1 y for every pB 1 , b1 q P H 2 r ‰ 0, p is an OSpin-structure on Y , and either k ‰ 0 (C3) n “ 3, ω pBq or b ‰ 0, or r ‰ 0, p is an OSpin-structure on Y , and there exists (C4) n “ 3, ω pBq a finite-order automorphism ψ of pX, ω, φ, Y q which restricts to an orientation-reversing diffeomorphism of Y and acts trivially r φ pX, Y q, or on H 2 r ‰ 0, b ‰ 0, p is an OSpin-structure (C5) n R 2Z, k “ 0, l˚ “ l, ω pBq 1 ě0 on Y , and ω pB q P 4Z for all spherical classes B 1 P H2φ pX; Zq with ωpB 1 q ą 0. The invariance in the first three cases above is established in the proof of Theorem 1.3 in [39]; see also the proofs of Theorems 1.6 and 1.7 in [14] and Proposition 5.2 in [6]. The proof of Proposition 1.3 in [8] establishes the invariance in the fourth setting and applies in the third as well. The last setting is a special case of that of Theorems 1.6 and 1.7 in [14]. We describe the underlining reason for the invariance in all cases in Section 13.5. An l-tuple h satisfying the first condition in (13.5) determines one or two possible values of l˚ so that the second condition in (13.5) is satisfied after a permutation of the elements of h. Under any of

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the conditions C1–C5, the associated sign sp;l˚ ;h pCq is same for the admissible choices of l˚ and can be defined for l “ 0; see Section 13.4. By Theorem 1.1 in [51], every homology class in a smooth manifold can be represented by a pseudocycle. In the five cases above, we thus obtain a multilinear symmetric functional @ Dφ,p ‘l p 2˚ ÝÑ Z (13.6) ¨, . . . , ¨ B;Y r ;k : H pX, Y q r curves enumerating real rational irreducible J-holomorphic degree B C Ă X. For y P Y , we denote by SpNy Y q a small sphere in the fiber of the normal bundle NY of Y in X. An OSpin-structure p on Y determines an orientation on SpNy Y q; see Section 2.5 in [8]. We denote the resulting homology class in Hn´1 pX´Y ; Zq by rSpNy Y, pqs and the Lefschetz–Poincare dual of rSpNy Y, pqs in H n`1 pX, Y ; Zq ˝ by γX,Y . The last class generates the kernel of the surjective homomorphism H n`1 pX, Y ; Zq ÝÑ H n`1 pX; Zq in the cohomology relative exact sequence for the pair pX, Y q. Since φ preserves (resp. reverses) the orientations of X and SpNy Y q if n P 2Z (resp. n R 2Z), ˝ ˝ “ γX,Y . φ˚ γX,Y

(13.7)

Theorem 13.1. Suppose pX, ω, φq is a compact connected real symplectic manifold of (real) dimension 2n, Y Ă X φ is a connected rPH r φ pX, Y q, and p is a relative Pin˘ - or component, k P Zě0 , B 2 φ OSpin-structure on Y Ă X . If one of the conditions C1–C5 above holds, then the multilinear functionals p13.6q satisfy the following properties: (RGW1) xγ1 , . . . , γl yφ,p r ;k “ 0 unless p13.1q with 2n´dim Hi replaced B;Y by deg γi holds. (RGW2) If φ˚ γi “ ´p´1qpdeg γi q{2 γi for some i P rls, then xγ1 , . . . , γl yφ,p r ;k “ 0. B;Y # xγ1 , pty, if pk, lq “ p1, 1q; φ,p r “ 0, xγ1 , . . . , γl y “ (RGW3) If B r ;k B;Y 0, otherwise.

Counts of Stable Real Rational Maps

# (RGW4) x1, γ2 , . . . , γl yφ,p r

B;Y ;k

(RGW5) If γ0 P γl yφ,p r

B;Y ;k

H 2 pX; Zq,

“

359

1,

r k, lq “ p0, 1, 1q; if pB,

0,

otherwise.

xγ0 , γ1 , . . . , γl yφ,p r

B;Y ;k

r 1, . . . , “ xγ0 , Byxγ

.

˝ , γ1 , . . . , γl yφ,p (RGW6) If n “ 3, xγX,Y r

B;Y ;k

“ 2xγ1 , . . . , γl yφ,p r

B;Y ;k`1

.

(RGW7) If n “ 3 and p is the OSpin-structure on Y corresponding to p as in the SpinPin 3 property in Section 1.2, then “ ´p´1qk x¨, . . . , ¨yφ,p . x¨, . . . , ¨yφ,p r r B;Y ;k

B;Y ;k

The vanishing property (RGW1) holds because the dimensions of the relevant moduli spaces and the constraints are different unless (13.1) with 2n ´ dim Hi replaced by deg γi holds. This in turn implies the vanishing statement in (RGW4). Since every degree 0 J-holomorphic map is constant, (RGW5) also leads to the vanishing statement in (RGW3). We discuss the remaining properties of the real GW-invariants (13.6) stated in Theorem 13.1 in Section 13.4, after defining the sign sp;l˚ ;h pCq. Let pX 1 , ω 1 , φ1 q and pX 2 , ω 2 , φ2 q be compact connected real sym1 plectic manifolds of dimensions 2 and 4, respectively, Y 1 Ă X 1φ and 2 Y 2 Ă X 2φ be connected components, pX, ω, φq ” pX 1 , ω 1 , φ1 qˆpX 2 , ω 2 , φ2 q,

and Y ” Y 1 ˆY 2 .

The restriction H 4 pX 2 , Y 2 ; Zq ÝÑ H 4 pX 2 ; Zq is then an isomorphism. ˝ Since φ1 acts by the multiplication by ´1 on H 2 pX 1 ; Zq and γX 1 ,Y 1 generates the kernel of the surjective homomorphism H 2 pX 1 , Y 1 ; Zq ÝÑ H 2 pX 1 ; Zq, the identity (13.7) with pX, Y, φq replaced by pX 1 , Y 1 , φ1 q implies that the induced homomorphism ˘ ( ˘ ` `  α P H 2 X 1 , Y 1 ; Zr1{2s : φ1˚ α “ ´α ÝÑ H 2 X 1 ; Zr1{2s is an isomorphism. Pairs pγ01 , γ42 q P H 0 pX 1 ; Zq‘H 4 pX 2 ; Zq and pγ21 , γ22 q P H 2 pX 1 ; Zq‘H 2 pX 2 ; Zq

Spin/Pin-Structures & Real Enumerative Geometry

360

thus determine elements ˘ ` γ01 ˆγ42 P H 4 X, X 1 ˆY 2 ; Z

˘ ` and γ21 ˆγ22 P H 4 X, Y 1 ˆX 2 ; Zr1{2s ,

respectively. Both restrict to elements of H 4 pX, Y ; Zr1{2sq, which we denote in the same way. Thus, the cross product induces a bilinear map p 2˚ pX 2 , Y 2 q “ H 2˚ pX 1 ; Zq ˆ : H 2˚ pX 1 ; Zqˆ H p 2˚ pX, Y qbZ Zr1{2s. ˆH 2˚ pX 2 ; Zq ÝÑ H Theorem 13.2. Suppose pX 1 , ω 1 , φ1 q, pX 2 , ω 2 , φ2 q, Y 1 , Y 2 , pX, ω, φq, and Y are as above, os1 and os2 are OSpin-structures on Y 1 and Y 2 , respectively, and γ11 , . . . , γl1 P H 2˚ pX 1 ; Zq,

p 2˚ pX 2 , Y 2 q. γ12 , . . . , γl2 P H

Let k P Zě0 , ` 1 2˘ 1 1 , Y 1 q‘ H r PH r φ2pX 2 , Y 2 q Ă H r φ pX, Y q, r” B r ,B r φpX B 2 2 2 os “ xxπ 1˚ os1 , π 2˚ os2 yy‘ P OSpinpY q, where π 1 , π 2 : Y ÝÑ Y 1 , Y 2 are the two projections. r 1 “ 0, then (RGW8) If B @

Dφ,os γ11 ˆγ12 , . . . , γl1 ˆγl2 B;Y r ;k ˙ $ˆ l & ś xγ 1 , pty @γ 2 , . . . , γ 2 Dφ,os2 , i 1 l B r 2 ;Y 2 ;1 “ % i“1 0,

if k “ 1, deg γi1 “ 0 @ i P rls; otherwise.

r 1 “ prP1 sZ , rS 1 sZ q, B r 2 “ 0, and (RGW9) If pX 1 , φ1 q “ pP1 , τ q, B 2 1 1 os “ os0 pT S q is the standard OSpin-structure for either orientation on S 1 , then @ 1 Dφ,os γ1 ˆγ12 , . . . , γl1 ˆγl2 B;Y r ;k $ ’ 0, ’ ’ &@ Dφ,os2 γ12 , . . . , γl2 Br 2 ;Y 2 ;3 , “ ’ ’ ’ %xγ 1 , rP1 sZ y@γ 2 , . . . , γ 2 Dφ,os2 , 1 1 l B r 2 ;Y 2 ;1

ˇ ˇ if k`2ˇti P rls : deg γi1 “ 2uˇ ă 3; if k “ 3, γi1 “ 1 @ i P rls; if k “ 1, γi1 “ 1 @ i P rls´t1u.

Counts of Stable Real Rational Maps

361

In the setting of Theorem 13.2, we can compute the invariants (13.6) using an almost complex structure J “ J 1 ˆ J 2 with 1 2 r 1 “ 0, every real rational irreducible JJ 1 P Jωφ1 and J 2 P Jωφ2 . If B r curve C Ă X is then of the form y 1 ˆ C 2 for holomorphic degree B 1 1 r2 some y P Y and a real rational irreducible J 2 -holomorphic degree B 2 2 curve C Ă X . This implies the second statement in (RGW8) as well as the first one up to sign, which is confirmed in Section 13.4. If pX 1 , φ1 q “ pP1 , τ q, we can take J 1 to be the standard complex strucr 1 “ prP1 sZ , rS 1 sZ q, every real rational irreducible ture JP1 on P1 . If B 2 r curve C Ă X is then the graph of a real J 2 J-holomorphic degree B r 2 map from P1 to X 2 . Since pP1 , τ q with k real holomorphic degree B 1 points and l conjugate pairs of points has a positive-dimensional family of automorphisms if k`2l1 ă 3, this establishes the first statement in (RGW9). Since any two configurations of k real points and l1 conjugate pairs of points on pP1 , τ q are equivalent if k`2l1 “ 3 and any such configuration has no automorphism, we also obtain the second and third statements in (RGW9) up to sign, which is confirmed in Section 13.7 as a corollary of Proposition 13.8. If n “ 2, (the tangent bundle of) Y admits a Pin´ -structure; see the SpinPin 1(a) property on page 9 and (11.12). By the RelSpinPin 1(a) property on page 81 and (11.12), Y Ă X also admits a relative Pin` -structure p with w2 ppq “ w2 pXq. In particular, the invariants (13.6) can be defined for every compact real symplectic fourfold. r Jq splits Similar to the situation in Section 12.3, each set Mp;h pB; ´1 r 1 1 into subsets Mp prB s; Jq with rB s P ΦX,Y pBq. By the same reasoning as for (13.4), each sum ÿ φ;p sp;l˚ ;h pCq NrB 1 s;k,l˚ ;h pY q ” CPMp;h prB 1 s;Jq

is well defined and independent of the choices of J, cycles hi in their p ˚ pX, Y q, and the points pR under respective homology classes in H i the same conditions as (13.4) and yields a multilinear symmetric functional @ Dφ,p p 2˚ pX, Y q‘l ÝÑ Z :H (13.8) ¨, . . . , ¨ 1 rB s;Y ;k

φ H 2 pX, Y

; Zq satisfying the analogs of the properties for each rB 1 s P in Theorems 13.1 and 13.2. By (13.36), this functional has a simple

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Spin/Pin-Structures & Real Enumerative Geometry

sign dependence on the choice of admissible relative Pin-structure p on Y Ă X in the settings C1 and C2. In the settings C3–C5, this functional similarly has a simple sign dependence on the choice of OSpin-structure on Y , but in these cases, the sign is determined by r “ ΦX,Y prB 1 sq; see (13.36). B 13.2

Orienting Moduli Spaces of Real Curves

Let PSLτ2 C Ă PSL2 C be the subgroup of automorphisms of P1 commuting with the conjugation τ on P1 . For k P Zě0 , let rks “ t1, . . . , ku τ the moduli as before. For k, l P Zě0 with k`2l ě 3, we denote by Mk,l space of equivalence classes of smooth real genus 0 curves with separating fixed locus, k real marked points, and l conjugate pairs of marked points. Thus, ` ˘ τ Mk,l « pxi qiPrks , pzi` , zi´ qiPrls : xi P S 1 , zi˘ P P1 ´S 1 , zi` “ τ pzi´ q, (L xi ‰ xj , zi` ‰ zj` , zj´ @ i ‰ j PSLτ2 C. τ are the equivalence classes of the marked symThe elements of Mk,l metric Riemann surfaces ` ˘ C ” Σ “ P1, σ “ τ, pxi qiPrks , pzi` , zi´ qiPrls (13.9)

modulo the reparametrizations by PSLτ2 C. This space is a smooth manifold of dimension k ` 2l ´ 3. Its topological components are indexed by the possible distributions of the points zi` between the interiors of the two disks cut out by the fixed locus S 1 of the standard involution τ on P1 and by the orderings of the real marked points xi on S 1 . If k`2l ě 4 and i P rks, let τ τ fR k,l;i : Mk,l ÝÑ Mk´1,l

(13.10)

be the forgetful morphism dropping the i-th real marked point. The associated exact sequence dfR k,l;i

τ R˚ τ 0 ÝÑ ker dfR k,l;i ÝÑ T Mk,l ÝÝÝÑ fk,l;i T Mk´1,l ÝÑ 0

induces an isomorphism ` ˘ ˘ ` ˘ ` τ ˚ τ R « fR λ T Mk,l k,l;i λ T Mk´1,l b ker dfk,l;i .

(13.11)

Counts of Stable Real Rational Maps

363

If k`2l ě 5 and i P rls, we similarly denote by τ τ ÝÑ Mk,l´1 fk,l;i : Mk,l

(13.12)

the forgetful morphism dropping the i-th conjugate pair of marked points. It induces an isomorphism ` ˘ ˘ ` ˘ ` τ ˚ τ λ T Mk,l´1 « fk,l;i bλ ker dfk,l;i . (13.13) λ T Mk,l For each CP Mk,l as in (13.9), ker d Cfk,l;i « Tz ` P1 i

is canonically oriented by the complex orientation of the fiber P1 at zi` . We denote the resulting orientation of the last factor in (13.13) by o` i . τ is as in (13.9). Let D2 Ă C Ă P1 be Suppose l P Z` and C P Mk,l ` the disk cut out by the fixed locus S 1 of τ which contains z1` . We orient S 1 Ă D2` Ă C in the standard way. If k`2l ě 4 and i P rks, this determines an orientation oR i of the fiber 1 ker d CfR k,l;i « Txi S

of the last factor in (13.11) over fR k,l;i p Cq. τ The space M1,1 is a single point; we take o1,1 ” `1 to be its orientation as a plus point. We identify the one-dimensional space τ with R` ´t1u via the cross ratio M0,2 ` ˘ τ ÝÑ R` ´t1u, ϕ rpz1` , z1´ q, pz2` , z2´ qs ϕ : M0,2 “

|1´z1` {z2´ |2 z2` ´z1´ z2` ´z1` : “ ; z2´ ´z1´ z2´ ´z1` |z1` ´z2` |2

(13.14)

see Figure 13.1. This identification, which is the opposite of [18, (3.1)] τ . and [20, (1.12)], determines an orientation o0,2 on M0,2

Figure 13.1. τ of M0,2 .

τ

The structure of the Deligne–Mumford compactification M0,2

364

Spin/Pin-Structures & Real Enumerative Geometry

τ for l P Z` and k`l ě 3 We now define an orientation ok,l on Mk,l inductively. If k ě 1, we take ok,l to be so that the i “ k case of the isomorphism (13.11) is compatible with the orientations ok,l , ok´1,l , and oR k on the three line bundles involved. If l ě 2, we take ok,l to be so that the i “ l case of the isomorphism (13.13) is compatible with the orientations ok,l , ok,l´1 , and o` l . By a direct check, the orientations τ τ τ via (13.13) are the on M1,2 induced from M0,2 via (13.11) and M1,1 τ are even-dimensional, it follows that same. Since the fibers of fk,l;l |Mk,l τ the orientation ok,l on Mk,l is well defined for all l P Z` and k P Zě0 with k ` 2l ě 3. This orientation is as above [14, Lemma 5.4]; for k “ 0, this orientation is the opposite of that taken in [18, Section 3] and [20, Section 1.4]. If l ě 2 and C is as in (13.9) with z1` “ 0 and z2` P R` , then the natural isomorphism ` ˘ τ « Tz ` R` ‘ Tz ` P1 ‘ ¨ ¨ ¨ ‘ Tz ` P1 ‘ Tx1 S 1 ‘ ¨ ¨ ¨ ‘ Txk S 1 Tr Cs Mk,l 2

3

l

(13.15) τ , the opposite o of the stanrespects the orientations ok,l on Tr Cs Mk,l R dard orientation on Tz ` R` , and the standard orientations on Tz ` P1 2

i

and Txi S 1 . If k ě 2 and C is as in (13.9) with z1` “ 0, then the natural isomorphism τ « Tz ` P1 ‘ ¨ ¨ ¨ ‘ Tz ` P1 ‘ Tx2 S 1 ‘ ¨ ¨ ¨ ‘ Txk S 1 Tr Cs Mk,l 2

l

τ and the standard orientarespects the orientations ok,l on Tr Cs Mk,l tions on Tz ` P1 and Txi S 1 . The claim concerning induced orientations i τ in the previous paragraph corresponds to the composite isoon M1,2 morphism τ « Tz ` P1 Tz ` R` ‘Tx1 S 1 « Tr Cs M1,2 2

2

being orientation-preserving with respect to the orientations oR and the standard orientations on Tx1 S 1 and Tz ` P1 if C is as in (13.9) 2

with pk, lq “ p1, 2q, z1` “ 0, and z2` P R` . τ is as in (13.9). Let px , x Suppose l P Z` and C P Mk,l 1 j2 p Cq , . . . , xjk p Cq q be the ordering of the real marked points of C starting with x1 and going in the direction of the standard orientation of

Counts of Stable Real Rational Maps

365

S 1 Ă D2` . We denote by δR p Cq P Z2 the sign of the permutation  (  (  C : 2, . . . , k ÝÑ 2, . . . , k ,  Cpiq “ ji p Cq. If k “ 0, we take δR p Cq “ 0. For l˚ P rls, let ˇ (ˇ δlc˚ p Cq “ ˇ i P rls´rl˚ s : zi` R D2` ˇ ` 2Z P Z2 .

(13.16)

In particular, δR p Cq “ 0 if k ď 2 and δlc p Cq “ 0. The functions δR τ . For l˚ P rls, we denote by o and δl˚ are locally constant on Mk,l k,l;l˚ τ τ if and only if the orientation on Mk,l which equals ok,l at r Cs P Mk,l δR p Cq “ δlc˚ p Cq, i.e. ˇ ˇ c (13.17) ok,l;l˚ ˇr Cs “ p´1qδR p Cq`δl˚ p Cq ok,l ˇr Cs . The following lemma is straightforward. τ with k, l P Zě0 and Lemma 13.3. The orientations ok,l;l˚ on Mk,l l˚ P rls such that k`2l ě 3 and satisfy the following properties:

(oM1) The isomorphism (13.13) with pl, iq replaced by pl ` 1, l˚ ` 1q respects the orientations ok,l`1;l˚ `1 , ok,l;l˚ , and o` l˚ `1 . (oM2) ok,l;l˚ is preserved by the interchange of two real points xi and xj with 2 ď i, j ď k. τ , o (oM3) If 2 ď i ď k and r Cs P Mk,l k,l;l˚ is preserved at r Cs by the interchange of the real points x1 and xji p Cq with 2 ď i ď k if and only if pk´1qpi´1q P 2Z. τ , and the marked points z ` and z ` of C (oM4) If 1 ď i, j ď l, r Cs P Mk,l i j lie on the same connected component of P1 ´S 1 , then ok,l;l˚ is preserved at r Cs by the interchange of conjugate pairs pzi` , zi´ q and pzj` , zj´ q. (oM5) If l˚ ă i ď l (resp. 1 ă i ď l˚ ), ok,l;l˚ is preserved (resp. reversed) by the interchange of the points in a conjugate pair pzi` , zi´ q. τ , o (oM6) If r Cs P Mk,l k,l;l˚ is preserved at r Cs by the interchange of the points in the conjugate pair pz1` , z1´ q if and only if ˆ ˙ k ˚ mod 2 or k ‰ 0 and l´l – 2 k “ 0 and l´l˚ – 1 mod 2.

Spin/Pin-Structures & Real Enumerative Geometry

366

13.3

Orienting Moduli Spaces of Real Maps

r P Suppose pX, ω, φq is a real symplectic manifold, k, l P Zě0 , B φ φ r pX, Y q is as in (12.19), and J P Jω . Let H 2 r Jq Ă Mk,l pB; r Jq, M˚k,l pB; r Jq Ă Mk,l pB; r Jq M˚k,l pB; be as in (12.29). For a tuple u as in (12.27), let φ DJ;u : Γpuq ÝÑ Γ0,1 J puq

be as in (12.30). Define λR u pXq “

k â ` ˘ λ Tupxi q Y ,

λC u pXq “

i“1

l â ` ˘ λ Tupz ` q X , i“1

i

` ˘ ˘ ` ru D φ , X “ λR pXq˚ bλC pXq˚ bλ D φ . λ u u J J;u

φ is surjective for every For a generic J P Jωφ , the real CR-operator DJ;u r Jq. Thus, rus P M˚k,l pB; ` ` φ ˘ φ ˘ “ λ ker DJ;u λ DJ;u

r Jq is a smooth manifold of dimension ω pBq r and the space M˚k,l pB; in this case. For each i P rks, let r evR i : Mk,l pB; Jq ÝÑ Y,

˘ ` ` ´ evR i ru : Σ ÝÑ X, σ, pxj qjPrks , pzj , zj qjPrls s “ upxi q, be the evaluation morphism for the i-th real marked point. For each i P rls, let r ev` i : Mk,l pB; J, νq ÝÑ X,

˘ ` ` ´ ` ev` i ru : Σ ÝÑ X, σ, pxj qjPrks , pzj , zj qjPrls s “ upzi q, be the evaluation morphism for the positive point of the i-th conjugate pair of marked points. We define ev ”

k l ź ź k l r evR ˆ ev` i i : Mk,l pB; Jq ÝÑ Y ˆX i“1

i“1

to be the total evaluation map.

(13.18)

Counts of Stable Real Rational Maps

Suppose B ‰ 0 or k`2l ě 5. If i P rls, let r Jq ÝÑ M˚ fk,l;i : M˚ pB;

r

k,l´1 pB; Jq

k,l

367

(13.19)

be the forgetful morphism dropping the i-th conjugate pair of marked points. Similar to (13.13), it induces an isomorphism ˘ ` ˘ ` ˘ ` r Jq « f ˚ λ T M˚ pB; r Jq bλ ker dfk,l;i . (13.20) λ T M˚ pB; k,l

k,l;i

k,l´1

r Jq with rus as in (12.27) for pΣ, σq “ pP1 , τ q, For each rus P M˚k,l pB; ker du fk,l;i « Tz ` P1 i

is canonically oriented by the complex orientation of the fiber P1 at zi` . We again denote the resulting orientation of the last factor in (13.20) by o` i . Suppose l P Z` . Given a tuple u as in (12.27) with pΣ, σq “ pP1 , τ q, let C as in (13.9) be its marked domain and D2` Ă P1 be the half-disk cut out by the fixed locus S 1 Ă P1 of τ so that z1` P D2` . A relative Pin˘ -structure (resp. OSpin-structure) p on Y Ă X pulls back to a relative Pin˘ -structure (resp. OSpin-structure) u˚ p on the fixed locus of the real bundle pair u˚ pT X, dφq over pP1 , τ q. If (13.3) holds, (12.13) implies that this real bundle pair is C-balanced as defined above Theorem 7.1. A relative Pin˘ -structure p on Y Ă X then determines φ ˚ an orientation on λR u pXq bλpDJ;u q as in Theorem 7.1(b). Along with the orientation of λC u pXq determined by the symplectic orientation oω of pX, ωq, it thus determines an orientation ˚ ˚ oD p;u ” o C;p pu T X, u dφq

(13.21)

ru pD φ , Xq λ J

for each tuple u as in (12.27) so that of the line r Jq; this orientation varies continuously with u. A relrus P Mk,l pB; ative OSpin-structure p on Y Ă X similarly determines an orientation λRCppq ” λRCpoq on λR u pXq as above the CROrient 3 property φ in Section 7.2, an orientation on λpDJ;u q as in Theorem 7.1(a), φ D r and an orientation op;u on λu pDJ , Xq varying continuously with r Jq. rus P Mk,l pB; If k`2l ě 3, let r Jq ÝÑ Mτ fk,l : M˚ pB; k,l

k,l

be the forgetful morphism dropping the map component u from each tuple u as in (12.27) with pΣ, σq “ pP1 , τ q. If J P Jωφ is generic and

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Spin/Pin-Structures & Real Enumerative Geometry

r Jq, the associated exact sequence u P M˚k,l pB; du fk,l

φ r Jq ÝÝÝÑ Tf puq Mτ ÝÑ 0 “ ker du fk,l ÝÑ Tu M˚k,l pB; 0 ÝÑ ker DJ;u k,l k,l

of vector spaces induces an isomorphism

˘ ` ˚ ˚ C ˚ r λu pevq ” λR u pXq bλu pXq bλu Mk,l pB; Jq ˘ ` ` ˘ ru D φ , X bλf puq Mτ . «λ k,l J k,l

(13.22)

By the previous paragraph, a relative Pin˘ -structure p (resp. OSpinφ r structure p) on Y Ă X determines an orientation oD p;u on λu pDJ , Xq ` φ ˚ D r if (13.3) holds (resp. oD p;u on λu DJ , Xq). For each l P rls, op;u and the orientation ok,l;l˚ of Lemma 13.3 on the second factor in (13.22) in turn determine an orientation op;l˚ ;u on λu pevq. These orientations vary continuously with u and thus determine a relative orientation op;l˚ of the smooth map ev|M˚ pB;Jq r . k,l

r Jq by interchangIf u1 is a marked map obtained from u P M˚k,l pB; ing the points in a conjugate pair pzi` , zi´ q, we identify λC u pXq with pXq via the isomorphism d φ in the i-th factor. This idenλC ` u1 upzi q tification is orientation-preserving with respect to oω if and only if r Jq with the associated marked n P 2Z. For i P rks and u P M˚k,l pB; curve C as in (13.9), let ji puq “ ji p Cq P rks ` ˘ be as in Section 13.2. We set ´1 2 ” 0 as before. The following two lemmas follow immediately from Lemma 13.3 and the CROrient 1os, 1p, and 2 properties in Section 7.2.

Lemma 13.4. Suppose pX, ω, φq is a real symplectic manifold of dimension 2n, Y Ă X φ is a topological component, k, l P Zě0 with rPH r φ pX, Y q, and J P Jωφ is generic. If k and k`2l ě 3, l˚ P rls, B 2 r ” pB, bq satisfy (13.3), then every relative Pin˘ -structure p on B with the Y Ă X determines a relative orientation op;l˚ of ev|M˚ pB;Jq r k,l

following properties: r Jq, the orientations op;l˚ (op 0) If η P H 2 pX, Y ; Z2 q and rus P M˚k,l pB; and oη¨p;l˚ at rus are the same if and only if xη, u˚ rD2` sZ2 y “ 0.

Counts of Stable Real Rational Maps

369

(op 1) The orientations op;l˚ `1 oω and o` l˚ `1 op;l˚ of the composition r Jq M˚k,l`1 pB; ev op;l˚ `1

fk,l`1;l˚ `1 o` l˚ `1



Y k ˆX l`1

(op 2) (op 3)

(op 4)

(op 5)

(op 6)

oω

r Jq / M˚ pB; k,l ev op;l˚

 / Y k ˆX l

are the same (the bottom arrow above drops the l˚ -th component of X l`1 ). The interchange of two real points xi and xj with 2 ď i, j ď k preserves op;l˚ . r Jq, op;l˚ is preserved at rus by the If 2 ď i ď k and rus P M˚k,l pB; interchange of the real points x1 and xji puq with 2 ď i ď k if and only if pk´1qpi´1qn P 2Z. r Jq, and the marked points z ` If 1 ď i, j ď l, rus P M˚k,l pB; i ` and zj of u lie on the same connected component of P1 ´S 1 , then op;l˚ is preserved at rus by the interchange of conjugate pairs pzi` , zi´ q and pzj` , zj´ q. r Jq, op;l˚ is preIf l˚ ă i ď l (resp. 1 ă i ď l˚ ) and rus P M˚k,l pB; served (resp. reversed) at rus by the interchange of the points in a conjugate pair pzi` , zi´ q if and only if n P 2Z. r Jq, op;l˚ is preserved at rus by the interchange If rus P M˚k,l pB; of the points in the conjugate pair pz1` , z1´ q if and only if ˆ ˙ k´1 xc1 pX, ωq, By`1´k ˚ ` l´l `n `xw2 ppq, By 2 2 # ´ pY q; 0, if p P PX “ ` xw2 pXq, By, if p P PX pY q.

r and J are in Lemma 13.5. Suppose pX, ω, φq, n, Y , k, l, l˚ , B, Lemma 13.4. Every relative OSpin-structure os on Y Ă X deterwith the following mines a relative orientation oos;l˚ of ev|M˚ pB;Jq r k,l properties: (oos 0) The statements pop 0q, pop 1q, pop 2q, pop 4q, and pop 5q in Lemma 13.4 with p replaced by os hold and oos;l˚ “ ´p´1qk oos;l˚ .

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r Jq, oos;l˚ is preserved at rus by (oos 3) If 2 ď i ď k and rus P M˚k,l pB; the interchange of the real points x1 and xji puq with 2 ď i ď k if and only if pk´1qpi´1qpn`1q P 2Z. r Jq, oos;l˚ is preserved at rus by the interchange (oos 6) If rus P M˚k,l pB; of the points in the conjugate pair pz1` , z1´ q if and only if ˆ ˙ xc1 pX, ωq, By k´1 `k´1`l´l˚ `pn`1q `xw2 posq, By “ 0. 2 2 The following two lemmas concern properties of the orientations op;l˚ and oos;l˚ in special cases. The two statements of the first lemma follow from the CROrient 3, 5(a), and 6(a) properties in Section 7.2. r J, and p are as Lemma 13.6. Suppose pX, ω, φq, n, Y , k, l, l˚ , B, r satisfies (13.3). in Lemma 13.4 and the pair pk, Bq r “ 0, op;1 is the orientation of λu pevq induced (op 7) If k, l “ 1 and B by the diffeomorphism evR 1 and the orientation oω . (op 8) If o is an orientation on T Y , s is the image of p under the isomorphism (6.11), and os ” po, sq, then op;l˚ “ oos;l˚ . r 2 , and B r Let pX 1 , ω 1 , φ1 q, pX 2 , ω 2 , φ2 q, Y 1 , Y 2 , pX, ω, φq, Y , k, l, B ˚ 1 2 be as in Theorem 13.2(RGW8), l P rls, and J “ J ˆ J for some 1 2 generic J 1 P Jωφ1 and J 2 P Jωφ2 . There is then a natural isomorphism r Jq « Y 1 ˆM˚ pB r 2 ; J 2 q. M˚k,l pB; k,l

(13.23)

r Jq, let ru1 s P M˚ p0; J 1 q and ru2 s P M˚ pB r 2 ; J 2 q be For rus P M˚k,l pB; k,l k,l the projections of rus to X 1 and X 2 . A relative OSpin-structure os1 1 on Y 1 Ă X 1 determines orientations o1 of λpTu1 pP1 q Y 1 q and λR u1 pos q R 1 1 R 1 of λu1 pX q and thus a homotopy class o λu1 pos q of isomorphisms ˘ ` 1 C 1 λ Tu1 pP1 q Y 1 « λR u1 pX qbλu1 pX q. r satisfies (13.3), a relative Pin˘ -structure p2 on If the pair pk, Bq 2 2 Y Ă X determines a homotopy class op2 ;l˚ of isomorphisms ˘ ` r 2 ; J 2 q « λR2 pX 2 qbλC2 pX 2 q. λu2 T M˚k,l pB u u r So does a relative OSpin-structure p2 , whether or not the pair pk, Bq satisfies (13.3).

Counts of Stable Real Rational Maps

371

Along with the isomorphisms (7.23) and (13.23), these homotopy 1 classes determine an orientation po1 λR u1 pos qqop2 ;l˚ of the line λu pevq in (13.22) as below (7.24). Let π 1 , π 2 : X ÝÑ X 1 , X 2 be the two projections. The following statement follows from the CROrient 5(b) property in Section 7.2. Lemma 13.7. With the assumptions as above and p “ xxπ 1˚ os1 , π 2˚ p2 yy‘ , ˆ ˙ ˇ ˇ ` 1 R ˘ pdim X 1 qpdim X 2 q k 1 ˇ ˇ P 2Z. op;l˚ u “ o λu1 pos q op2 ;l˚ u2 iff 4 2

13.4

Definition of Curve Signs

r and J be as in Lemma 13.4. Suppose Let pX, ω, φq, n, Y , k, l, l˚ , B, that either p is a relative Pin˘ -structure on Y Ă X and the pair pk, Bq satisfies the condition (13.3) or p is a relative OSpin-structure on Y . For l1 P rls and a tuple h ” phi : Hi ÝÑ XqiPrl1 s

(13.24)

of maps, define ˚ r Jq Zk,l;h pB; + # l1 ź ˘ ` ` ˚ r 1 “ u, pyi qiPrl1 s P Mk,l pB; Jqˆ Hi : evi puq “ hi pyi q @ i P rl s . i“1

We denote by

` ˘ ˚ r J ÝÑ Y k ˆX l´l1 B; evk,l;h : Zk,l;h

(13.25)

the map induced by (13.18). If J P Jωφ is generic and the maps hi are smooth and generically chosen, (13.25) is a smooth map from a smooth manifold of dimension l1 ÿ ` ˘ ˚ r r J “ ω pBq`k`2l ´ codim hi , dim Zk,l;h B; i“1

where codim hi ” dim X ´dim Hi . Orientations on Hi determine an orientation oh on 1

l ź Hi . Mh ” i“1

(13.26)

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Spin/Pin-Structures & Real Enumerative Geometry

Along with oh and the symplectic orientation oω of X, the relative orientation op;l˚ of Lemma 13.4 or 13.5 determines a relative orientation op;l˚ ;h of the map (13.25). The precise definition of op;l˚ ;h generally depends on the orientation conventions, such as those in Section 5.1 of [7], but there is no ambiguity if the dimensions of all Hi are even. We now restrict to the case l1 “ l so that (13.1) holds. If J P Jωφ is generic and the maps hi are smooth and generically chosen, then (13.25) is a smooth map between manifolds of the same dimen˚ r Jq is a regular point of (13.25), we set sp;l˚ pr r P Zk,l;h pB; u; hq sion. If u to be `1 if the isomorphism ` ˘ ˚ r J ÝÑ Tev pruq Y k B; (13.27) dur evk,l;h : Tur Zk,l;h k,l;h lies in the homotopy class determined by op;l˚ ;h and to be ´1 otherwise. Suppose k`2l ě 5, i P rl˚ s´r1s, the codimension of hi is 2, ` ˘ h1 “ h1 , . . . , hi´1 , hi`1 , . . . , hl , and ` ˘ ` ˘ ˚ r r 1 ” fk,l;i puq, py1 , . . . , yi´1 , yi`1 , . . . , yl q P Zk,l´1;h u 1 B; J . By the genericity assumptions above, the homomorphism Tz ` P1 ‘Tyi Hi ÝÑ Tupz ` q X “ Thi pyi q X, i

i

pv, wq ÝÑ dz ` upvq`dyi hi pwq, i

(13.28)

uq to be `1 if this isomorphism is then an isomorphism. We set si pr is orientation-preserving and to be ´1 if it is orientation-reversing. By (op 1), (op 4), and (op 5) in Lemma 13.4 or the corresponding statements in Lemma 13.5, uq “ si pr uqsp;l˚ ´1;h1 pr u1 q ; sp;l˚ ;h pr

(13.29)

see the proof of the real divisor relation of [6, Proposition 5.2]. Suppose that k`2l ě 5, l1 P rl˚ s, and the codimension of hi is 2 for every i P rl1 s. Let ` ˘ h1 “ hl1 `1 , . . . , hl

Counts of Stable Real Rational Maps

373

˚ r Jq be the image of u r Jq under the r P Zk,l;h and u1 P M˚k,l´l1 `1 pB, pB; composition ˚ r Jq Ă M˚ pB, r Jq Zk,l;h pB; k,l f

˝¨¨¨˝f

k,l´l1 `2;2 k,l;l1 r Jq ÝÝ r Jq ÝÝÝÝÝÝÝÝÝÝÑ M˚k,l´l1 `1 pB, ˆMh ÝÑ M˚k,l pB,

and u “ fk,1`l;1 pu1 q. By (13.29), the sign uqs2 pr uq . . . sl1 pr uqsp;l˚ ;h pr uq P sp;l˚ ´l1 ;h1 puq ” s1 pr



( ˘1

(13.30)

˚ r Jq r P Zk,l;h is independent of generic choices of h2 , . . . , hl1 and u pB; with the pair pu1 , y1 q fixed. By Lemma 13.4(op 4) or the corresponding statement in Lemma 13.5, this sign is also independent of the choices of h1 and pu1 , y1 q with u fixed if l1 ě 2 and the marked points z1` r lie in the same component of P1 ´S 1 . and z2` of u If φ1 is an involution on H1 so that

φ˝h1 “ h1 ˝φ1 : H1 ÝÑ X,

(13.31)

p obtained from u r by exchanging the marked points z1` then the tuple u ˚ r Jq. If in addition and z1´ and replacing y1 by φ1 py1 q lies in Zk,l;h pB; the isomorphism dy1 φ :

Tφph1 py1 qq X Th1 py1 q X ÝÑ dy1 h1 pTy1 H1 q dφ1 py1 q h1 pTφ1 py1 q H1 q

(13.32)

on the normal bundle of h1 induced by φ and φ1 is orientationuq “ s1 pr uq. Along with the conclusions of the previreversing, then s1 pp ous paragraph, this implies that the sign (13.30) is independent of the choices of h1 and pu1 , y1 q with u fixed if the relative orientation op;l˚ r by the interchange of the points in the conjugate is reversed at u uq “ sp;l˚ ;h pp uq). By Lemmas 13.4(op 6) pair pz1` , z1´ q (so that sp;l˚ ;h pr and 13.5(oos 6), the latter is the case if and only if ˆ ˙ k´1 ˚ `xw2 ppq, By l´l `n 2 $ xc pX,ωq,By´1´k ´ 1 ’ `2Z, if p P PX pY q; ’ 2 & ` “ xc1 pX,ωq,By´1´k `xw2 pXq, By, if p P PX pY q; 2 ’ ’ ` ˘ % xc1 pX,ωq,By k´1 ` 2 `k`2Z, if p P OSpX pY q. (13.33) 2

374

Spin/Pin-Structures & Real Enumerative Geometry

The sign sp;l˚ ;h1 pupP1 qq ” sp;l˚ ;h1 puq of upP1 q is then independent of generic choices of the codimension 2 maps hi with i P rl1 s and of l1 (as long as k`2l ě 3). Under the assumptions in C1 and C2 in Section 13.1, (13.33) is equivalent to r ω pBq´k mod 2. l´l˚ – 2 Under the assumptions in C3–C5 in Section 13.1, (13.33) is equivalent to r ω pBq l´l˚ – `k mod 2. 2 By (13.1) and (13.5), both congruences hold. This in particular defines the signs sp;l˚ ;h pCq in (13.4). Suppose pX, ω, φq is a real symplectic fourfold and p1 , p2 are relative Pin´ -structures (resp. Pin` -structures) on Y Ă X satisfying (C1) (resp. (C2)). By the RelSpinPin 2 property on page 81, there exists η P H 2 pX, Y ; Z2 q such that p2 “ η¨p1

and

r φ pX, Y q. xη|X , By “ 0 @ pB, 0q P H 2

(13.34)

r Jq, u is as in (12.18) and (12.20), and D2` Ă P1 is a disk If C P Mp pB; cut out by S 1 , then 2

sp2 ;l˚ ;h pCq “ p´1qxη,u˚ prD` sZ2 qy sp1 ;l˚ ;h pCq ;

(13.35)

this follows immediately from Lemma 13.4(op 0) and the construction of the sign sp;l˚ ;h pCq above. By the condition on η in (13.34), the exponent in (13.35) does not depend on the choice of D2` . This yields a simple sign relation between the functionals (13.8): Dφ,p @ Dφ,p 1 @ (13.36) ¨, . . . , ¨ rB 12s;Y ;k “ p´1qxη,B y ¨, . . . , ¨ rB 11s;Y ;k , if X is compact. If pX, ω, φq is a compact real symplectic sixfold and Y Ă X, the first claim of Lemma 13.5(oos 0) and the RelSpinPin 8 property on page 83 similarly imply that @ Dφ,η¨os @ Dφ,p1 xη,by ¨, . . . , ¨ B;Y ¨, . . . , ¨ B;Y r ;k r ;k “ p´1q

@ os P OSppXq, η P H 1 pY ; Z2 q , r ” pB, bq P H r φ pX, Y q, B 2

(13.37)

Counts of Stable Real Rational Maps

375

if k ‰ 0 or b ‰ 0. The same relation holds in the setting (C5) on page 357 with k “ 0. By the SpinPin 2(b) property on page 9, any Spin-structure on Y determining the same orientation as os equals η ¨os for some η P H 1 pY ; Z2 q. By the last claim of Lemma 13.5(oos 0), r Jq . sos;l˚ ;h pCq “ ´p´1qk sos;l˚ ;h pCq @ os P OSppXq, C P Mp pB; This implies Theorem 13.1(RGW7). Theorem 13.1(RGW5) follows from (13.29). The first statement in (RGW8) in Theorem 13.2 follows from Lemmas 13.7 and 13.6(op 7) (with pX, ω, φq replaced by pX 1 , ω 1 , φ1 q in the latter case). For the purposes of establishing the first cases of the properties (RGW3) r “ 0, and (RGW4) in Theorem 13.1, we take k “ 1, l “ l˚ “ 1, B and h1 “ idX . The claims in these two cases then follow from Lemma 13.6(op 7). The motivation behind (RGW6) is that a Jholomorphic curve passing though a point pR k`1 P Y intersects an infinitesimal sphere SpNpR Y q at two points. By a straightfork`1 ward computation, the two resulting real rational irreducible Jholomorphic curves C Ă X through SpNpR Y q contribute to the k`1 left-hand side in (RGW6) with the same sign as the nearby curve through pR k`1 contributes to the right-hand side; see the proof of Proposition 2.1 in [8]. Suppose i P rl˚ s´r1s (resp. i P rls´rl˚ s) and hi is an automorphism of Hi with the sign of p´1qn (resp. p´1qn`1 ) which satisfies (13.31) ˚ r Jq, the tuple u p obtained r P Zk,l;h pB; with 1 replaced by i. For each u ` ´ r by exchanging the marked points zi and zi and replacing yi from u ˚ r Jq. The isomorphism in (13.32) with 1 by φi pyi q then lies in Zk,l;h pB; replaced by i is orientation-preserving (resp. orientation-reversing) in this case. Along with Lemma 13.4(op 5) or the corresponding statement of Lemma 13.5, this implies that ˘ ˘ ` ` ppP1 q “ ´sp;l˚ ;h u rpP1 q . sp;l˚ ;h u ˚ r Jq vanishes. Combining this pB; Thus, the signed cardinality of Zk,l;h conclusion with (13.5), we obtain Theorem 13.1(RGW2).

376

13.5

Spin/Pin-Structures & Real Enumerative Geometry

Proof of Invariance

The proof of the invariance of the sums (13.4) under suitable topological conditions goes back to [39]. The argument of [39] uses the moduli space of stable nodal disk maps to pX, Y q but can be reformur Jq of stable real genus 0 lated in terms of the moduli space Mk,l pB; r maps to pX, φq as done in [14]. It applies J-holomorphic degree B the general principle, used in [44] and followed in Section 12.5, of showing directly that the sums (13.4) are invariant along generic r ” pht qtPr0,1s of constraints r ” ppt qtPr0,1s of points in Y , h paths p in X or X ´Y , as appropriate, and J ” pJt qtPr0,1s of almost complex structures in Jωφ . After first summarizing the invariance argument in [14,39] under the positivity assumptions in the following, we recast it in terms of pseudocycles as in [6, 8]. In the n “ 2 settings of (C1) and (C2), we assume that (12.26) holds. In the n “ 3 settings of (C3)–(C5), we modify (12.26) to the assumption B ‰ B1 `mB2

if

m ě 2, B1 P H2φ pX; Zqτω , B2 P H2φ pX; ZqRS ω , # D t´1, 0u, if B2 P H2φ pX; Zqτω ; @ c1 pX, ωq, B2 R t´1u, if B2 P H2φ pX; Zqηω . (13.38)

Without these assumptions, strata of multiply covered maps as in Figure 12.3 can have larger than expected dimensions. On the other hand, such strata create no issues in the virtual class approach of [39, Section 7]. We continue with the notation and setup as at the beginning of Section 13.1 and denote by p either a relative Pin´ -structure as in (C1) or (C2) in Section 13.1 or an OSpin-structure as in (C3)–(C5). Let ď ` ˘ ` ˘ r J “ r Jt , ttuˆM˚k,l B; M˚k,l B; tPr0,1s

ď ˘ ˚ `r ˘ ˚ `r Mk,l B; ttuˆMk,l B; J “ Jt , tPr0,1s

ď ` ˘ ` ˘ r J “ r Jt ; Mk,l B; ttuˆMk,l B; tPr0,1s

Counts of Stable Real Rational Maps

377

the last space is compact. The total evaluation map (13.18) extends to a continuous map ˘ ` ˘ ` ˘ ` r J ÝÑ r0, 1sˆY k ˆ r0, 1sˆX l . ev : Mk,l B;

(13.39)

˚ r r Jq of Jq Ă Mk,l pB; For a generic path J in Jωφ , the subspace Mk,l pB; simple maps is stratified by smooth manifolds. The top-dimensional r Jq, consists of maps from P1 . stratum of this subspace, M˚k,l pB; If k ‰ 0 or b ‰ 0 (as is the case under the assumptions in (C1)–(C3) and (C5)), all codimension 1 strata S consist of pairs of real maps from pP1 , τ q with a real point in common, as in (W3) on page 237. The relative orientations op;l˚ of Lemmas 13.4 and 13.5 extend across some codimension 1 strata (those with l˚ pSq – 0, 1 mod 4 below) but r Jq with the codimension 1 not others. We denote the union of M˚k,l pB; r Jq. strata S over which op;l˚ extends by M‹ pB; k,l

r and J as above, the preimr , h, For generic choices of p r under the extended total evalr Jq of the paths p r and h age Mpr ;hr pB; uation map (13.39) is a one-dimensional manifold with ` ˘ r J1 q \ Mp ;h pB; r J0 q. r J “ Mp ;h pB; (13.40) BMpr ;hr B; 1 1 0 0

r Jq with a codimension c stratum of The intersection of Mpr ;hr pB; ˚ r r Jq of codimension c and is Mk,l pB; Jq is a submanifold of Mpr ;hr pB; r Jq with in particular empty for c ě 2. The intersection of M r pB; r ;h p

r Jq inherits an orientation o ˚ r from the relative orientaM˚k,l pB; p;l ;r p;h tion op;l˚ of Lemma 13.4 or 13.5, as appropriate. If op;l˚ extends over ˚ r Jq, then op;l˚ ;rp;hr extends over a codimension 1 stratum S of Mk,l pB; r the intersection of M r pB; Jq with S. A key observation originatr ;h p

r Jq does not intersect the codimension 1 ing in [39] is that Mpr ,hr pB; strata S over which op;l˚ does not extend. By the positivity assumptions in (12.26) and (13.38), ˘ ` ˘ ` ˘ ` r Jq´M˚k,l pB; r Jq Ă r0, 1sˆY k ˆ r0, 1sˆX l ev Mk,l pB; is covered by a smooth map from a manifold of dimension 2 less than r Jq in the setting of (C1)–(C3) and (C5); see the proof that of M˚k,l pB;

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Spin/Pin-Structures & Real Enumerative Geometry

of Proposition 5.2 in [6]. Thus, ` ˘ ` ˘ r J Ă M‹ B; r J Mpr ;hr B; k,l is a compact oriented one-dimensional manifold so that (13.40) r Jq, the orientarespects the orientation op;l˚ ;rp;hr on Mpr ;hr pB; r J1 q, and the opposite of the orientation op;l˚ ;p1 ;h1 on Mp1 ;h1 pB; r tion op;l˚ ;p0 ;h0 on Mp0 ;h0 pB; J0 q. This implies that ÿ ÿ sp;l˚ ;h0 pCq “ sp;l˚ ;h1 pCq r 0q CPMp0 ;h0 pB;J

r 1q CPMp1 ;h1 pB;J

and shows that the functionals (13.6) are well defined. The only case in the setting (C4) not covered by (C3) is k “ 0 and b “ 0. The sum (13.4) then vanishes; see the proof of Proposition 1.3(1) in [8]. ˚ r A codimension 1 two-disk boundary stratum S of Mk,l pB; Jq or ˚ r r of the Mk,l pB; Jq is characterized by the distributions of the degree B map components u of its elements rus, the k real marked points, and the l conjugate pairs of marked points between the two irreducible components of the domains. Supposing that l P Z` , we define r φ pX, Y q, K1 pSq Ă rks, and L1 pSq Ă rls r1 pSq ” pB1 pSq, b1 pSqq P H B 2

to be the degree of the restriction of the map u of the elements rus of S to the irreducible component P11 of the domain carrying the marked points z1˘ , the set of real marked points carried by P11 , and the set of conjugate pairs of marked points carried by P11 , respectively. We define r φ pX, Y q, K2 pSq Ă rks, and L2 pSq Ă rls r2 pSq ” pB2 pSq, b2 pSqq P H B 2

to be the analogous objects for the irreducible component P12 of the domain not carrying the marked points z1˘ . Let rpSq “

$ &1, if 1 R K2 pSq; %2, if 1 P K pSq; 2

ˇ ˇ˘ @ D ` l˚ pSq “ c1 pX, ωq, B2 pSq ´ pn´1qˇK2 pSq| ` 2|L2 pSq ´ rl˚ sˇ .

If (C1) or (C2) holds, Corollary 7.5 implies that the orientaφ r tion oD p;u on the line λu pDJ , Xq extends across S if and only if |K2 pSq|p|K2 pSq|´1q xc1 pX, ωq, B2 pSqypxc1 pX, ωq, B2 pSqy´1q `n 2 2 ˇ ` ˘` ˘` ˘ ˇ ˇ ˇ ` rpSq´1 |K1 pSq|´1 |K2 pSq|´1 ` K2 pSq xc1 pX, ωq, B2 pSqy P 2Z.

Counts of Stable Real Rational Maps

379

If (C3) or (C5) holds, the same is implied by the CROrient 1os(1) property in Section 7.2 and the definition of the orientation λRCposq ” λRCpoq on λR u pXq above the CROrient 3 property. If |K2 pSq| ` τ defined in Section 13.2 2|L2 pSq| ě 2, the orientation ok,l;l˚ on Mk,l τ extends across the image of S in Mk,l under the forgetful morphism fk,l dropping the map component if and only if ˇ ˘` ˘` ˘ ˇ |K2 pSq|p|K2 pSq|´1q ` ` rpSq´1 |K1 pSq|´1 |K2 pSq|´1 ` ˇK2 pSqˇ 2 ˇ ˇ ˇ ` L2 pSq´rl˚ sˇ P 2Z. If any of the conditions (C1)–(C3) or (C5) holds and |K2 pSq| ` 2|L2 pSq| ě 2, the relative orientation op;l˚ of (13.18) defined via (13.22) thus extends across the codimension 1 stratum S of ˚ r ˚ r Mk,l pB; Jq or Mk,l pB; Jq if and only if l˚ pSq ” 0, 1

mod 4.

(13.41)

In light of Lemma 13.4(op 1) or the corresponding statement of Lemma 13.5, as appropriate, the condition |K2 pSq| ` 2|L2 pSq| ě 2 is not necessary for the last conclusion. For J P Jωφ generic (resp. generic path J in Jωφ ), we denote by ‹ r ˚ r r M‹ k,l pB; Jq (resp. Mk,l pB; Jq) the union of Mk,l pB; Jq (resp. r Jq) with the codimension 1 strata S of M˚k,l pB; r Jq (resp. M˚ pB; k,l ˚ r Mk,l pB; Jq)

satisfying (13.41). For tuples ˘ ` ˘ ` r” r r i ÝÑ r0, 1sˆX hi : H h ” hi : Hi ÝÑ X iPrls and h iPrls

of maps, define

#

‹ r Jq ” Zk,l;h pB;

+ l ź ˘ ` r Hi : ev` u, pyi qiPrls P M‹ i puq “ hi pyi q @ i P rls , k,l pB; Jqˆ i“1

# + l ź ` ˘ ‹ r ` r r r u, pyi qiPrls P Mk,l pB; Jqˆ Hi : evi puq “ hi pyi q @ i P rls , r pB; Jq ”

‹ Zk,l; h

i“1

where ev` i is the composition of (13.39) with the projection to the i-th r0, 1sˆX component. Let ‹ r Jq ÝÑ Y k pB; evk,l;h : Zk,l;h

and

r Jq ÝÑ Y k evk,l;hr : Z‹ r pB; k,l;h

(13.42)

380

Spin/Pin-Structures & Real Enumerative Geometry

be the maps induced by the evaluations at the points x1 , . . . , xk of the domains. We define r Jq ÝÑ Y k and ev r : Z r pB; r Jq ÝÑ Y k evk,l;h : Zk,l;h pB; k,l;h k,l;h ‹ r r r as above with M‹ k,l pB; Jq replaced by Mk,l pB; Jq and Mk,l pB; Jq r Jq. replaced by Mk,l pB; Suppose in addition that h is a tuple of smooth maps from oriented even-dimensional manifolds in general position satisfying (13.1) r is a tuple of smooth maps from oriented odd-dimensional borand h dered manifolds in general position so that r “ t1uˆh1 ´ t0uˆh0 Bh

for some tuples h0 , h1 also satisfying the above conditions for h. The map evk,l;h in (13.42) is then a smooth map between manifolds of the same dimension; it inherits a relative orientation op;l˚ ;h from the relative orientation op;l˚ of Lemma 13.4 or 13.5, as appropriate. The domain of the map evk,l;hr in (13.42) is a bordered manifold; this map inherits a relative orientation op;l˚ ;hr from the relative orientation op;l˚ of Lemma 13.4 or 13.5 so that ˘ ` ˘ ` ˘ ` B evk,l;hr , op;l˚ ;hr “ evk,l;h1 , op;l˚ ;h1 ´ evk,l;h0 , op;l˚ ;h0 . (13.43) From now on, we also assume that h, h0 , h1 are generic tuples r is a generic tuple of pseudocycle of pseudocycles satisfying (13.5), h equivalences between the components of t0u ˆ h0 and t1u ˆ h1 as r being pseudodefined in [51] with the dimension n components of h cycle equivalences to X ´Y , and one of the conditions C1–C3 or C5 in Section 13.1 holds. We show in the following that the limit set of the map evk,l;h in (13.42), ď ˘ ` ‹ r Jq´K , evk,l;h Zk,l;h pB; Ωpevk,l;h q ” ‹ r KĂ Zk,l;h pB;Jq

cmpt

is then covered by a smooth map from a manifold of dimension nk´2, ‹ r Jq. This map is thus a codimension 0 pB; i.e. 2 less than that of Zk,l;h pseudocycle; its degree, ÿ sp;l˚ ;h pCq ” N φ;p pY q, deg evk,l;h ” ˚ r r CPMp;h pB;Jq

B;k,l ;h

Counts of Stable Real Rational Maps

381

is independent of a generic choice of p P Y k . We also show that the limit set of the map evk,l;hr in (13.42), Ωpevk,l;hr q, is covered by a smooth map from a manifold of dimension nk´1, i.e. 2 less than that r Jq. This map is thus a pseudocycle equivalence between of Z‹ r pB; k,l;h evk,l;h0 and evk,l;h1 , and so the above degree is also independent of the choice of pseudocycle representatives for the even-dimensional p ˚ pX, Y q and of the choice of a generic J P Jωφ . This elements of H shows that the functionals (13.6) are well defined. By the proof of the second case in Theorem 13.1(RGW4), we can assume that dim hi ď 2n´2 for every i P rls. By the proof of Theorem 13.1(RGW5) in Section 13.4, we can also assume that codim h1 “ 2. Along with (13.5), this implies that # 2, if i P rl˚ s; mod 4 . (13.44) codim hi “– 0, if i P rls´rl˚ s; With dim h denoting the sum of the dimensions of the domains Hi , let l ÿ codim hi . codim h “ 2nl ´ dim h “ i“1

Z ÝÑ X l

be a smooth map from a manifold of dimension Let h : dim h´2 covering Ωph1 ˆ¨ ¨ ¨ˆhl q, ev` ” 

ź

˘ ` l r ev` i : Mk,l B; J ÝÑ X ,

iPrls

( r Jq ” pu, zq P Mk,l pB; r JqˆZ : ev` puq “ hpzq ÝÑ Y k . evk,l;h : Zk,l;h pB; ˚

r Jq, let For a stratum S of Mk,l pB;

ˆ ź ˙ l r Sh ” Zk,l;hpB; Jq X Sˆ Hi i“1

cpSq P Zě0

be the number of nodes of the domains of the elements and of S. r Jq is compact, Since the space Mk,l pB; ˇ ˘ ˘ ` ` r Jq´ Z‹ pB; r Jq Ă evk,l;h Zk,l;h pB; Ω evk,l;h ˇZ‹ pB;Jq r k,l;h k,l;h

` ˘ r Jq . Yevk,l;h Zk,l;h pB;

(13.45)

Spin/Pin-Structures & Real Enumerative Geometry

382

The subspace l ź ‹ r r r Zk,l;hpB; Jq´ Zk,l;hpB; Jq Ă Mk,l pB; Jqˆ Hi i“1

r Jq consists of the spaces Sh corresponding to the strata S of Mk,l pB; with either cpSq ě 2 nodes or l˚ pSq – 2, 3 mod 4 and of the subspace r Jq´M˚k,l pB; r Jq of multiply covered maps meeting the pseuMk,l pB; ˚ r docycles in h. By (13.1) and S being the codimension c in Mk,l pB; Jq, the dimension of Sh is at most nk ´2 if cpSq ě 2. By the positivity assumption (12.26) or (13.38), as appropriate, (13.1), and the reasoning in the proof of Proposition 5.2 in [6], the image under evk,l;h of the r Jq´M˚k,l pB; r Jq meeting the pseudocycles in h can subspace Mk,l pB; also be covered by a smooth map from a manifold of dimension nk´2. The same applies to the last set in (13.45). ˚ r Jq with l˚ pSq – Suppose S is a codimension 1 stratum of Mk,l pB; 2, 3 mod 4. For r “ 1, 2, let rr “ B rr pSq, kr “ |Kr pSq|, lr “ |Lr pSq|, L˚r “ Lr pSqXrl˚ s, B Br “ Br pSq, hr “ phi qiPLr pSq . In particular, k1 ` k2 “ k, l1 ` l2 “ l, codim h1 ` codim h2 “ codim h; D @ D @ c1 pX, ωq, B1 ` c1 pX, ωq, B2 D @ (13.46) “ c1 pX, ωq, B “ pn´1qk`codim h ´ 2l ` 3 ´ n; the last equality holds by (13.1). By the definition l˚ pSq and (13.44), ` ˘ D @ c1 pX, ωq, B2 ´pn´1qk2 ´2 l2 ´|L˚2 | – 2, 3 mod 4, ` D ˘ @ c1 pX, ωq, B2 ´pn´1qk2 ´ codim h2 ´2l2 – 2, 3 mod 4. (13.47) Along with (13.46) and xc1 pX, ωq, B 1 y P 2Z if n “ 3, the second equality in (13.47) gives ` D ˘ @ c1 pX, ωq, B1 ´pn´1qk1 ´ codim h1 ´2l1 – 2, 3 mod 4, ` ˘ D @ c1 pX, ωq, B1 ´pn´1qk1 ´2 l1 ´|L˚1 | – 2, 3 mod 4. (13.48) By the first statement in (13.47) and the second in (13.48), ˘ ` Br , kr , lr , |L˚r | ‰ p0, 0, 0, 0q, p0, 0, 1, 1q @ r “ 1, 2.

Counts of Stable Real Rational Maps

383

Since the image of hi with i P rls´rl˚ s is disjoint from Y , Sh “ H if ˘ ` Br , kr , lr , |L˚r | “ p0, 0, 1, 0q. Thus, we can assume that either Br ‰ 0, or kr `2lr ě 3, or kr “ 2 for each r “ 1, 2. By the definition of P11 , l1 ě 1. Suppose B2 “ 0, l2 , l2˚ “ 0, and k2 “ 2. The restriction of evk,l;h to Sh then factors as ‹ r JqˆM3,0 p0; Jq ÝÑ Y k´2 ˆΔY ÝÑ Y k , Sh ÝÑ Zk´1,l;h pB;

where ΔY Ă Y 2 is the diagonal. Thus, evk,l;h pSh q is again contained in a smooth manifold of dimension nk´2. Suppose now that either Br ‰ 0 or 2lr ` kr ě 3 for each r “ 1, 2. The restriction of evk,l;h to Sh then factors as

Sh

‹ / Z‹ r r k1 ,l1 ;h1 pB1 ; Jq ˆ Zk2 ,l2 ;h2 pB2 ; Jq evk1 ,l1 ;h1





evk2 ,l2 ;h2

Y k1 ˆY k2

/ Y k.

Thus, evk,l;hpSh q is covered by a smooth map from a manifold of dimension rr ; Jq`dim Y k3´r dim Zk‹r ,lr ;hr pB rr q ` kr ` 2lr ´ codim hr ` nk3´r “ ω pB @ D “ c1 pX, ωq, Br ´ pn ´ 1qkr ` 2lr ´ codim hr ` nk ` n ´ 3

(13.49)

for r “ 1, 2. By (13.46), the second statement in (13.47), and the first in (13.48), D @ c1 pX, ωq, Br ´ pn ´ 1qkr ` 2lr ´ codim hr ` n ´ 3 ď ´2 (13.50) for either r “ 1 or r “ 2. Thus, evk,l;hpSh q is covered by a smooth map from a manifold of dimension nk ´ 2 in this case as well. The above shows that the limit set of the map evk,l;h in (13.42) is covered by a smooth map from a manifold of dimension nk ´ 2. The same reasoning shows that the limit set of the map evk,l;hr in (13.42) is covered by a smooth map from a manifold of dimension nk ´ 1. Since codim r hi “ codim hi @ i P rls,

Spin/Pin-Structures & Real Enumerative Geometry

384

(13.46), the second statement in (13.47), the first statement in r Since r (13.48), and (13.50) hold with h replaced by h. hi is a pseuhi docycle equivalence in X ´Y if n “ 3 and dim r hi “ 3, the image of r ˚ with i P rls´ rl s is disjoint from r0, 1sˆ Y in this case. The analog of (13.49) is now rr ; Jq ` dim Y k3´r dim Zk‹ ,l ;hr pB r r r @ D r r ` nk ` n ´ 2. “ c1 pX, ωq, Br ´ pn ´ 1qkr ` 2lr ´codim h 13.6

Signs at Immersions

r J, and p are as at the beginning of Suppose pX, ω, φq, n, Y , k, l, l˚ , B, r Jq Section 13.4 with l ě 2. Let u be as in (12.27) so that rus P M˚k,l pB; and u is an immersion. We denote by C the marked domain of u as in (13.9). The conjugation dφ on T X descends to a conjugation ϕ on the normal bundle u˚ T X ÝÑ T P1 Nu ” Im du for the immersion u so that 0 ÝÑ pT P1 , dτ q ÝÑ u˚ pT X, dφq ÝÑ pNu, ϕq ÝÑ 0 q

du

(13.51)

is an exact sequence of real bundle pairs over pP1 , τ q. Define ˇ ˘ ` ˇ ˇ ˘ ` ˇ Nu ” NuC ‘ NuR ” Nuˇz ` ‘¨ ¨ ¨‘ Nuˇz ` ‘ Nuϕ ˇx1 ‘¨ ¨ ¨‘ Nuϕ ˇx . 3

l

k

We denote by B¯ the real CR-operator on the rank 1 real bun¯ dle pT P1 , dτ q over pP1 , τ q determined by the holomorphic B-operator φ 1 ¯ on T P . The linearization Du ” DJ;u of the BJ -operator on the space of real maps from pP1 , τ q descends to a real CR-operator Du2 on the real bundle pair pNu, ϕq and induces an exact triple 0 ÝÑ B¯ ÝÑ Du ÝÑ Du2 ÝÑ 0

(13.52)

of real CR-operators. Let 2 C ev2C u : ker Du ÝÑ Nu

2 R and ev2R u : ker Du ÝÑ Nu

be the evaluations at the marked points z3` , . . . , zl` and x1 , . . . , xk , respectively.

Counts of Stable Real Rational Maps

385

Let os0 pT S 1 q be the OSpin-structure on S 1 Ă D2` Ă P1 determined by the standard orientation of S 1 as in Example 1.8 and eR be the real part of the short exact sequence (13.51) as in (7.22). By the RelSpinPin 7 property in Section 6.2, the relative Pin˘ structure (resp. OSpin-structure) p on Y Ă X determines a relative Pin˘ -structure (resp. OSpin-structure) p2u on the vector bundle Nuϕ over S 1 Ă P1 so that  ˘ @@ DD (˚ ` os0 pT S 1 q, p2u e “ u|S 1 p P PP˘1 tu|S 1 u˚ T Y R ` ` ˘˘ ˆ resp. OSpP1 tu|S 1 u˚ T Y . (13.53) By Theorem 7.1, p2u in turn determines a homotopy class ` ˘ o2p;u ” o C;p2u Nu, ϕ

(13.54)

of isomorphisms λpker Du2 q « λpNuR q. Let h as in (13.24) with l1 “ l be a tuple of smooth maps from oriented manifolds in general position so that (13.1) holds. Suppose in addition that `

˘

r ” rus, pyi qiPrls P u

˚ r Jq Zk,l;h pB;

Ă

M˚k,l ˆ

l ź Hi

(13.55)

i“1

with u as above is a regular point of (13.25) and the homomorphisms (13.28) with i “ 1, 2 are isomorphisms. Let gur :

l à

Tyi Hi ÝÑ NuC

i“3

be the direct sum of the compositions of the differentials dyi hi with the projections to Nu|z ` and i # + l à 2 2C Kur ” pξ, wq P pker Du q‘ Tyi Hi : evu pξq “ gur pwq . (13.56) i“3

The homotopy class o2p;u of isomorphisms λpker Du2 q « λpNuR q and the orientations of Hi and Nu determine a homotopy class op;h;ru of isomorphisms λpKur q « λpNuR q. By (13.1) and the transversality assumptions, the homomorphism R ev2R r ÝÑ Nu r : Ku u

386

Spin/Pin-Structures & Real Enumerative Geometry

induced by ev2R u is an isomorphism. We set sp;h puq to be `1 if this isomorphism lies in op;h;ru and ´1 otherwise. The following proposition compares this sign with the sign sp;l˚ ;h puq defined by (13.30) uq ” δlc˚ p Cq be as in (13.16). with l1 “ 0. Let δlc˚ pr Proposition 13.8. Suppose pX, ω, φq is a real symplectic manifold of dimension 2n, Y Ă X φ is a topological component, k, l P Zě0 with rPH r φ pX, Y q, J P Jωφ is generic, and that either p l ě 2, l˚ P rls, B 2 ˘ is a relative Pin -structure on Y Ă X and the pair pk, Bq satisfies the condition (13.3) or p is a relative OSpin-structure on Y . Let h as in (13.24) with l1 “ l be a tuple of smooth maps from oriented even-dimensional manifolds in general position so that (13.1) ˚ r Jq be as in (13.55) so that the homomorr P Zk,l;h pB; holds and u phisms (13.28) with i “ 1, 2 are orientation-preserving isomorphisms. The signs sp;l˚ ;h puq and sp;h puq are the same if and only if ˆ ˙ @ D k`1 c r P Z2 . (13.57) uq`pn´1q `pn´1q ` 2Z “ w2 pXq, B δl˚ pr 2 Remark 13.9. Let h2 “ ph3 , . . . , hl q. The sign sp;h puq depends only r Jq obtained by forgetting on the image ru2 s of rus in M˚k,l´2 pB; the marked points z1˘ and z2˘ and possibly on the choice of the disk D2` Ă P1 bounding the real locus S 1 of τ (as determined by the condition z1` P D2` ). If the sign sp;l˚ ´2;h2 pupP1 qq is well defined as in Section 13.4, Proposition 13.8 provides a comparison between sp;l˚ ´2;h2 pupP1 qq and the sign sp;h2 pu2 q ” sp;h puq obtained from the normal bundle to the curve upP1 q and a choice of the half of this curve bounding upS 1 q. With the assumption as in Proposition 13.8, let h1 “ ph1 , h2 q and ` ˘ ˚ ˚ ˚ r r 1 ” rus, py1 , y2 q P Zk,l;h u 1 ” Zk,l;h1 pB; Jq Ă Mk,l ˆY1 ˆY2 r JqˆY1 ˆY2 . ” M˚k,l pB, We can assume that z1` “ 0 and z2` P R` . Let Vu1 ” Tz ` P1 ‘¨ ¨ ¨‘Tz ` P1 ‘Tx1 S 1 ‘¨ ¨ ¨‘Txk S 1 , 3

l

u1 q P Xk,l´2 ” Y k ˆX l´2 . p ” evk,l;h1 pr

(13.58)

Counts of Stable Real Rational Maps

387

By the short exact sequence above (13.22), there is a natural isomorphism ` ˘ (13.59) Trus M˚k,l « ker Du ‘Tz ` R` ‘Vu1 . 2

By the transversality of u to H1 and H2 , ` ˘ ˚ ˚ ` ¯ Tur 1 Zk,l;h “ t0u, Tur 1 Zk,l;h 1 X ker B “ t0u, 1 XT ` R z 2

˚ ˚ and the image of the projection of Tur 1 Zk,l;h 1 ÝÑ Trus Mk,l is of codimension 4. Thus, the isomorphism (13.59) and the exact triple (13.52) of real CR-operators induce a short exact sequence ˚ 2 0 ÝÑ Vu1 ÝÑ Tur 1 Zk,l;h 1 ÝÑ ker Du ÝÑ 0

(13.60)

˚ of vector spaces and a homotopy class of isomorphisms λpTur 1 Zk,l;h 1q « 1 2 λpVu qbλpDu q. Combining the latter with the homotopy class o2p;u of isomorphisms λpker Du2 q « λpNuR q above, the orientation of NuC , and an identification as in (7.23), we obtain a homotopy class op;ru1 of isomorphisms ` ˘ ˘ ` ˚ 1 2 1 λ Tur 1 Zk,l;h 1 « λpVu qbλpDu q « λpVu q b λpN u q “ λ Tp Xk,l´2 (13.61) ˚ q to λpT X q. Another homotopy class o from λpTur 1 Zk,l;h 1 p k,l´2 p;l˚ ;h1 |u r1 of such isomorphisms is obtained at the beginning of Section 13.4 with l1 “ 2. The following lemma is proved in Section 13.7.

Lemma 13.10. If the homomorphisms (13.28) with i “ 1, 2 are orientation-preserving isomorphisms, then the relative orientations r1 op;l˚ ;h1 |ur 1 and op;ru1 of the map evk,l;h1 as in (13.25) with l1 “ 2 at u are the same if and only if (13.57) holds. Proof of Proposition 13.8. By (13.30) with l1 “ 0 and the sentence containing (13.27), sp;l˚ ;h puq is the sign of the differential of r with respect to the relative the map evk,l;h in (13.25) with l1 “ l at u orientation op;l˚ ;h |ur described at the beginning of Section 13.4. Since the dimensions of H3 , . . . , Hl are even, op;l˚ ;h |ur is the relative orien˚ r Jq as r induced by op;l˚ ;h1 |ur 1 by viewing Zk,l;h pB; tation of evk,l;h at u the fiber product of ` ˚ l´2 pev` 3 , . . . , evl q : Zk,l;h1 ÝÑ X

with h3ˆ¨ ¨ ¨ˆhl . We denote by op;h |ur the relative orientation of evk,l;h r similarly induced by op;ru1 . The sign of dur evk,l;h with respect to at u

388

Spin/Pin-Structures & Real Enumerative Geometry

this relative orientation is sp;h puq. Along with Lemma 13.10, this  implies the claim. Proof of Theorem 13.2(RGW9). We continue with the notation and setup in the statement of this theorem. It remains to verify the signs in the last two cases. Let π 1 , π 2 : X ÝÑ X 1 ” P1 , X 2 be the component projections. We take the almost complex structure J on X to be JP1 ˆ J 2 , with JP1 denoting the standard com2 plex structure on P1 and J 2 P Jωφ2 generic. For each i P rls, we take hi “ h1i ˆh2i with h1i “ idP1 if k “ 3 or i ě 2 and h1i being the inclusion of a generic point p11 P P1 if k “ 1 and i “ 1. Let p as in (13.2) be 1R 2R generic with pR i “ ppi , pi q. Define h ” phi qiPrls ,

h2 ” ph2i qiPrls ,

2k p2 ” pp2R . i qiPrks P Y

r Jq. Fix C P Mp;h pB; r be tuples as in (12.27) with pΣ, σq “ pP1 , τ q Let u and u and (13.55) so that their map component u ” pidP1 , u2 q satisfies (12.18) and (12.20). The map u2 then satisfies (12.18) and (12.20) r 2 , and r and C replaced by X 2 , φ2 , u2 , B with X, φ, u, B, r 2 ; J 2 q, C 2 ” π 2 pCq P Mp2 ;h2 pB respectively. Let C as in (13.9) be the marked domain of u. We denote r 2 the tuples obtained from u and u r by replacing the map by u2 and u component u by u2 and the points yi P Hi by their projections yi2 to the domains Hi2 of h2i . Define l ` ˘ à Kur Ă ker Du2 ‘ Tyi Hi i“1

as in (13.56) with the lower limit i “ 3 for the direct sum replaced by i “ 1. The inclusion π 2˚ T X 2 ÝÑ T X induces an isomorphism ` u2˚ pT X 2 , dφ2 q ÝÑ Nu, ϕq of real bundle pairs over pP1 , τ q. This isomorphism identifies the real CR-operator Du2 with Du2 , the OSpin-structures u2˚ os2 with os2u , the differential dyi2 h2i with the composition of the restriction of dyi hi to Tyi2 X 2 with the projection to Nu|z ` , and Tp2 Y 2k with NuR . i

Counts of Stable Real Rational Maps

389

Suppose k “ 3 and thus Hi “ P1 ˆHi2 for every i P rls. The natural isomorphism τ Tr Cs Mk,l « Tz ` P1 ‘¨ ¨ ¨‘Tz ` P1 1

(13.62)

l

is then orientation-preserving if δl˚ p Cq P 2Z. The homomorphism

# l l à à ` 2 2˘ ` ˘ ˚ r B ” ξ, pwi1 qiPrls , pwi2 qiPrls P pker Du2 q‘ Φ : Tu Z ; J Tz ` P1 ‘ Ty 2 Hi2 : 2 r k,l;h2 i“1

i

i“1

+

ξpzi` q`dz ` u2 pwi1 q “ dy 2 h2i pwi2 q @ i P rls i

i

i

ÝÑ Ku r ,

˘ ` ˘ ` Φ ξ, pwi1 qiPrls , pwi2 qiPrls “ ξ, p´wi1 , wi2 qiPrls ,

is an isomorphism. By the previous statement, it identifies the homo˚ R 2 topy class oos2 ;l˚ ;h2 |ur 2 of isomorphisms λpTrru2 s Zk,l;h 2 q « λu2 pX q, defined at the beginning of Section 13.4 with l1 “ l, with the homotopy class oos;h;ru of isomorphisms λpKur q « λpNuR q, defined as above Proposition 13.8, if and only if δl˚ p Cq P 2Z. Furthermore, the diagram ` 2 2˘ ˚ r Tur 2 Zk,l;h 2 B ;J

du r 2 evk,l;h2

/ Tp2 Y 2k

Φ «





ev2R r u

Kur

«

/ NR u

commutes. Thus, the signs sos;l˚ ;h2 pu2 q and sos;h puq of the top and bottom isomorphisms, respectively, in this diagram with respect to the relative orientations oos2 ;l˚ ;h2 |ur 2 and oos;h;ru satisfy sos2 ;l˚ ;h2 pC 2 q ” sos;l˚ ;h2 pu2 q “ p´1qδl˚ p Cq sos;h puq. By Proposition 13.8 and Remark 13.9, sos;l˚ ;h pCq ” sos;l˚ ;h puq “ p´1qδl˚ p Cq sos;h puq. Combining the last two statements, we obtain sos;l˚ ;h pCq “ sos2 ;l˚ ;h2 pC 2 q. This establishes the equality in the second case of Theorem 13.2(RGW9). Suppose k “ 1. Thus, H1 “ tp11 uˆH12 and Hi “ P1 ˆHi2 for every i ě 2. The analog of the isomorphism (13.62) in this case is τ « Tz ` P1 ‘¨ ¨ ¨ ‘ Tz ` P1 . Tr Cs Mk,l 2

l

390

Spin/Pin-Structures & Real Enumerative Geometry

The isomorphism Φ is as in the k “ 3 case, but with w11 “ 0. The rest of the proof of the k “ 3 case applies without any changes and yields sos;l˚ ;h pCq “ sos2 ;l˚ ;h2 pC 2 q. This establishes the equality in the third case of Theorem 13.2(RGW9).  13.7

Proof of Lemma 13.10

We continue with the notation and setup above Lemma 13.10. We orient the individual summands of Vu1 in (13.58) by the complex orientation of P1 and the standard orientation of S 1 Ă D2` . We denote by o1u the orientation of Vu1 obtained from these orientations by ordering the summands Txi S 1 by the position of the marked points xi on S 1 starting from x1 . Fix an orientation o2u on Nu . Let ou be the orientation of Tp Xk,l´2 determined by the orientations o1u and o2u via the last identification in (13.61). The orientations ou on Tp Xk,l´2 and o2u on Nu determine orientao2u on ker Du2 via the homotopy classes oD tions p oD p;u on ker Du and p p;u 2 and op;u of isomorphisms ` ˘ ` ˘ λ Du « λpTp Xk,l´2 q and λ Du2 « λpNu q, as in (13.21) and (13.54), respectively. Let p o1u ” o0 pT P1 , dτ ; oS 1 q be the orientation of ker B¯ determined by the OSpin-structure os0 pT S 1 q on S 1 as in Theorem 7.1(a). The orientation ou on Tp Xk,l´2 and the symplectic orientation oω ˚ on X determine orientations p op;l˚ on Trus M˚k,l and p op;l˚ ;h1 on Tur 1 Zk,l;h 1 via the homotopy classes op;l˚ |rus and op;l˚ ;h1 |ur 1 of isomorphisms ˘ ` ˘ ` λ Trus M˚k,l « λ Th1 py1 q X ‘Th2 py2 q X ‘Tp Xk,l´2 ˘ ` ˘ ` ˚ and λ Tur1 Zk,l;h 1 « λ Tp Xk,l´2 , as below (13.22) and (13.26), respectively. We denote by p op the ori˚ similarly determined by the orientation ou entation on Tur 1 Zk,l;h 1 on Tp Xk,l´2 and the relative orientation op;ru1 of the second isomorphism above defined in the sentence containing (13.61). The relative r 1 are the orientations op;l˚ ;h1 |ur 1 and op;ru1 of (13.25) with l1 “ 2 at u

Counts of Stable Real Rational Maps

391

˚ same if and only if the orientations p op;l˚ ;h1 and p op of Tur 1 Zk,l;h 1 are the same. By the sentence containing (13.15), the isomorphism in the right column of the first diagram in Figure 13.2 respects the opposite oR of the standard orientation on Tz ` R` , the orientation o1u on Vu1 , and the 2 τ if and only if δ c p Cq is even. By (13.53) orientation ok,l;l˚ on T CMk,l l˚ and the CROrient 5 property in Section 7.2, the left column in this diagram respects the orientations p o1u , p oD o2u if and only if p;u , and p `k ˘ pn´1q 2 is even. By the definition of the relative orientation op;l˚ on λu pevq via (13.22), the middle row in this diagram respects the op;l˚ , and ok,l;l˚ . Along with Lemma 8.13 and orientations p oD p;u , p

dim ker Du2 “ n ´ 1 ` xc1 pX, ωq, By ´ 2 , this implies that the middle column in the first diagram of Figure 13.2 op;l˚ , and p o2u if and only if the respects the orientations p o1u ‘oR ‘o1u , p

Figure 13.2. Commutative squares of exact rows and columns for the proof of Lemma 13.10.

392

Spin/Pin-Structures & Real Enumerative Geometry

number

ˆ ˙ ` ˘ k ´ 1q ` dim ker Du2 pk ` 1q 2 ˆ ˙ k`1 c – δl˚ p Cq`pn ´ 1q `pn ´ 1q ` pk ` 1qxc1 pX, ωq, By mod 2 2 (13.63)

δlc˚ p Cq`pn

is even. For i “ 1, 2, let πi : Thi pyi q X ÝÑ Nyi Hi ”

Thi pyi q X dyi ui pTyi Hi q

be the projection to the normal bundle of Hi at yi . We denote by oN;ru1 the orientation on Ny1 H1 ‘ Ny2 H2 induced by the symplectic orientation oω on X and the given orientations on H1 and H2 . Define ` ˘ g0 : ker B¯ ‘Tz ` R` ÝÑ Ny1 H1 ‘ Ny2 H2 , 2 ` ` ˘ ` ˘˘ g0 pξ, vq “ π1 dz ` upξpz1` qq , π2 dz ` upξpz2` q`vq , 1

2

Trus M˚k,l

ÝÑ Ny1 H1 ‘ Ny2 H2 , ˘ ` ˘˘ ` ` ` gpξq “ π1 drus ev` 1 pξq , π2 drus ev2 pξq . g:

Since the isomorphisms (13.28) with i “ 1, 2 are orientationpreserving, the conclusion of Example 7.8 implies that g0 is an isomorphism which respects the orientations p o1u ‘ oR and oN;ru1 . The middle row in the second diagram of Figure 13.2 respects ˚ op;l˚ on Trus M˚k,l , and oN;ru1 the orientations p op;l˚ ;h1 on Tur 1 Zk,l;h 1, p on Ny1 H1 ‘ Ny2 H2 . The left column in this diagram respects the ˚ op on Tur 1 Zk,l;h o2u on ker Du2 . By the orientations o1u on Vu1 , p 1 , and p last sentence of the previous paragraph, the isomorphism g0 respects the orientations p o1u ‘ oR and oN;ru1 . Combining this with the above conclusion concerning the middle column in the two diagrams and op if and only if the rightLemma 8.13, we conclude that p op;l˚ ;h1 “ p hand side of (13.63) is an even number. Along with (13.3), this establishes Lemma 13.10.

Chapter 14

Counts of Real Rational Curves vs. Maps

We conclude by comparing Welschinger’s counts of real rational maps in real symplectic fourfolds and sixfolds pX, ω, φq, as defined intrinsically in [44, 45], with the corresponding invariant counts (13.4) obtained from the moduli theoretic perspective of Chapter 13. These comparisons are provided by Theorems 14.1 and 14.2 in Section 14.1 and illustrated on basic examples of counts of real rational curves in Sections 14.2 and 14.3. In Section 14.4, we deduce Theorems 14.1 and 14.2 from Proposition 13.8. 14.1

Comparison Theorems

Let pX, ω, φq be a real symplectic fourfold and Y Ă X φ be a connected rPH r φ pX, Y q as in (12.19), component of the fixed locus of φ. For B 2 r 2 “ B 2 . Define let B ` ˘ r φ pX, Y q, dY pBq “ B ´φ˚ B, 0 . dY : H2 pX; Zq ÝÑ H 2 Since φ˚ ω “ ´ω, dY pBq2 P 2Z for every B P H2 pX; Zq. r PH r φ pX, Y q is as in (12.19), k, l P Zě0 Suppose in addition B 2 satisfy (12.23), and p is a tuple of distinct points as in (12.24). r Jq of real rational irreFor a generic J P Jωφ , the set Mp pB; r curves passing through the points pR ducible J-holomorphic degree B i ` and pi is then finite. Furthermore, the only singularities of any 393

394

Spin/Pin-Structures & Real Enumerative Geometry

r Jq are simple nodes. As in Section 12.3, we denote curve C P Mp pB; by δE pCq the number of nodes of C of type E as on page 226. As noted in Section 13.1 and justified in Section 13.4, a relative Pin´ -structure p on Y Ă X satisfying (C1) on page 250 determines a sign sp pCq ” sp;0;pp` qiPrls pCq P t˘1u i

r Jq. Let for every C P Mp pB; @ Dφ,p pY q ” pt, . . . , pt B;Y N φ,p r ;k ” loooomoooon r B,l l

ÿ

sp pCq

(14.1)

r CPMp pB;Jq

be the associated invariant curve count (13.6) with point insertions only. If p is a Pin´ -structure on Y , let φ,ιX ppq pY B,l

N φ,p r pY q ” N r B,l

q.

(14.2)

Theorem 14.1. Let pX, ω, φq be a compact real symplectic fourfold and Y Ă X φ be a connected component. There exists a collection ` ˘ r φ pX, Y q ÝÑ Z2 wsp : H 2 pPP´ pY q of group homomorphisms so that ` ˘ wsp dY pB 1 q “ dY pB 1 q2 {2`2Z, wsη¨p pB, bq “ wsp pB, bq`xη, by,

(14.3)

sp pCq “ p´1qwsp pB,bq`l´1`δE pCq

(14.4)

r φ pX, Y q for all p P P´ pY q, η P H 1 pY ; Z2 q, B 1 P H2 pX; Zq, pB, bq P H 2 with B ‰ 0, k, l P Zě0 satisfying p12.23q, p P Y k ˆ X l and J P Jωφ generic, and C P Mp ppB, bq; Jq. Let μp be as in Theorem 11.1. We show in Section 14.4 that (14.4) holds with xc1 pX, ωq, By2 `B 2 ` μp pbq. 2 (14.5) By (11.5), (12.13), and (12.21), this map is a group homomorphism. By (11.5) and (12.16), it satisfies the first property in (14.1). By (11.6), it also satisfies the second property in (14.1). r φ pX, Y q ÝÑ Z2 , wsp : H 2

wsp pB, bq “

Counts of Real Rational Curves vs. Maps

395

By (14.1), (14.4), and (12.25), r

wsp pBq`l´1 φ N r pY q N φ,p r pY q “ p´1q B,l

B,l

r PH r φ pX, Y q, l P Zě0 . @B 2 (14.6)

Thus, Theorem 14.1 establishes a comparison between the counts of real genus 0 J-holomorphic curves in pX, ω, φq with the intrinsic signs of [44] and the counts of such curves with signs dependent on a Pin´ -structure p on Y as in [6, 14, 39]. Suppose next that pX, ω, φq is a compact connected real symplectic sixfold, Y Ă X φ is a connected component, and os is an OSpinr PH r φ pX, Y q as in (12.19) be structure on Y . Let l P Zě0 and B 2 so that k ” ω pBq{2´2l P Zě0

(14.7)

and the conditions in (C3) on page 250 are satisfied. For a generic r Jq of rational irreducible real J-holomorphic J P Jωφ , the set Mp pB; r degree B curves C Ă X passing through a generic tuple p of points as r Jq is an embedded in (12.24) is then finite. Each element C P Mp pB; ˚ curve and is assigned an intrinsic sign sos pCq P t˘1u in [45,46]; see the proof of Theorem 14.2 in Section 14.4. According to [45, Theorem 0.1] in the projective case and [46, Theorem 4.1] in general, the sum ÿ pY q ” s˚os pCq (14.8) W φ,os r B,l

r CPMp pB;Jq

is independent of generic choices of p P Y k ˆX l and J P Jωφ . By the proof of Proposition 1.3 in [8] or Theorem 14.2, this is also the case if the conditions in (C4) on page 250, instead of (C3), are satisfied. As noted in Section 13.1 and justified in Section 13.4, an OSpinstructure os on Y determines a sign sos pCq ” sos;0;pp` qiPrls pCq P t˘1u i

r Jq. Let for every C P Mp pB; @ Dφ,ιY posq pY q ” loooomoooon pt, . . . , pt B;Y N os,φ r ;k ” r B,l l

ÿ

sos pCq

r CPMp pB;Jq

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Spin/Pin-Structures & Real Enumerative Geometry

be the associated invariant curve count (13.6) with point insertions only. The following theorem relates this count to the one in (14.8). Theorem 14.2. Let pX, ω, φq be a compact real symplectic sixfold, Y Ă X φ be a connected component, and os be an OSpin-structure on Y . Then, k`1 sos pCq “ p´1qp 2 q`1 s˚os pCq

(14.9)

rPH r φ pX, Y q satisfying p14.7q and pC3q on for all k, l P Zě0 and B 2 r Jq. page 250, p P Y k ˆX l and J P Jωφ generic, and C P Mp pB; Before establishing Theorems 14.1 and 14.2 in Section 14.4, we deduce some conclusions from them in Sections 14.2 and 14.3. For a compact real symplectic manifold pX, ω, φq with connected fixed locus X φ , let ` ˘ r φ X, X φ . r φ pXq “ H H 2 2 We also omit Y “ X φ from the notation in (12.25), (14.2), and (14.8) in Sections 14.2 and 14.3. 14.2

Basic Examples: Fourfolds

Let pX, ω, φq be a compact real symplectic fourfold with connected rPH r φ pXq fixed locus X φ . Suppose p is a Pin´ -structure on X φ , B 2 ě0 is as in (12.19), and k, l P Z satisfy (12.23). If pX, ω, Jq is projective and Fano, then the number NB of (complex) rational irreducible J-holomorphic degree B curves passing through k`2l generic points in X is finite and independent of the choice of the points. Furthermore, ˇ φ ˇ ˇ φ,p ˇ ˇN ˇ, ˇN ˇ ď NB and N φ , N φ,p – NB mod 2. (14.10) r r r r B,l

B,l

B,l

B,l

Example 14.3. The complex projective plane P2 with the Fubini– Study symplectic form and the standard conjugation ˘ “ ` ‰ τ2 rZ0 , Z1 , Z2 s “ Z0 , Z1 , Z2 , τ2 : P2 ÝÑ P2 , is a compact real symplectic fourfold. The fixed locus of τ2 is the r τ2 pP2 q is freely generated real projective plane RP2 . The group H 2

Counts of Real Rational Curves vs. Maps

397

by the pair pL, LR q consisting of the homology class L of a linearly embedded P1 Ă P2 and the non-zero element LR of H1 pRP2 ; Z2 q. We r τ2 pP2 q with Z. Let p´ ” p´ pRP2 q and p´ ” p´ pRP2 q thus identify H 0 0 1 1 2 be the two Pin´ -structures on RP2 as in Example 1.16. By (14.5) and (11.10), wsp´ pdq “ d,

wsp´ pdq “ 0

0

@ d P Z.

1

Along with (14.6) and (12.7), this implies that τ ,p´ 0

2 N1,0

τ ,p´ 1

2 N1,0

“ 1,

τ ,p´ 0

2 N1,1

τ ,p´ 1

2 “ ´1, N1,1

τ ,p´ 0

2 “ ´1, N2,0

τ ,p´ 1

“ 1,

2 N2,0

τ ,p´ 0

2 “ ´1, N2,1

τ ,p´ 1

2 “ ´1, N2,1

τ ,p´ 0

2 “ 1, N2,2

τ ,p´ 1

2 “ 1, N2,2

“ ´1; “ ´1.

With ωFS denoting the Fubini–Study symplectic form on P1 and π1 , π2 : P1 ˆP1 ÝÑ P1 denoting the component projections, let ω “ π1˚ ωFS `π2˚ ωFS . The group H2 pP1 ˆ P1 ; Zq is freely generated by the homology classes L1 and L2 of the slices P1 ˆ q2 and q1 ˆ P1 , respectively, with q1 , q2 P P1 . We thus identify this homology group with Z2 . By symmetry, 1

1

1

P ˆP P ˆP “ Npb,aq Npa,bq

1

@ a, b P Zě0 .

Since a degree pa, 0q-curve in P1ˆP1 is a degree a cover of a horizontal section, 1

1

P ˆP “ 0 @ a ě 2. Npa,0q

(14.11)

An irreducible degree p1, bq-curve in P1ˆP1 corresponds to the graph of a ratio of two degree b polynomials on C. Since every such rational function is determined by its values at 2b`1 points, 1

1

P ˆP “1 Np1,bq

@ b P Zě0 .

(14.12)

With τ1 : P1 ÝÑ P1 as in (12.6), the involutions ` ˘ 1 : P1 ˆP1 ÝÑ P1 ˆP1 , τ1,1 pq1 , q2 q “ τ1 pq1 q, τ1 pq2 q , τ1,1 , τ1,1 ` ˘ 1 pq1 , q2 q “ τ1 pq2 q, τ1 pq1 q , τ1,1

398

Spin/Pin-Structures & Real Enumerative Geometry

on P1 ˆ P1 are anti-symplectic with respect to ω. Their fixed loci are RP1 ˆRP1 and (  grpτ1 q ” pq, τ1 pqqq : q P P1 « S 2 , respectively. r τ1,1pP1 ˆP1 q is freely generated by the Example 14.4. The group H 2 R pairs pL1 , LR 1 q and pL2 , L2 q, where ` 1 ˘ R 1 LR 1 , L2 P H1 RP ˆRP ; Z2 are the homology classes of the slices RP1 ˆq2R and q1R ˆRP1 , respecr τ1,1pP1 ˆP1 q with Z2 . By tively, with q1R , q2R P RP1 . We thus identify H 2 symmetry, τ

@ a, b, l P Zě0 .

τ

1,1 1,1 “ Npb,aq,l Npa,bq,l

(14.13)

By (14.11) and (14.12), 1,1 “ 0 @ a ě 2, l P Zě0 Npa,0q,l

τ

and (14.14)

1,1 1,1 1,1 , Np1,bq,1 , . . . , Np1,bq,b “ 1 @ b P Zě0 , Np1,bq,0

τ

τ

τ

´ 1 2 1 2 respectively. Let p´ 0 be the Pin -structure on pRP q “ pS q corresponding to the canonical OSpin-structure os0 of Example 11.4 via the bijection (1.15). By (14.5) and (11.9),

wsp´ pa, bq “ a`b

@ a, b P Z.

0

Along with (14.6), (14.13), and (14.14), this implies that ,p´

,p´

1,1 0 1,1 0 “ Npb,aq,l @ a, b, l P Zě0 , Npa,bq,l

τ

τ

τ

,p´

1,1 0 Npa,0q,l “0

,p´

@ a ě 2, l P Zě0 ,

1,1 0 “ p´1qb`l @ b, l P Zě0 , l ď b. Np1,bq,l

τ

(14.15)

1

r τ1,1pP1 ˆ P1 q is freely generated by Example 14.5. The group H 2 1 r τ1,1pP1 ˆP1 q with Z. By (14.12), pL1 `L2 , 0q. We thus identify H 2

1 τ1,1

τ1

1,1 “ 1. N1,0 , N1,1

(14.16)

Counts of Real Rational Curves vs. Maps

399

´ 2 Let p´ 0 be the unique Pin -structure on S . By (14.5) and the conclusion of Example 11.3,

wsp´ pdq “ d

@ d P Z.

0

Along with (14.6) and (14.16), this implies that τ 1 ;p´ 0

1,1 N1,0

14.3

“ 1,

τ 1 ;p´ 0

1,1 N1,1

“ ´1.

Basic Examples: Sixfolds

We next obtain some counts of real rational curves in P3 with the standard conjugation and in pP1 q3 with two different conjugations. For a compact connected real symplectic sixfold pX, ω, φq with connected fixed locus X φ , we identify an element of H˚ pX; Qq (resp. H˚ pX ´ X φ ; Qq) and its Poincare dual in H ˚ pX; Qq (resp. H ˚ pX, X φ ; Qq). If ψ is an involution of pX, ω, φq satisfying the conditions in (C4) on page 250, Dφ,os Dφ,ψ˚ os @ @ A1 , . . . , Al B;Y r ;k “ ψ˚ pA1 q, . . . , ψ˚ pAl q B;Y r ;k ` ˘ p ˚ X, X φ bZ Q. @ A1 , . . . , Al P H (14.17) 14.3.1

The Projective Space P3

The complex projective space P3 with the Fubini–Study symplectic form and the standard conjugation τ3 as in (12.6) is a compact real symplectic sixfold. The fixed locus of τ3 is the real projective plane RP3 . We denote by L P H2 pP3 ; Zq the homology class of a linearly embedded P1 Ă P3 and by LR P H1 pRP3 ; Z2 q the non-zero r τ3 pP3 q are freely generated by L element. The groups H2 pP3 ; Zq and H 2 R and pL, L q, respectively; we thus identify them with Z. The involution ` ˘ ψ : P3 ÝÑ P3 , ψ rZ0 , Z1 , Z2 , Z3 s “ rZ0 , Z1 , Z2 , ´Z3 s, satisfies the conditions in (C4) on page 250 with Y “ RP3 ; the same is the case of any automorphisms of P3 exchanging a pair of its homogeneous coordinates. By the n “ 3 case of (1.35) and (1.13), ψ ˚ os ‰ os

@ os P OSppRP3 q,

(14.18)

400

Spin/Pin-Structures & Real Enumerative Geometry

i.e. ψ reverses both the orientation and the Spin component of every OSpin-structure on RP3 . As explained in Section 2.6 of [8], the lines P1 Ă P3 contained in P3 ´ RP3 determine two distinct homology classes Lc` , Lc´ in P3 ´ RP3 so that “ ‰ (14.19) Lc` ´Lc´ “ SpNy RP3 , osq ; for an OSpin-structure os on RP3 , the right-hand side in (14.19) is as above (13.7). If H1 , H2 Ă P3 are two projective hyperplanes intersecting transversally in P3 ´RP3 so that H1 XH2 represents Lc` , then H1 X τ3 pH2 q represents Lc´ . By (14.19), Lc` and Lc´ freely generate H2 pP3 ´RP3 ; Zq and are interchanged by ψ. For d, a, b P Zě0 with a`2b “ 4d, let Nd;a,b P Zě0 be the number of (complex) rational irreducible holomorphic degree d curves passing through a generic lines and b generic points in P3 . For d, a, b P Zě0 and an OSpin-structure os on RP3 , let A Eτ3 ,os τ3 ,os c c c c “ pt, . . . , pt , pL `L q{2, . . . , pL `L q{2 Nd;a,b ` ´ ` ´ loooomoooon loooooooooooooooooomoooooooooooooooooon d;RP3 ;2d´a´2b . b

a

τ3 ,os P Z. Similar to (14.10), By Theorem 13.1(RGW6) and (14.19), Nd;a,b ˇ τ3 ,os ˇ Nd;0,2d ´ ˇWd;b ˇ P 2Zě0 if b ď d, ˇ ˇ Nd;2a,2d´a ´ ˇN τ3 ,os ˇ P 2Zě0 if a ` 2b ď 2d. d;a,b

By (14.18), (13.37), and Theorem 13.1(RGW7), τ3 ,ψ Nd;a,b

˚ os

τ3 ,os “ ´p´1qd´a Nd;a,b .

Along with (14.17) and Theorem 14.2, this gives τ3 ,os τ3 ,os “ 0 if d´a P 2Z, Wd;b “ 0 if d P 2Z, Nd;a,b τ3 ,ψ Nd;a,b

˚ os

τ3 ,os τ3 ,ψ “ Nd;a,b , Wd;b

˚ os

τ3 ,os “ Wd;b .

(14.20)

Since there is a unique line passing through 2 general points in P3 , N1;0,1 “ 1. Along with the definition of the sign s˚os pCq in [45, 46] and the k “ 2, 0 cases of (14.9), this implies that τ3 ,os τ3 ,os “ W1;1 “ ˘1, W1;0

τ3 ,os τ3 ,os τ3 ,os ´N1;0,0 “ N1;0,1 “ ´W1;1 “ ¯1.

The first equality in the second equation above is also implied by the extended WDVV equations for pP3 , ω, τ3 q predicted in [2] and established in [8].

Counts of Real Rational Curves vs. Maps

401

Let os0 ” os0 pRP3 q and os1 ” os1 pRP3 q be the two OSpinstructures on RP3 defined in Example 1.17. By the third statement τ3 ,os0 τ3 ,os1 and Nd;a,b are invariant under linin (14.20), the numbers Nd;a,b 3 ear reparametrizations of pP , τ3 q; this implication is a special case of Theorem 1.6 in [21]. It also follows from the extended WDVV τ3 ,os from equations for pP3 , ω, τ3 q, which determine all numbers Nd;a,b τ3 ,os the single input N1;0,1 ; see Section 1.4 in [2] and Section 7 in [9]. The holomorphic normal bundle NP3 P1 of any real holomorphic line P1 Ă P3 is isomorphic to two copies of a rank 1 real bundle pair pV, ϕq over pP1 , τ q of degree 1. In particular, NP3 P1 is balanced in the sense of [45, Section 2.2] and [46, Section 1.2.2]. As noted in Remark 5.2, the base OSpin-structure on T pRP3 q|RP1 in [45,46] is the OSpin-structure os0 |RP1 . Along with the definition of the sign s˚os pCq in [45, Section 2.2] and [46, Section 3.2], this gives τ3 ,os0 τ3 ,os0 τ3 ,os1 τ3 ,os1 “ W1;1 “ 1 and W1;0 “ W1;1 “ ´1. W1;0

Along with the k “ 2, 0 cases of (14.9), this implies that τ3 ,os0 τ3 ,os0 “ N1;0,1 “ ´1 and ´N1;0,0

τ3 ,os1 τ3 ,os1 ´ N1;0,0 “ N1;0,1 “ 1.

(14.21)

By the paragraph above Theorem 1.5 in [20], the value of `1 for τ3 ,os0 obtained in Example 6.3 in [11] corresponds to the orientation N1;0,1 of (13.22) with the orientation on the last factor on the right-hand side as in [18, 20]. As noted in Section 13.2, the last orientation is the opposite of the orientation o0,l used in this manuscript. Thus, the second equality in (14.21) is consistent with the m, l “ 1, t1 “ 3 statement of Example 6.3 in [11]. 14.3.2

The Sixfold pP1 q3

We now take X “ pP1 q3 with the symplectic form ω “ π1˚ ωFS `π2˚ ωFS `π3˚ ωFS , where π1 , π2 , π3 : pP1 q3 ÝÑ P1 are the component projections. The group H2 ppP1 q3 ; Zq is freely generated by the homology classes L1 , L2 , L3 of the slices P1 ˆq2 ˆq3 ,

q1 ˆP1 ˆq3 ,

respectively, with q1 , q2 , q3 P P1 .

and q1 ˆq2 ˆP1 ,

(14.22)

402

Spin/Pin-Structures & Real Enumerative Geometry

For d ” pd1 , d2 , d3 q with each di P Z, let |d| “ d1 `d2 `d3 ,

˘ ` Bpdq “ d1 L1 `d2 L2 `d3 L3 P H2 pP1 q3 ; Z .

If in addition a ” pa1 , a2 , a3 q with each ai P Zě0 and b P Zě0 are so that pa1 `a2 `a3 q`2b “ 2pd1 `d2 `d3 q,

(14.23)

we define Nd;a,b P Zě0 to be the number of (complex) rational irreducible holomorphic degree Bpdq curves passing through a1 , a2 , a3 generic slices (14.22) representing L1 , L2 , L3 , respectively, and b generic points in pP1 q3 . 1 The involutions φ ” τ1 ˆτ1,1 and φ1 ” τ1 ˆτ1,1 on pP1 q3 are antisymplectic with respect to ω. Their fixed loci Y are pRP1 q3 and RP1 ˆ grpτ1 q, respectively. In both cases, we take Y to be the fixed locus. The involution ψ : pP1 q3 ÝÑ pP1 q3 , ˘ ` ψ rZ1 , W1 s, rZ2 , W2 s, rZ3 , W3 s ˘ ` “ rZ1 , ´W1 s, rZ2 , W2 s, rZ3 , W3 s , satisfies the conditions in (C4) on page 250. We take the OSpin-structures on the two fixed loci to be DD @@ os ” π1˚ os0 pT S 1 q, xxπ2˚ os0 pT S 1 q, π3˚ os0 pT S 1 qyy‘ ‘ and @@ DD ˚ os0 pT S 2 q ‘ , os1 ” π1˚ os0 pT S 1 q, π23 where os0 pT S 1 q is as in Theorem 13.2(RGW9), os0 pT S 2 q is the unique OSpin-structure for either orientation on S 2 , and π1 , π2 , π3 : pRP1 q3 ÝÑ RP1 « S 1 1

and

1

π1 , π23 : RP ˆgrpτ1 q ÝÑ RP , grpτ1 q « S 2 are the component projections. Since ψ ˚ os “ os and ψ ˚ os1 “ os1 , Theorem 13.1(RGW7) and (14.17) imply that @ Dφ,os Dφ1 ,os1 @ and A1 , . . . , Al B;RP A1 , . . . , Al B;pRP 1 q3 ;k “ 0 1ˆgrpτ q;k “ 0 r r 1 ` ˘ p 2˚ pP1 q3 , Y s.t. ψ˚ pAi q “ Ai @ i. @ k P 2Z, A1 , . . . , Al P H (14.24)

Counts of Real Rational Curves vs. Maps

403

r φ ppP1 q3 q is freely generated by the pairs Example 14.6. The group H 2 ` ` ` ˘ ˘ ˘ r 1 ” L1 , LR r 2 ” L2 , LR r 3 ” L3 , LR L L and L 1 , 2 , 3 , R R 1 3 where LR 1 , L2 , L3 P H2 ppRP q ; Z2 q are the homology classes of the slices

RP1 ˆq2R ˆq3R ,

q1R ˆRP1 ˆq3R ,

and

q1R ˆq2R ˆRP1 ,

respectively, with q1R , q2R , q3R P RP1 . The space of the slices (14.22) of each kind contained in pP1 q3 ´pRP1 q3 is path-connected and thus determines a homology class Lci in pP1 q3 ´pRP1 q3 . For d ” pd1 , d2 , d3 q and a ” pa1 , a2 , a3 q with di , ai P Zě0 and b P Zě0 , define r r 1 `d2 L r 2 `d3 L r 3, Bpdq “ d1 L

kd pa, bq “ |d|´pa1 `a2 `a3 q´2b,

φ,os φ,os “ WBpdq,b , Wd;b r

A Eφ,os φ,os Nd;a,b “ loooomoooon pt, . . . , pt, looooomooooon Lc1 , . . . , Lc1 , looooomooooon Lc2 , . . . , Lc2 , looooomooooon Lc3 , . . . , Lc3 1 q3 ;k r Bpdq;pRP b

a1

a2

a3

d pa,bq

.

Similar to (14.10), ˇ φ,os ˇ ˇ P 2Zě0 if 2b ď |d|, Nd;0,|d| ´ ˇWd;b ˇ φ,os ˇ ˇ P 2Zě0 if kd pa, bq ě 0. Nd;2a,2b`kd pa,bq ´ ˇNd;a,b If h is the diffeomorphism interchanging a pair of components of pP1 q3 , h˚ os “ os. Combining this with Theorem 13.1(RGW7) and (14.24), we obtain φ,os φ,os φ,os “ Npd “ Npd . Npd 1 ,d2 ,d3 q;pa1 ,a2 ,a3 q,b 2 ,d1 ,d3 q;pa2 ,a1 ,a3 q,b 1 ,d3 ,d2 q;pa1 ,a3 ,a2 q,b (14.25) ´ With p0 as in Example 14.4, $ &N τ1,1 ,p´ 0 φ,os pd2 ,d3 q,a1 , if a1 “ d2 `d3 ´1, a2 , a3 , b “ 0; Np0,d2 ,d3 q;pa1 ,a2 ,a3 q,b “ %0, otherwise;

(14.26)

404

Spin/Pin-Structures & Real Enumerative Geometry

by Theorem 13.2(RGW8). By Theorems 13.2(RGW9) and 13.1 (RGW5), φ,os Np1,d 2 ,d3 q;pa1 ,a2 ,a3 q,b $ 0, ’ ’ ’ ’ ´ ’ ’ &N τ1,1 ,p0 , pd2 ,d3 q,a1 “ τ ,p´ 0 ’ ’ , Npd1,1 ’ 2 ,d3 q,a1 `1 ’ ’ ´ ’ %d N τ1,1 ,p0 , 2

pd2 ,d3 q,a1

if d2 `d3 ą 0, a1 ą d2 `d3 `a2 `a3 ´2; if a1 “ d2 `d3 ´2, a2 , a3 , b “ 0; if a1 “ d2 `d3 ´2, a2 , a3 “ 0, b “ 1; if a1 “ d2 `d3 ´1, a2 , b “ 0, a3 “ 1.

The holomorphic normal bundle N of any real slice Li Ă pP1 q3 is isomorphic to two copies of a rank 1 real bundle pair pV, ϕq over pP1 , τ q of degree 0 and is thus balanced. The base OSpinstructure on T ppP1 q3 q|LR in [45,46] is the OSpin-structure os|LR or its i i conjugate os|LR . Along with the definition of the sign s˚os pCq in [45,46] i and the k “ 1 cases of Theorem 13.1(RGW7) and (14.9), this gives φ,os “ 1, Wp1,0,0q;0

The

same

applies

to

φ,os Np1,0,0q;p0,0,0q,0 “ 1.

the

numbers

(14.27)

φ,os φ,os , Wp0,0,1q;0 Wp0,1,0q;0

and

φ,os φ,os , Np0,0,1q;p0,0,0q,0 . These statements agree with (14.25), Np0,1,0q;p0,0,0q,0 (14.26), and the b, l “ 0 case of (14.15).

Example 14.7. Let L1 , L2 , L3 be as in Example 14.7 and LR 1 P H1 pRP1 ˆ grpτ1 q; Z2 q be the homology class of the slice RP1 ˆ q2 ˆ τ1 pq2 q with q2 P RP1 . The pairs ` ˘ r 1 ” L1 , LR L 1

and

` ˘ r 23 ” L2 `L3 , 0 L

r φ1ppP1 q3 q. The spaces of the slices then freely generate the group H 2 P1 ˆq2 ˆq3 with q2 , q3 P P1 and (  q1 ˆ pq2 , hpq2 qq : q2 P P1 with q1 P P1 , h P AutpP1 q contained in pP1 q3 ´RP1 ˆgrpτ1 q are path-connected and thus determine homology classes Lc1 and Lc23 in pP1 q3 ´RP1 ˆgrpτ1 q.

Counts of Real Rational Curves vs. Maps

405

For d ” pd1 , d23 q and a ” pa1 , a23 q with di , ai P Zě0 and b P Zě0 , define r 1 `d23 L r 23 , kd pa, bq “ pd1 `2d23 q´pa1 `a23 q´2b, r Bpdq “ d1 L 1

1

1

1

φ ,os Wd;b “ W φr ,os , Bpdq,b

Dφ1 ,os1 @ φ1 ,os1 “ pt, . . . , pt, loooomoooon Lc1 , . . . , Lc1 , loooooomoooooon Lc23 , . . . , Lc23 Bpdq;RP Nd;a,b 1 ˆgrpτ q;k pa,bq . r loooomoooon 1 d a1

b

a23

Similar to (14.10),

ˇ φ1 ,os1 ˇ ˇ P 2Zě0 Npd1 ,d23 ,d23 ;2a1 ,2a2 ,2a3 ,2b`kd pa,bq ´2´a23 ˇNpd 1 ,d23 q;pa1 ,a23 q,b a2 , a3 P Zě0 , a2 `a3 “ a23 , kd pa, bq ě 0.

if

With p´ 0 as in Example 14.5, $ 1 ´ & τ1,1 ,p0 1 1 Nd23 ,a1 , if a1 “ 2d23 ´1, a23 , b “ 0; φ ,os “ Np0,d 23 q;pa1 ,a23 q,b %0, otherwise; by Theorem 13.1(RGW5),

1

13.2(RGW8).

1

φ ,os “ Np1,d 23 q;pa1 ,a23 q,b

$ ’ ’0, ’ ’ ’ τ 1 ,p´ ’ &N 1,1 0 ,

By

Theorems

13.2(RGW9)

and

if d23 ą 0, a1 ą 2d23 `a23 ´2;

d23 ,a1 1 ,p´ τ1,1 0 ’ ’ Nd23 ’ ,a1 `1 , ’ ’ ’ 1 ,p´ τ1,1 % 0 2d23 Nd23 ,a1 ,

if a1 “ 2d23 ´2, a23 , b “ 0; if a1 “ 2d23 ´2, a23 “ 0, b “ 1; if a1 “ 2d23 ´1, a23 “ 1, b “ 0.

Similar to (14.27), 1

1

φ ,os “ 1, Wp1,0q;0

14.4

1

1

φ ,os Np1,0q;p0,0q,0 “ 1.

Proofs of Theorems 14.1 and 14.2

We deduce (14.4) and (14.9) from Proposition 13.8. We take r ” pB, bq, J, pX, ω, φq, Y , k, l, B ˘ ` R ` ` k φ l p ” pp1 , . . . , pR k q, pp1 , . . . , pl q P Y ˆpX ´X q ,

406

Spin/Pin-Structures & Real Enumerative Geometry

r Jq, and p or os as in Theorems 14.1 and 14.2. Let hi C P Mp pB; r be be the inclusion of p` i into X, h ” ph1 , . . . , hl q, and u and u tuples as in (12.27) with pΣ, σq “ pP1 , τ q and (13.55) so that their map component u satisfies (12.18) and (12.20). Let C as in (13.9) be the marked domain of u. In particular, u is an immersion. We 2R 2 2 define pNu, ϕq, Nu , NuR , Du2 , ev2C u , evu , pu , op;u , and sp;h puq as in Section 13.6 but take ˇ ˇ NuC “ Nuˇz ` ‘ ¨ ¨ ¨ ‘ Nuˇz ` . 1

l

p2u ,

o2p;u ,

In the case of Theorem 14.2, we denote and sp;h puq by os2u , 2 oos;u , and sos;h puq, respectively. By definition, sp;h puq “ `1 if and only if the isomorphism ` 2C 2R ˘ (14.28) evu , evu : ker Du2 ÝÑ Nu lies in the homotopy class of isomorphisms λpker Du2 q « λpNu q determined by o2p;u and the symplectic orientation of X. Proof of (14.4). Since p is generic, the loop uR ” u|S 1 in Y is admissible in the sense of Theorem 11.1. Thus, p2u “ p´ 0 pNuR q if and only if 1`μp pbq`δH pCq “ 0 P Z2 .

(14.29)

Along with the CROrient 2(b) property on page 113, this implies that the homotopy class o2p;u of isomorphisms λpker Du2 q « λpNuR q is the same as the “canonical” homotopy class o´ C;0 pNu, ϕq determines ´ by p0 pNuR q if and only if (14.29) holds. By Corollary 7.7, the isomorphism (14.28) lies in the homotopy class of such isomorphisms determined by o´ C;0 pNu, ϕq and the symplectic orientation of X if and c only if δ0 p Cq P 2Z. Thus, c

sp;h puq “ p´1qδ0 p Cq`1`μp pbq`δH pCq .

(14.30)

By Proposition 13.8 and Remark 13.9, sp pCq ” sp;0;h puq “ p´1qδ0 p Cq`p c

q`1`xw2 pXq,By s

k`1 2

p;h puq.

Combining this with (14.30), we obtain sp pCq “ p´1q

kpk`1q `xw2 pXq,By`μp pbq`δH pCq 2

.

Along with (12.22) and (12.23), this establishes (14.4).



Counts of Real Rational Curves vs. Maps

407

Proof of (14.9). By (14.7), the degree of Nu is 2pk ` 2l ´ 1q in this case. By [4, Proposition 4.1], the real bundle pair pNu, ϕq is thus isomorphic to 2pV, ψq for any real line bundle pair pV, ψq of degree k ` 2l ´ 1. This isomorphism can be chosen so that it identifies the orientation of Nϕ determined by os2u with the canonical orientation on 2V ψ as in Example 1.13(a). By the SpinPin 2(b) property on page 9, there are two OSpinstructures compatible on Nuϕ “ 2V ψ with its canonical orientation. If k R 2Z, we denote by os0 pNuϕ q the canonical OSpin-structure on Nuϕ as in Example 1.13(b). This is the preferred OSpin-structure os‹pNuϕ q in the first case in Section 1.2.2 in [46]. If k P 2Z, we denote by os0 pNuϕ q the OSpin-structure os0 p2V ψ , o´ q as in Examples 3.7 Vψ q corresponds to the and 5.1; under the bijection (1.20), os0 p2V ψ , o´ Vψ ´ ´ ψ ψ canonical Pin -structure p0 pV q on V with respect to the orientation of S 1 as the boundary of D2` Ă P1 . As noted in Remark 5.2, the preferred OSpin-structure os‹pNuϕ q in the second case in [46, Section 1.2.2] is os1 p2V ψ , o´ q. By the definition in [46, Section 3.2], Vψ s˚os pCq “ `1 if and only if p2u “ os‹pNuϕ q. Since the degree of the line bundle V is at least ´1, a real CRoperator DpV,ψq on the real bundle pair pV, ψq is surjective and the evaluation homomorphism ˇ ˘ ` ˘ ` ˇ ker DpV,ψq ÝÑ Vu ” VuC‘VuR ” Vz ` ‘¨ ¨ ¨‘Vz ` ‘ V ψ ˇx1 ‘¨ ¨ ¨‘V ψ ˇx 1

k

l

is an isomorphism. By the surjectivity of DpV,ψq , λpker DpV,ψq q “ λpDpV,ψq q. By Proposition 8.9(2), the natural isomorphism ` ˘ ` ˘ ` ˘ ker D2pV,ψq “ ker DpV,ψq ‘DpV,ψq « ker DpV,ψq ‘ ker DpV,ψq is orientation-preserving with respect to the orientation o0 pNuϕ q on the left-hand side induced by the OSpin-structure os0 pNuϕ q on Nuϕ and the canonical orientation on the right-hand side. Thus, the sign of the evaluation isomorphism ker D2pV,ψq ÝÑ Nu

(14.31)

with respect to the orientation o0 pNuϕ q on the left-hand side and the orientation on the right-hand side induced by the orientations of Nu k and Nuϕ is p´1qp2q .

408

Spin/Pin-Structures & Real Enumerative Geometry

By the last two paragraphs and the CROrient 2(a) property on page 112, the sign of the evaluation isomorphism (14.31) with respect to the orientation on the left-hand side induced by the OSpinstructure os‹pNuϕ q and the orientation on the right-hand side induced k by the orientations of Nu and Nuϕ is p´1qp2q`k`1 . Along with the definition of the sign s˚os pCq and the sentence containing (14.28), this gives k`1 s˚os pCq “ p´1qp 2 q`1 sos;h puq.

(14.32)

By Proposition 13.8 and Remark 13.9, sos pCq ” sos;l;h puq “ sos;h puq. Combining this with (14.32), we obtain (14.9).



Appendices

A

ˇ Cech Cohomology

ˇ This appendix contains a detailed review of Cech cohomology, including for sheaves of non-abelian groups, describes its connections with singular cohomology and principal bundles, and classifies oriented vector bundles over bordered surfaces. We carefully specify the assumptions required in each statement. We generally follow the perspective of [43, Chapter 5]. In particular, a sheaf S over a topological space Y is a topological space along with a projection map π : S ÝÑ Y so that π is a local homeomorphism. For a sheaf S of modules over a ring R as in [43] and in Section A.1, Sy ” π ´1 pyq is a module over R for every y P Y and the module operations are continuous with respect to the topology of S. For a sheaf S of groups (not necessarily abelian), as in Sections A.2–A.4, Sy is a group for every y P Y and the group operations are continuous with respect to the topology of S. For a collection tUα uαPA of subsets of a space Y and α0 , α1 , . . . , αp P A, we set Uα0 α1 ...αp “ Uα0 XUα1 X¨ ¨ ¨XUαp Ă Y. A.1

Identification with singular cohomology

For a sufficiently nice topological space Y and a module M over ˇ q p pY ; M q of Y with coefa ring R, the Cech cohomology group H ficients in the sheaf Y ˆ M of germs of locally constant functions on Y with values in M is well known to be canonically isomorphic 409

410

Spin/Pin-Structures & Real Enumerative Geometry

to the singular cohomology group H p pY ; M q of Y with coefficients in M . Proposition A.1 makes this precise in the M “ Z2 case relevant to our purposes, making use of the locally H k -simple notion of Definition 3.2. The statement and proof of this proposition apply to an arbitrary module M over a ring R. The p “ 1 case of the isomorphism of Proposition A.1 is described explicitly at the end of this section. Proposition A.1. Let k P Zě0 . For every paracompact locally H k -simple space Y, there exist canonical isomorphisms « qp pY ; Z2 q, p “ 0, 1, . . . , k. (A.1) ΦY : H p pY ; Z2 q ÝÑ H If Y is another paracompact locally H k -simple space and f : Y ÝÑ Y 1 is a continuous map, then the diagram H p pY 1 ; Z2 q f˚



H p pY ; Z2 q

ΦY 1

ΦY

q p pY 1 ; Z2 q /H 

f˚

/H q p pY ; Z2 q

(A.2)

commutes for every p ď k. Proof. Let p P Zě0 and Y be a topological space. Denote by SYp ÝÑ Y the sheaf of germs of Z2 -valued singular p-cochains on Y as in [43, 5.31], by dp : SYp ÝÑ SYp`1

(A.3)

the homomorphism induced by the usual differential in the singular cohomology theory, and by ` ˘ ` ˘ dp;Y : Γ Y ; SYp ÝÑ Γ Y ; SYp`1 the resulting homomorphism between the spaces of global sections. Let ZYp Ă SYp be the kernel of the sheaf homomorphism (A.3) so that dp

t0u ÝÑ ZYp ÝÑ SYp ÝÑ ZYp`1

(A.4)

is an exact sequence of sheaves. If Y is locally path-connected, ZY0 “ Y ˆZ2 . From now on, we assume that Y is paracompact. By the exactness of (A.4), ` ˘ ` ˘ q 0 Y ; Zp . (A.5) ker dp;Y “ Γ Y ; Zp “ H

Appendices

411

By [43, p. 193], each sheaf SYp is fine. By [43, p. 202], this implies that q q pY ; Sp q “ 0 @ p P Zě0 , q P Z` . H Y

(A.6)

Each Z2 -valued singular p-cochain  on Y determines a section pρy,Y pqqyPY of SYp over Y . By [43, 5.32], the induced homomorphism ` ˘ “ ‰ H p pY ; Z2 q ÝÑ H p ΓpY ; SY˚ q, d˚;Y , rs ÝÑ pρy,Y pqqyPY , (A.7) is an isomorphism. Combining the p “ 0 cases of this isomorphism and of the identification (A.5), we obtain an isomorphism (A.1) for p “ 0. Suppose Y is locally H k -simple and p P Z` with p ď k. The sequence t0u ÝÑ ZYp´q´1 ÝÑ SYp´q´1 ÝÑ ZYp´q ÝÑ t0u

(A.8)

of sheaves is then exact for every q P Zě0 with q ă p. From the ˇ exactness of the associated long sequence in Cech cohomology, (A.5), and (A.6), we obtain isomorphisms ` ˘ ker dp;Y δqY : H p ΓpY ; SY˚ q, d˚;Y ” Im dp´1;Y q 0 pY ; Zp q ˘ H « q 1` ÝÑ H Y ; ZYp´1 , q 0 pY ; Sp´1 qq dp´1 pH ˘ « q q`1 ` q q pY ; Zp´q q ÝÑ H Y ; ZYp´q´1 @ q P Z` , q ă p. δqY : H Y “

Putting these isomorphisms together, we obtain an isomorphism ` ˘ ` ˘ ` ˘ q p Y ; ZY0 “ H q p Y ; Z2 . (A.9) δqYp : H p ΓpY ; SY˚ q, d˚;Y ÝÑ H Combining (A.7) with this isomorphism, we obtain an isomorphism as in (A.1) with p ą 0. Suppose Y is another paracompact locally H k -simple space. A continuous map f : Y ÝÑ Y 1 induces commutative diagrams

412

Spin/Pin-Structures & Real Enumerative Geometry

for all p, q P Zě0 with q ă p ď k. Combining the p “ 0 cases of the first diagram above and of the identifications (A.5) for Y and for Y 1 , we obtain a commutative diagram (A.2) for p “ 0. The second commutative diagram above induces a commutative diagram ` ˘ « ` ˘ /H q p Y 1 ; Z2 H p ΓpY 1 ; SY˚ 1 q, d˚;Y 1 f˚

f˚



` ˘ H p ΓpY ; SY˚ q, d˚;Y

«

`  ˘ q p Y ; Z2 /H

with the horizontal isomorphisms as in (A.9). Combining this with the first commutative diagram in this paragraph, we obtain (A.2)  with p ą 0. Let Y be a paracompact locally H 1 -simple space. We now describe the p “ 1 case of the isomorphism (A.1) explicitly. Suppose  is a Z2 -valued singular 1-cocycle on Y. Since Y is locally H 1 -simple, there exist an open cover tUα uαPA of Y and a Z2 -valued singular 0-cochain μα on Uα for each α P A so that ˇ d0;Uα μα “ ˇUα @ α P A. ˇ We define a Cech 1-cocycle η on Y by ˇ ˇ ` ˘ @ α, β P A. ηαβ “ μβ ˇU ´ μα ˇU P SY1 Uαβ αβ

αβ

(A.10)

Since d0;Uαβ ηαβ “ 0 and Y is locally path-connected, ηαβ is a locally constant function on Uαβ . Thus, η takes values in the sheaf of germs of Z2 -valued continuous functions on Y and so defines an element rηs q 1 pY ; Z2 q. This is the image of rs under the p “ 1 case of the of H isomorphism ΦY in (A.1). Suppose Y is a CW complex and  is a Z2 -valued singular 1-cocycle on Y as above. For each vertex α P Y0 of Y , let Uα Ă Y denote the (open) star of α, i.e. the union of all open cells ˚ e of Y so that α is contained in the closed cell e. In particular, Uα is an open neighborhood of α and the collection tUα uαPY0 covers the 1-skeleton Y1Ă Y . We take  tUα uαPA ” tUα uαPY0 \ Y ´Y1 u; this is an open cover of Y . By adding extra vertices to Y1 , we can ensure that no closed 1-cell is a cycle. This implies that every closed

Appendices

413

1-cell e of Y runs between distinct vertices α and β with e Ă Uα YUβ ,

e XUαβ “ ˚ e,

e X Uγ “ H @ γ P A´tα, βu. (A.11)

For every α P A, there then exists a Z2 -valued singular 0-cochain μα on Uα so that ˇ d0;Uα μα “ ˇUα , μα pαq “ 0 @ α P A. Every closed 1-cell e of Y is cobordant to the difference of a singular 1-simplex exβ running from a point x P˚ e to β and a singular 1-simplex exα running from x to α. By (A.11), exα Ă Uα and exβ Ă Uβ . Since  is a cocycle, it follows that ˘ ` ˘ ` ˘  (` ˘ ` peq “  exβ ´exα “  exβ ´  exα “ d0;Uβ μβ exβ  (` ˘ ´ d0;Uα μα exα ˘ ` ˘ ` “ μβ pβq´μβ pxq ´ μα pαq´μα pxq “ μα pxq´μβ pxq. ˇ Along with (A.10), this implies that the Cech cohomology class rηs ” ΦY prsq corresponding to rs under the isomorphism (A.1) is represented by a collection tηαβ uα,βPA associated with an open cover tUα uαPA of Y such that ˇ “ peq P Z2 ˚ e Ă Uαβ , ηαβ ˇ˚ e for all α, β P Y0 and every closed 1-cell e with vertices α and β. A.2

Sheaves of groups

ˇ q p are normally defined for sheaves or Cech cohomology groups H q 0 and H q1 presheaves of (abelian) modules over a ring. The sets H can be defined for sheaves or presheaves of non-abelian groups as well. The first set is still a group, while the second is a pointed set, i.e. it has a distinguished element. A short exact sequence of such ˇ sheaves gives rise to an exact sequence of the corresponding Cech pointed sets, provided the kernel sheaf R lies in the center ZpSq of the ambient sheaf S; see Proposition A.3. The main examples of interest are the sheaves S of germs of continuous functions over a topological space Y with values in a Lie group G, as in Section A.3.

414

Spin/Pin-Structures & Real Enumerative Geometry

We denote the center of a group G by ZpGq. We call a collection ˘ ˘ `` δp : C p ÝÑ C p`1 p“0,1,2 , ˚ : C 0 ˆC 1 ÝÑ C 1 consisting of maps δp between groups C p with the identity element 1p and a left action ˚ a short cochain complex if δp 1p “ 1p`1 ,

δp`1 ˝δp “ 1p`2 , ˘ ` δ1 pf ˚gq “ δ1 g @ f P C 0 , g P δ1´1 ZpC 2 q , δ0 pf ¨f 1 q “ f ˚pδ0 f 1 q, f ˚g “ pδ0 f qg @ f, f 1 P C 0 , g P ZpC 1 q, ` ˘ ` ˘` ˘ @ g P C p , g1 P ZpC p q, p “ 1, 2. δp g¨g1 “ δp g δp g 1 By the second condition in (A.13), ` ˘ C 0 ˚tgu “ Im δ0 ¨tgu

@ g P ZpC 1 q.

By both conditions in (A.13), ˘ ` H 0 pC ˚ q ” H 0 pC p , δp qp“0,1,2 , ˚ ” ker δ0 ” δ0´1 p11 q

(A.12) (A.13) (A.14)

(A.15)

(A.16)

is a subgroup of C 0 . By the last property in (A.12), ˚ restricts to an action on ker δ1 ” δ1´1 p12 q. We can thus define ˘ ` L (A.17) H 1 pC ˚ q ” H 1 pC p , δp qp“0,1,2 , ˚ “ ker δ1 C 0 ; this is a pointed set with the distinguished element given by the image of Im δ0 Q 11 in H 1 pC ˚ q. By (A.13) and the p “ 1 case of (A.14), δ0 and δ1 are group homomorphisms if the group C 1 is abelian and ˚ is the usual action of the 1-coboundaries on the 1-cochains via the group operation. In this case, (A.17) agrees with the usual definition and the last condition in (A.12) is automatically satisfied. If in addition the group C 2 is also abelian, as happens for the kernel complex B ˚ in Lemma A.2, then the map δ2 is a group homomorphism as well and ˘ ` L H 2 pC ˚ q ” H 2 pC p , δp qp“0,1,2 , ˚ “ ker δ2 Im δ1 is a well-defined abelian group.

Appendices

415

A morphism of short cochain complexes ` ˘ ` ˘ ι ” pιp qp“0,1,2,3 : pB p , δp qp“0,1,2 , ˚ ÝÑ pC p , δp qp“0,1,2 , ˚ is a collection of group homomorphisms ιp : B p ÝÑ C p that commute with the maps δp and the actions ˚. Such a homomorphism induces morphisms ι˚ : H p pB ˚ q ÝÑ H p pC ˚ q,

p “ 0, 1,

of pointed sets, i.e. ι˚ takes the distinguished element of the domain to the distinguished element of the target; the map ι0 is a group homomorphism. The kernel of such a morphism is the preimage of the distinguished element of the target. The following lemma is an analog of the Snake Lemma [43, Proposition 5.17] for short cochain complexes of groups. Lemma A.2. For every short exact sequence ˘ ι ` ˘ ` t1u ÝÑ pB p , δp qp“0,1,2 , ˚ ÝÑ pC p , δp qp“0,1,2 , ˚ ˘ j ` ÝÑ pD p , δp qp“0,1,2 , ˚ ÝÑ t1u of short cochain complexes of groups such that ιp pB p q Ă ZpC p q for p “ 1, 2, there exist morphisms Bp : H p pD ˚ q ÝÑ H p`1 pB ˚ q,

p “ 0, 1,

(A.18)

of pointed sets such that the sequence j˚

B

j˚

B

0 t1u ÝÑ H 0 pB ˚ q ÝÑ H 0 pC ˚ q ÝÑ H 0 pD ˚ q ÝÑ

ι˚

B

0 1 ÝÑ H 1 pB ˚ q ÝÑ H 1 pC ˚ q ÝÑ H 1 pD ˚ q ÝÑ H 2 pB ˚ q

ι˚

(A.19)

of morphisms of pointed sets is exact. The maps Bp are natural with respect to morphisms of short exact sequences of short cochain complexes of groups. Proof. We proceed as in the abelian case. Given dp P ker δp Ă D p , let cp P C p be such that jp pcp q “ dp . Since ˘ ˘ ` ` jp`1 δp pcp q “ δp jp pcp q “ δp pdp q “ 1p`1 P D p`1 , there exists a unique bp`1 P B p`1 such that ιp`1 pbp`1 q “ δp pcp q. By the second condition in (A.12), bp`1 P ker δp`1 . We set ˘ “ ` ‰ Bp rdp s “ bp`1 P H p`1 pB ˚ q.

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Spin/Pin-Structures & Real Enumerative Geometry

By the first condition in (A.13), rb1 s is independent of the choice of c0 P C 0 such that j0 pc0 q “ d0 . By the p “ 1 case of (A.14) and the assumption that ιp pB p q Ă ZpC p q for p “ 1, 2, rb2 s is independent of the choice of c1 P C 1 such that j1 pc1 q “ d1 . By the last condition in (A.12) and the assumption that ι2 pB 2 q Ă ZpC 2 q, rb2 s does not depend on the choice of representative d1 for rd1 s. Thus, the maps (A.18) are well defined. By the first condition in (A.12), Bp pr1p sq “ r1p`1 s, i.e. Bp is a morphism of pointed sets. By the construction, the maps Bp are natural with respect to morphisms of exact sequences of short cochain complexes. It is immediate that (A.19) is exact at H 0 pB ˚ q and H 0 pC ˚ q and that j˚ ˝ι˚ “ r11 s : H 1 pB ˚ q ÝÑ H 1 pD ˚ q, Bp ˝j˚ “ r1p`1 s : H p pC ˚ q ÝÑ H p`1 pB ˚ q. The exactness of (A.19) at H 1 pB ˚ q is immediate from (A.15) with g “ 11 P ZpC 1 q. The exactness at H 1 pC ˚ q follows from (A.15) with g “ 11 P ZpD 1 q, the second condition in (A.13), and the assumption that ι1 pB 1 q Ă ZpC 1 q. The exactness at H 0 pD ˚ q follows from (A.15) with g “ 11 P ZpB 1 q, both conditions in (A.13), and the assumption that ι1 pB 1 q Ă ZpC 1 q. The exactness at H 1 pD ˚ q follows from the assumption that ιp pB p q Ă ZpC p q for p “ 1, 2 and the p “ 1 case  of (A.14). q0 We next review the definitions and key properties of the group H 1 q and pointed set H for a sheaf S of groups over a topological space Y. We denote by ZpSq Ă S the subsheaf consisting of the centers ZpSy q of the groups Sy with y P Y and by 1y P Sy the identity element of Sy . Let U ” tUα uαPA be an open cover of Y. As in the abelian case, the set ź ` ˘ q p pU ; Sq ” Γ Uα α ...αp ; S C α0 ,α1 ,...,αp PA

0 1

ˇ of Cech p-cochains is a group under pointwise multiplication of sections: q p pU ; Sq ÝÑ C q p pU ; Sq, q p pU ; Sq ˆ C ¨:C th ¨ h1 uα0 α1 ...αp pyq “ hα0 α1 ...αp pyq ¨ h1α0 α1 ...αk pyq @ α0 , α1 , . . . , αp P A, y P Uα0 α1 ...αp .

Appendices

417

q p pU ; Sq is given by The identity element 1p P C p1p qα0 α1 ...αp pyq “ 1y

@ α0 , α1 , . . . , αp P A, y P Uα0 α1 ...αp .

Define the boundary maps by q 0 pU ; Sq ÝÑ C q 1 pU ; Sq, δ0 : C

pδ1 gqα0 α1 α2

ˇ pδ0 f qα0 α1 “ fα0 ˇUα

0 α1

ˇ ˇ ¨ fα´1 1 Uα

0 α1

q 1 pU ; Sq ÝÑ C q 2 pU ; Sq, δ1 : C ˇ ˇ ˇ ˇ “ gα1 α2 ˇUα α α ¨ gα´1 ¨ gα0 α1 ˇUα 0 α2 Uα α α 0 1 2

0 α1 α2

0 1 2

,

.

q 1 pU ; Sq by q 0 pU ; Sq on C We also define a left action of C q 0 pU ; Sq ˆ C q1 pU ; Sq ÝÑ C q 1 pU ; Sq, ˚: C ˇ ˇ ˇ tf ˚guα0 α1 “ fα0 ˇUα α ¨ gα0 α1 ¨ fα´1 P ΓpUα0 α1 ; Sq. 1 Uα α 0 1

0 1

We now construct a short cochain complex. Let # q p pU ; Sq, ` ˘ if p “ 0, 1, 2; C p C U; S “ q p pU ; Sqq if p “ 3. AbelpC For p “ 0, 1, we take

` ˘ ` ˘ δp : C p U ; S ÝÑ C p`1 U ; S

to be the maps defined above. We take δ2 to be the composition of the map q 2 pU ; Sq ÝÑ C q 3 pU ; Sq, δ2 : C ˇ ˇ ` ˘ ˇ δ2 h α0 α1 α2 α3 “ hα1 α2 α3 ˇUα α α α hα´1 0 α2 α3 Uα α α α 0 1 2 3 0 1 2 3 ˇ ˇ ´1 ˇ ˇ ˆ hα0 α1 α3 Uα α α α hα0 α1 α2 Uα α α α , ` 3

0 1 2 3

0 1 2 3

˘ q 3 pU ; Sq ÝÑ C U ; S . The tuple with the projection C ˘ `` δp : C p pU ; Sq ÝÑ C p`1 pU ; Sq p“0,1,2 , ˚ : C 0 pU ; Sq ˘ ˆ C 1 pU ; Sq ÝÑ C 1 pU ; Sq

is then a short cochain complex of groups. We denote the associated q 1 pU ; Sq, q 0 pU ; Sq and H group (A.16) and the pointed set (A.17) by H respectively.

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Spin/Pin-Structures & Real Enumerative Geometry

Let U 1 ” tUα1 uαPA1 be an open cover of Y refining U , i.e. there exists a map μ : A1 ÝÑ A such that Uα1 Ă Uμpαq for every α P A1 . Such a refining map induces group homomorphisms ` ˚ ˘ μp h α0 ...αp

qp pU ; Sq ÝÑ C q p pU 1 ; Sq, (A.20) μ˚p : C ˇ qp pU ; Sq, α0 , . . . , αp P A1 . “ hμpα0 q...μpαp q ˇU 1 @ hPC α0 ...αp

These homomorphisms commute with δ0 , δ1 , and the action of q 1 p¨; Sq. Thus, μ induces maps q 0 p¨; Sq on C C 0 q0 q0 1 RU 1 ,U : H pU ; Sq ÝÑ H pU ; Sq 1 q1 q1 1 and RU 1 ,U : H pU ; Sq ÝÑ H pU ; Sq

(A.21)

of pointed sets; the first map above is a group homomorphism. If μ1 : A1 ÝÑ A is another refining map, then Uα1 Ă Uμpαqμ1 pαq for every α P A1 and thus ˇ ˇ ˇ q0 1 μ˚0 ˇker δ0 “ μ1˚ 0 ker δ0 : ker δ0 ÝÑ ker δ0 Ă C pU ; Sq. q1 pU ; Sq, define For g P C q 0 pU 1 ; Sq h1 g P C q 1 pU ; Sq, then If g P ker δ1 Ă C ˇ gμpα qμpα q ˇ 0

1

Uμ1 pα

1 qμpα0 qμpα1 q

“ gμ´1 1 pα qμpα q 1 0 ˇ gμ1 pα0 qμpα0 q ˇU 1

ˇ ˇ

ph1 gqα “ gμ1 pαqμpαq |Uα1 .

by

ˇ ˇ ¨ gμ´1 1 pα qμpα q U 1 1

Uμ1 pα

1 qμpα0 qμpα1 q

μ pα1 qμ1 pα0 qμpα0 q

“ gμ´1 1 pα qμ1 pα q 1 0

ˇ ˇ

μ1 pα1 qμpα0 qμpα1 q

,

ˇ ˇ ¨ gμ´1 1 pα qμpα q U 1 0

μ1 pα1 qμ1 pα0 qμpα0 q

Uμ1 pα

1 1 qμ pα0 qμpα0 q

for all α0 , α1 P A. From this, we find that ˚ q1 μ1˚ 1 g “ ph1 gq ˚ pμ1 gq @ g P ker δ1 Ă C pU ; Sq.

From the previous paragraph, the pointed maps (A.21) are independent of the choice of refining map μ : A1 ÝÑ A. We can thereq 1 pY ; Sq as the q 0 pY ; Sq and the pointed set H fore define the group H 0 q 1 pU ; Sq, q pU ; Sq and of the pointed sets H direct limits of the groups H

Appendices

419

respectively, over open covers of Y . The map ` ˘ q 0 pY ; Sq, f ÝÑ f |Uα , ΓpY ; Sq ÝÑ H αPA

(A.22)

is a group isomorphism. If S is a sheaf of abelian groups, as happens for the kernel sheaf R q 0 pY ; Sq and H q 1 pY ; Sq above in Proposition A.3, the definitions of H reduce to the ones in [43, Section 5.33]. Furthermore, q2 q3 q 2 pU ; Sq ” kerpδ2 : C pU ; Sq ÝÑ C pU ; Sqq H q 1 pU ; Sq ÝÑ C q 2 pU ; Sqq Impδ1 : C is a well-defined abelian group for every open cover U of Y . The group homomorphisms q 2 pU ; Sq ÝÑ H q 2 pU 1 ; Sq R2 1 : H U ,U

induced by refining maps still depend only on the covers U and U 1 . q 2 pY ; Sq is again the direct limit of the groups The abelian group H q 2 pU ; Sq over all open covers U of Y. H A homomorphism ι : R ÝÑ S of sheaves of groups over Y induces maps q 0 pY ; Rq ÝÑ H q 0 pY ; Sq, ι˚ : ΓpY ; Rq ÝÑ ΓpY ; Sq, ι˚ : H q 1 pY ; Rq ÝÑ H q 1 pY ; Sq ι˚ : H between pointed spaces. The first two maps are group homomorphisms which commute with the identifications (A.22). Proposition A.3. Let Y be a paracompact space. For every short exact sequence ι

j

t1u ÝÑ R ÝÑ S ÝÑ T ÝÑ t1u

(A.23)

of sheaves of groups over Y such that ιpRq Ă ZpSq, there exist morphisms q p pY ; Tq ÝÑ H q p`1 pY ; Rq, p “ 0, 1, (A.24) δqp : H of pointed sets such that the sequence j˚

δq

0 q 0 pY ; Sq ÝÑ H q 0 pY ; Tq ÝÑ q 0 pY ; Rq ÝÑ H t1u ÝÑ H

ι˚

q

q

ι˚ q 1 j˚ q 1 δ0 q 1 δ1 q 2 ÝÑ H pY ; Rq ÝÑ H pY ; Sq ÝÑ H pY ; Tq ÝÑ H pY ; Rq (A.25)

420

Spin/Pin-Structures & Real Enumerative Geometry

of morphisms of pointed sets is exact. The maps δqp are natural with respect to morphisms of short exact sequences of sheaves of groups over Y. Let U ” tUα uαPA be an open cover of Y, ` ˘ ` ˘ ` ˘ B p pU q “ C p U ; R , C p pU q “ C p U ; S , D p pU q “ C p U ; T .

Proof.

Since ιpRq Ă ZpSq, ι˚ pB p pU qq Ă ZpC p pU qq for all p. By the exactness of (A.23), the sequence j˚

ι˚

t1u ÝÑ B p pU q ÝÑ C p pU q ÝÑ D p pU q p

of groups is exact. For p “ 0, 1, 2, 3, we denote by D pU q Ă D p pU q p ˇ pointed sets the image of j˚ . For p “ 0, 1, let H pU ; Tq be the Cech ˚ determined by the short cochain complex D pU q. The sequence j˚

ι˚

p

t1u ÝÑ B p pU q ÝÑ C p pU q ÝÑ D pU q ÝÑ t1u

(A.26)

of short cochain complexes is exact. By Lemma A.2, there thus exist morphisms p q p`1 pU ; Rq, δqp : H pU ; Tq ÝÑ H

p “ 0, 1,

(A.27)

of pointed sets such that the sequence q

ι˚ q 0 j˚ 0 δ0 q 0 pU ; Rq ÝÑ H pU ; Sq ÝÑ H pU ; Tq ÝÑ t1u ÝÑ H q

q

ι˚ q 1 j˚ q 1 δ0 q 1 δ1 q 2 ÝÑ H pU ; Rq ÝÑ H pU ; Sq ÝÑ H pU ; Tq ÝÑ H pU ; Rq (A.28)

of morphisms of pointed sets is exact. Let U 1 ” tUα1 uαPA1 be an open cover of Y refining U and μ : A1 ÝÑ A be a refining map. By the naturality of the morphisms (A.27), the group homomorphisms (A.20) induce commutative diagrams p

H pU ; Tq RpU 1 ,U p



H pU 1 ; Tq

δqp

δqp

q p`1 pU ; Rq /H 

Rp`1 U 1 ,U

/H q p`1 pU 1 ; Rq

Appendices

421

of pointed sets. Taking the direct limit of the morphisms (A.27) over all open covers of Y , we thus obtain morphisms p q p`1 pY ; Rq, δqp : H pY ; Tq ÝÑ H

p “ 0, 1,

(A.29)

q ˚ pY ; Tq replaced of pointed sets such that the sequence (A.25) with H p`1 pY ; Tq is exact. by H p The inclusions ip : D pU q ÝÑ D p pU q of short cochain complexes commute with the refining homomorphisms (A.20) and induce morphisms p q p pU ; Tq i˚ : H pU ; Tq ÝÑ H

p q p pY ; Tq and i˚ : H pY ; Tq ÝÑ H (A.30) of pointed sets. By the paracompactness of Y and the reasoning in [43, p204], for every open cover U ” tUα uαPA of Y and every element dp of D p pU q there exist an open cover U 1 ” tUα1 uαPA1 refining U , p a refining map μ : A1 ÝÑ A, and an element d1p of D pU 1 q such that ip pd1p q “ μ˚p pdp q. This implies that the second map in (A.30) is a bijection. Composing (A.29) with this bijection, we obtain a morphism as in (A.24) so that the sequence (A.25) is exact. A morphism of short exact sequences of sheaves of groups over Y as in (A.23) induces morphisms of the corresponding exact sequences of short cochain complexes as in (A.26) and of the inclusions ip above. Thus, it also induces morphisms of the corresponding maps as in (A.27) and as on the left-hand side of (A.30). These morphisms commute with the associated maps (A.21) and thus induce morphisms of the maps as in (A.24). This establishes the last  claim.

A.3

Sheaves determined by Lie groups

For a Lie group G and a topological space Y , we denote by SY pGq the sheaf of germs of continuous G-valued functions on Y and let ˘ ` q p Y ; SY pGq @ p “ 0, 1. q p pY ; Gq “ H H If G is abelian, we use the same notation for all p P Z. We begin this section by applying Proposition A.3 to short exact sequences of sheaves arising from short exact sequences ι

j

t1u ÝÑ K ÝÑ G ÝÑ Q ÝÑ t1u

(A.31)

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Spin/Pin-Structures & Real Enumerative Geometry

of Lie groups. For certain kinds of exact sequences (A.31), the topological condition on Y of Definition A.4 appearing in the resulting statement of Corollary A.5 reduces to the locally H 1 -simple notion of Definition 3.2. For such exact sequences of Lie groups and topological spaces, we combine Proposition A.1 and Corollary A.5 to ˇ obtain an exact sequence mixing Cech and singular cohomology; see Proposition A.6. A homomorphism ι : K ÝÑ G of Lie groups induces a homomorphism ι : SY pKq ÝÑ SY pGq of sheaves over every topological space and thus morphisms q p pY ; Kq ÝÑ H q p pY ; Gq ι˚ : H of pointed sets for p “ 0, 1; the p “ 0 case of ι˚ is a group homomorphism. A continuous map f : Y ÝÑ Y 1 induces group homomorphisms ˘ ˘ ` ` q p U ; SY 1 pGq ÝÑ C q p f ´1 pU q; SY pGq , p P Z, f˚ : C ˇ coboundfor every open cover U of Y 1 that commute with the Cech aries and group actions for the sheaves SY 1 pGq and SY pGq constructed in Section A.2 and with the refining homomorphisms as in (A.20). Thus, f induces morphisms q p pY 1 ; Gq ÝÑ H q p pY ; Gq f˚ : H of pointed sets for p “ 0, 1; the p “ 0 case of f ˚ is a group homomorphism. If G is abelian, then f induces such a morphism for every p P Z and this morphism is a group homomorphism. If in addition ι is a homomorphism of Lie groups as above, then the diagram q p pY 1 ; Kq H f˚



q p pY ; Kq H

ι˚

ι˚

q p pY 1 ; Gq /H 

f˚

/H q p pY ; Gq

commutes. Definition A.4. Let (A.31) be a short exact sequence of Lie groups. A topological space Y is locally simple with respect to (A.31) if it is locally path-connected and for every neighborhood U Ă Y of

Appendices

423

a point y P Y and a continuous map fU : U ÝÑ Q, there exist a neighborhood U 1 Ă U of y and a continuous map fU1 : U 1 ÝÑ G such that fU |U 1 “ j˝fU1 . For any topological space Y , a short exact sequence (A.31) of Lie groups induces an exact sequence j

ι

t1u ÝÑ SY pKq ÝÑ SY pGq ÝÑ SY pQq of sheaves over Y . The last map above is surjective if and only if Y is locally simple with respect to (A.31). If the restriction of j to the identity component G0 of G is a double cover of Q0 and π1 pQ0 q is (possibly infinite) cyclic, then the condition of Definition A.4 is equivalent to Y being locally H 1 -simple. This follows from the lifting property for covering projections [36, Lemma 79.1], Hurewicz isomorphism for π1 [42, Proposition 7.5.2], and the Universal Coefficient Theorem for Cohomology [35, Theorem 53.3]. Corollary A.5. Let Y be a paracompact space and (A.31) be a short exact sequence of Lie groups such that ιpKq Ă ZpGq. If Y is locally simple with respect to (A.31), then there exist morphisms q p pY ; Qq ÝÑ H q p`1 pY ; Kq, δqp : H

p “ 0, 1,

(A.32)

of pointed sets such that the sequence q

ι˚ q 0 j˚ q 0 δ0 q 0 pY ; Kq ÝÑ H pY ; Gq ÝÑ H pY ; Qq ÝÑ t1u ÝÑ H δq

j˚

δq

0 1 q 1 pY ; Kq ÝÑ H q 1 pY ; Gq ÝÑ H q 1 pY ; Qq ÝÑ q 2 pY ; Kq ÝÑ H H (A.33)

ι˚

of morphisms of pointed sets is exact. The maps δqp are natural with respect to morphisms of short exact sequences of Lie groups and with respect to continuous maps between paracompact spaces that are locally simple with respect to (A.31). Proof. Since Y is locally simple with respect to (A.31), the sequence ι

j

t1u ÝÑ SY pKq ÝÑ SY pGq ÝÑ SY pQq ÝÑ t1u

(A.34)

of sheaves over Y is exact. Since ιpKq Ă ZpGq, ιpSY pKqq Ă ZpSY pGqq. The existence of morphisms (A.32) so that the sequence (A.33) is exact thus follows from the first statement of Proposition A.3.

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Spin/Pin-Structures & Real Enumerative Geometry

A morphism of short exact sequences of Lie groups as in (A.31) satisfying the conditions at the beginning of the statement of the proposition induces a morphism of the corresponding short exact sequences of sheaves as in (A.34). Thus, the naturality of (A.32) with respect to morphisms of short exact sequences of Lie groups follows from the second statement of Proposition A.3. A continuous map f : Y ÝÑ Y 1 between paracompact spaces that are locally simple with respect to (A.31) induces a morphism of the corresponding exact sequences of short cochain complexes as in (A.26) and of the inclusions ip as in the proof of Proposition A.3. Thus, it also induces morphisms of the corresponding maps as in (A.27) and as on the left-hand side of (A.30). These morphisms commute with the associated maps (A.21) and thus induce morphisms of the maps as in (A.32). This establishes the naturality  of (A.32) with respect to continuous maps. Proposition A.6. Let Y be a paracompact locally H 1 -simple space and ι

j

t1u ÝÑ Z2 ÝÑ G ÝÑ Q ÝÑ t1u

(A.35)

be an exact sequence of Lie groups such that ιpZ2 q Ă ZpGq and π1 pQ0 q is cyclic. Then there exist morphisms q 0 pY ; Qq ÝÑ H 1 pY ; Z2 q and δq1 : H q 1 pY ; Qq ÝÑ H q 2 pY ; Z2 q δq0 : H (A.36) of pointed sets such that the sequence j˚

δq

0 q 0 pY ; Gq ÝÑ H q 0 pY ; Qq ÝÑ t1u ÝÑ H 0 pY ; Z2 q ÝÑ H

ι˚

ι˚ q 1 j˚ q 1 δq0 δq1 q 2 ÝÑ H 1 pY ; Z2 q ÝÑ H pY ; Gq ÝÑ H pY ; Qq ÝÑ H pY ; Z2 q (A.37)

of morphisms of pointed sets is exact. If in addition Y is locally q 2 pY ; Z2 q replaced by H 2 -simple, then the same statement with H H 2 pY ; Z2 q also holds. The maps δq0 and δq1 are natural with respect to morphisms of exact sequences of Lie groups as in (A.6) and with respect to continuous maps between paracompact locally H 1 -simple spaces. Proof. Since Y is locally H 1 -simple, it is locally simple with respect to the exact sequence (A.35) in the sense of Definition A.4.

Appendices

425

q p pY ; Z2 q is Thus, this proposition with all H p pY ; Z2 q replaced by H a specialization of Corollary A.5. By Proposition A.1, we can then q 1 pY ; Z2 q by H 1 pY ; Z2 q. If in q 0 pY ; Z2 q by H 0 pY ; Z2 q and H replace H q 2 pY ; Z2 q can also be replaced addition Y is locally H 2 -simple, then H 2 by H pY ; Z2 q.  A.4

Relation with principal bundles

Let G be a Lie group and Y be a topological space. We recall in the following the standard identification of the set PrinY pGq of equivalence (isomorphism) classes of principal G-bundles over Y with the q 1 pY ; Gq. This identification is key for applying Propopointed set H sition A.6 to principal G-bundles, including to study Spin- and Pin˘ structures in the classical perspective of Definition 1.1. Suppose πP : P ÝÑ Y is a principal G-bundle. Let U ” pUα qαPA be an open cover of Y so that the principal G-bundle P |Uα is trivializable for every α P A. Thus, for every α P A there exists a homeomorphism ˇ Φα : P ˇUα ÝÑ Uα ˆG s.t. ` ` ˘ ` ˘˘ πα;1 ˝Φα “ πP , πα;2 Φα ppuq “ πα;2 Φα ppq ¨u @ p P P |Uα , u P G, where πα;1 , πα;2 : UαˆG ÝÑ Uα , G are the two projection maps. Thus, for all α, β P A there exists a continuous map ˇ ˘ ˘` ˘˘ ` ` ` gαβ : Uαβ ÝÑ G s.t. πα;2 Φα ppq “ gαβ πP ppq ¨ πβ;2 Φβ ppq @ p P P ˇU

αβ

These continuous maps satisfy ˇ ˇ ˇ ´1 ˇ ˇ ¨g gβγ ˇU ¨gαγ αβ U U αβγ

αβγ

αβγ

.

“ 1 @ α, β, γ P A.

q 1 pU ; SY pGqq and thus Therefore, gP ” pgαβ qα,βPA lies in ker δ1 Ă C defines an element q 1 pY ; Gq. rgP s P H We show in the following that rgP s depends only on the isomorphism class of P . Suppose U 1 ” tUα1 uαPA1 is a refinement of U . If μ : A1 ÝÑ A is a refining map, then ˇ ˇ : P ˇ 1 ÝÑ Uα1 ˆG Φα ” Φμpαq ˇ P |U 1

α

Uα

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Spin/Pin-Structures & Real Enumerative Geometry

is a trivialization of the principal G-bundle P |Uα1 for every α P A1 . The corresponding transition data are  ( 1 ˚ 1 : U gP1 ” gμpαqμpβq |Uαβ αβ ÝÑ G α,βPA1 “ μ1 gP . Since q 1 pY ; Gq, rgP s “ rμ˚1 gP s P H it is thus sufficient to consider trivializations of isomorphic vector bundles over a common cover (otherwise we can simply take the intersections of open sets in the two covers). Suppose Ψ : P ÝÑ P 1 is an isomorphism of principal G-bundles over Y and the principal G-bundle P 1 |Uα is trivializable for every α P A. Thus, for every α P A there exists a homeomorphism ˇ Φ1α : P 1 ˇUα ÝÑ Uα ˆG s.t. ˘ ` ˘˘ ` ` πα;1 ˝Φ1α “ πP 1 , πα;2 Φ1α pp1 uq “ πα;2 Φ1α pp1 q ¨u @ p1 P P |Uα , u P G. For every α P A, there then exists a continuous map fα : Uα ÝÑ G

s.t.

˘` ˘˘ ` ` ˘˘ ` ` πα;2 Φ1α Ψppq “ fα πP ppq ¨ πα;2 Φα ppq

ˇ @ p P P ˇU . α

1 q The transition data gP 1 ” pgαβ α,βPA determined by the collection 1 tΦα uαPA of trivializations of P 1 then satisfies ˇ ˇ 1 “ fα ˇ ¨gαβ ¨f ´1 ˇ @ α, β P A. gαβ Uαβ

β

Uαβ

Thus, gP 1 “ f ˚gP , where f ” pfα qαPA, and “ ‰ “ ‰ q 1 pY ; Gq. gP 1 “ gP P H q 1 pY ; Gq constructed above We conclude that the element rgP s P H depends only on the isomorphism class of the principal G-bundle P over Y . q 1 pY ; Gq. Let U ” pUα qαPA be an open Conversely, suppose rgs P H q 1 pU ; SY pGqq cover of Y and g ” tgαβ uα,βPA be an element of ker δ1 Ă C representing rgs. Define ˜ ¸ M ğ tαuˆUα ˆG „g ÝÑ Y, πPg : Pg “ `

αPA

` ˘ ˘ α, y, gαβ pyqu „g β, y, u

@ α, β P A, py, uq P Uβ ˆG .

Appendices

427

This is a principal G-bundle over Y with trivializations ˇ ` ˘ Φα : Pg ˇUα ÝÑ Uα ˆG, Φα rα, y, uqs “ py, uq, for α P A and the associated transition data g. Thus, “ ‰ q 1 pY ; Gq. gPg “ rgs P H

(A.38)

We show in the following that the isomorphism class rPg s of Pg depends only on rgs. Suppose U 1 ” tUα1 uαPA1 is a refinement of U and μ : A1 ÝÑ A is a refining map. The map ˜ ¸ ˜ ¸ M M ğ ğ 1 tαuˆUα ˆG „μ˚ g ÝÑ Pg ” tαuˆUα ˆG „g , Ψ : Pμ˚ g ” αPA1

αPA

` ˘ “ ‰ Ψ rα, y, us “ μpαq, y, u ,

is then an isomorphism of principal G-bundles. Thus, it is sufficient to show that if ˘ ˘ ` ` q 1 U ; SY pGq and rgs “ rg1 s P H q 1 U ; SY pGq , g, g 1 P ker δ1 Ă C then the principal G-bundles Pg and Pg1 are isomorphic. By definition, there exists ` ˘ q 0 U ; SY pGq s.t. g1 “ f ˚g. f ” pfα qαPA P C The map Ψ : Pg “

˜

ğ

¸ ˜ ¸ M M ğ tαuˆUα ˆG „g ÝÑ Pg1 “ tαuˆUα ˆG „g1 ,

αPA

αPA

‰ ˘ “ Ψ rα, y, us “ α, y, fα pyq¨u , `

is then an isomorphism of principal G-bundles. Let P be a principal G-bundle over Y, tΦα uαPA be a collection of trivializations of P over an open cover U ” pUα qαPA, and gP ” pgαβ qα,βPA be the corresponding transition data. The map ˜ ¸ M ğ tαuˆUα ˆG „g , Ψ : P ÝÑ PgP ” αPA

‰ “ Ψppq “ α, Φα ppq @ p P P |Uα , α P A,

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Spin/Pin-Structures & Real Enumerative Geometry

is then an isomorphism of principal G-bundles. Along with (A.38), this implies that the maps q 1 pY ; Gq, PrinY pGq ÝÑ H

rP s ÝÑ rgP s,

q 1 pY ; Gq ÝÑ PrinY pGq, H

rgs ÝÑ rPg s,

(A.39)

q 1 pY ; Gq. are mutual inverses that identify PrinY pGq with H If f : Y ÝÑ Y 1 is a continuous map and P ÝÑ Y 1 is a principal G-bundle, then “ ‰ “ ‰ q 1 pY ; Gq. gf ˚ P “ f ˚ gP P H Thus, the identifications (A.39) are natural with respect to continuous maps. Corollary A.7. Let Y be a paracompact locally H 1 -simple space and ΦY be as in (A.1). For every real line bundle V over Y, ˘ “ ` ˘ ` ‰ ` ˘ q 1 Y ; Z2 Q ΦY w1 pV q “ gOpV q P H q 1 Y ; Op1q H under the canonical identification of the groups Z2 and Op1q. Proof. By the Universal Coefficient Theorem for Cohomology [35, Theorem 53.5], the homomorphism ` ˘ κ : H 1 pY ; Z2 q ÝÑ Hom π1 pY q, H 1 pS 1 ; Z2 q , ˘  (` κpηq f : S 1 ÝÑ Y “ f ˚ η, is injective. By the naturality of w1 , ΦY , and (A.39), it is thus sufficient to show that ˘ “ ˘ ` ˘ ` ‰ ` q 1 RP1 ; Op1q q 1 RP1 ; Z2 Q ΦRP1 w1 pf ˚ V q “ gOpf ˚ V q P H H (A.40) for every continuous map f : RP1 ÝÑ Y . Since every line bundle over the interval r0, 1s is trivializable, the line bundle f ˚ V is isomorphic to either the trivial line bundle τ1 or the real tautological line bundle γR;1 . Both sides of (A.40) vanish in the first case. Since (A.39) is a bijection, this implies that the right-hand side of (A.40) does not vanish in the second case. The left-hand side of (A.40) does not vanish in this case by the Normalization Axiom for Stiefel–Whitney  classes [34, p38].

Appendices

A.5

429

Orientable vector bundle over surfaces

We now combine the description of complex line bundles in terms ˇ ˇ of Cech cohomology and the identification of some Cech cohomology groups with the singular ones to characterize orientable vector bundles over surfaces and their trivializations. Lemma A.8. Let Y be a paracompact locally contractible space. The homomorphism c1 : LBC pY q ÝÑ H 2 pY ; Zq,

L ÝÑ c1 pLq,

from the group of isomorphism classes of complex line bundles is an isomorphism. Proof.

By Section A.4, there is a natural bijection q 1 pY ; C˚ q; LBC pY q ÝÑ H

it is a group isomorphism in this case. By the proof of Proposition A.1, there are natural isomorphisms q p pY ; Zq « H p pY ; Zq @ p P Z. H By the reasoning in [43, Section 5.10], SY pCq is a fine sheaf. Along with [43, p202], this implies that ˘ ` q p Y ; SY pCq “ t0u @ p P Z` . H Since Y is locally contractible, it is locally simple with respect to the short exact sequence t0u ÝÑ Z ÝÑ C ÝÑ C˚ ÝÑ t0u exp

of abelian Lie groups in the sense of Definition A.4. Thus, we obtain a commutative diagram q 1 pCP8 ; Cq t0u “ H



f˚

q 1 pY ; Cq t0u “ H

/ LBC pCP8 q 

δq1

f˚

/ LBC pY q

δq1

/ H 2 `CP8 ; Z˘ 

f˚

/ H 2 pY ; Zq

/H q 2 pCP8 ; Cq “ t0u 

f˚

/H q 2 pY ; Cq “ t0u

(A.41) of group homomorphisms for every continuous map f : Y ÝÑ CP8 .

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Spin/Pin-Structures & Real Enumerative Geometry

Let γC ÝÑ CP8 be the complex tautological line bundle. By [34, Theorem 14.5], H 2 pCP8 ; Zq is freely generated by c1 pγC q. Along with the exactness of the top row in (A.41), this implies that δq1 “ ˘c1 in this row. By [34, Theorem 14.6], for every complex line bundle L over Y there exists a continuous map f : Y ÝÑ CP8 such that L “ f ˚ γC . Along with the commutativity of (A.41), these statements imply that δq1 “ ˘c1 in the bottom row in (A.41) as well. The claim now follows from the exactness of this row.  Remark A.9. The statement and proof of Lemma A.8 apply to any paracompact space Y satisfying the k “ 2 case of Definition 3.2 with H p p¨; Z2 q replaced by H p p¨; Zq. Corollary A.10. Let Y be a CW complex with cells of dimension at most 2. If H 2 pY ; Zq “ t0u, then every orientable vector bundle V over Y is trivializable. Proof. Let n “ rk V . If n “ 1, then V is an orientable line bundle and is thus trivializable. Suppose n ě 2. Since the cells of Y are of dimension at most 2, there exists a rank 2 orientable vector bundle L over Y such that ˘ ` (A.42) V « L ‘ Y ˆRn´2 . The real vector bundle L admits a complex structure i. It can be obtained by fixing an orientation and a metric on L and defining iv P L for v P L non-zero to be the vector which is orthogonal to v and has the same length as v so that v, iv form an oriented basis for a fiber of L. By Lemma A.8, pL, iq is trivializable as a complex line  bundle. Along with (A.42), this establishes the claim. Corollary A.11. Let Σ be a surface, possibly with boundary, and n ě 3. The map OVBn pΣq ÝÑ H 2 pΣ; Z2 q,

V ÝÑ w2 pV q,

from the set of isomorphism classes of rank n-oriented vector bundles over Σ is a bijection. Proof. We can assume that Σ is connected. If Σ is not compact or has boundary, then H 2 pY ; Zq, H 2 pΣ; Z2 q “ t0u.

Appendices

431

By Corollary A.10, we can thus assume that Σ is closed and so H 2 pΣ; Z2 q « Z2 . Let C Ă Σ be an embedded loop separating Σ into two surfaces, Σ1 and Σ2 , with boundary C. By Corollary A.10, a rank n-oriented vector bundle V over Σ is isomorphic to the vector bundle obtained by gluing Σ1 ˆ Rn and Σ2 ˆ Rn along C ˆ Rn by a clutching map ϕ : C ÝÑ SOpnq. Since n ě 3, π1 pSOpnqq « Z2 . It thus remains to show that there exists an orientable vector bundle V over Σ with w2 pV q ‰ 0. Let γC;1 ÝÑ CP1 be the complex tautological line bundle. Since w2 pγC;1 q is the image of c1 pγC;1 q under the reduction homomorphism ` ` ˘ ˘ H 2 CP1 ; Z ÝÑ H 2 CP1 ; Z2 , w2 pγC;1 q ‰ 0 by the proof of Lemma A.8. If f : Σ ÝÑ CP1 is a degree 1 map with respect to the Z2 -coefficients, then @

D @ D @ D w2 pf ˚ γC;1 q, rΣsZ2 “ w2 pγC;1 q, f˚ rΣsZ2 “ w2 pγC;1 q, rCP1 sZ2 ‰ 0.

Thus, w2 of the orientable vector bundle ` ˘ V ” f ˚ γC;1 ‘ ΣˆRn´2 ÝÑ Σ is non-zero.



r is a compact surface with two boundary Corollary A.12. Suppose Σ p is a closed surface obtained from Σ r by identifying components and Σ ` p p be the orithese components with each other. Let n P Z and V ÝÑ Σ n r entable vector bundle obtained from ΣˆR by identifying its restricr via a clutching map ϕ : S 1 ÝÑ SOpnq. If Σ p is connected tions to BΣ and n ě 3, then ϕ is homotopically trivial if and only if w2 pVp q “ 0. Proof. By Corollary A.10, every rank n orientable vector bundle r is trivializable. Thus, every rank n orientable vector bundle Vp over Σ n by identifying its restrictions to the p is obtained from ΣˆR r over Σ r via a clutching map ϕ : S 1 ÝÑ SOpnq. Since two components of BΣ ` ˘ p Z2 q « Z2 , π1 SOpnq « Z2 and H 2 pΣ; the claim thus follows from Corollary A.11.



Corollary A.13. Let Σ be a compact connected surface with boundary components C, C1 , . . . , Ck and V be an orientable vector bundle over Σ. If rk V ě 3, then every trivialization of V over C1 Y¨ ¨ ¨YCk

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Spin/Pin-Structures & Real Enumerative Geometry

extends to a trivialization Ψ of V over Σ and the homotopy class of the restriction of Ψ to V |C is determined by the homotopy class of its restriction to V |C1 Y¨¨¨YCk . Proof. Let n “ rk V and choose an orientation on V . Denote by p the connected surface with one boundary component C obtained Σ from Σ by attaching the 2-disks Di2 along the boundary compop obtained by nents Ci . Let Vp be the oriented vector bundle over Σ identifying each Di2 ˆRn with V over Ci via the chosen trivialization φi . By Corollary A.10, the oriented vector bundle Vp admits a trivialization Ψ. Since there is a unique homotopy class of trivializations of Vp |Di2 , the restriction of Ψ to V |Ci is homotopic to φi and thus can be deformed to be the same. Suppose Ψ, Ψ1 are trivializations of V ÝÑ Σ restricting to the p same trivializations φi of V |Ci for every i “ 1, . . . , k. Denote by Σ r the closed (resp. compact) surface obtained from two copies (resp. Σ) of Σ, Σ and Σ1 , by identifying them along the boundary components r has two corresponding to C, C1 , . . . , Ck (resp. C1 , . . . , Ck ). Thus, Σ p can boundary components, each of which corresponds to C, and Σ r be obtained from Σ by identifying these two boundary components. Let r ÝÑ Σ and qp: Σ p ÝÑ Σ qr: Σ be the natural projections. The trivializations Ψ and Ψ1 induce a r which restricts to Ψ and Ψ1 over Σ, Σ1Ă r of qr˚ V over Σ trivialization Ψ p is obtained from qr˚ V by identifying the p Σ. r The bundle qp˚ V over Σ Σ, copies of V |C via the clutching map ϕ : S 1 ÝÑ SOpnq determined by the difference between the trivializations of V |C induced by Ψ and Ψ1 . Since ˘ ` p Z2 q, w2 qp˚ V “ qp˚ w2 pV q “ 0 P H 2 pΣ; Corollary A.12 implies that ϕ is homotopically trivial. Thus, Ψ and Ψ1 determine the same homotopy class of trivializations of V |C .  For an oriented vector bundle V ÝÑ Y, let TrivpV q denote the set of homotopy classes of trivializations of V. For an oriented vector bundle V over a surface Σ, we define the map ˘ ` (A.43) εV : Triv V |BΣ ÝÑ Z2 by setting εV pφq “ 0 for the trivializations φ of V |BΣ that extend to trivializations of V over Σ and εV pφq “ 1 for the trivializations φ that do not.

Appendices

433

Corollary A.14. Let Σ be a compact connected surface with BΣ ‰ H and V be an oriented vector bundle over Σ. If rk V ě 3, then the map (A.43) is surjective and changing the homotopy class of a trivialization φ over precisely one component of BΣ changes the value εV pφq. Proof. B

This follows from π1 pSOpnqq « Z2 and Corollary A.13.



Lie Group Covers

This appendix reviews basic statements concerning covers of Lie groups by Lie groups that are Lie group homomorphisms. Lemma B.1 describes the structure of connected Lie group covers. Lemma B.2 and Proposition B.3 do the same for covers of disconnected Lie groups with connected restrictions to the identity component of the base. We conclude with examples involving the groups Spinpnq and Pin˘ pnq defined in Sections 2.1 and 2.2, respectively. B.1

Terminology and summary

r ÝÑ G a Lie group covering if G r We call a covering projection q : G and G are Lie groups and q is a Lie group homomorphism. We call r is connected; this implies that so is G. such a cover connected if G Lie group coverings r ÝÑ G q: G

r1 ÝÑ G and q 1 : G

r ÝÑ G r1 are equivalent if there exists a Lie group isomorphism ρ : G 1 such that q “ q ˝ρ. For a connected Lie group G, we denote by CovpGq the set of equivalence classes of connected Lie group coverings of G and by π1 pGq its fundamental group based on the identity 1. For any group H, we denote by SGpHq the set of subgroups of H. The following lemma is established in Section B.2. Lemma B.1. (a) Let G be a connected Lie group. The map ˘ “ ‰ ` ˘ ` r ÝÑ G ÝÑ q˚ π1 G r , q: G CovpGq ÝÑ SG π1 pGq , is a bijection. For every rqs P CovpGq as above, q ´1 p1q is conr tained in the center of G.

434

Spin/Pin-Structures & Real Enumerative Geometry

r ÝÑ G and q 1 : G r1 ÝÑ G1 be connected Lie group cover(b) Let q : G ings. A Lie homomorphism ι : G ÝÑ G1 lifts to a Lie group homor ÝÑ G r1 if and only if morphism r ι: G ˘ ` r Ă q 1 π1 pG r1 q. ι˚ q˚ π1 pGq (B.1) ˚ If such a lift r ι exists, it is unique. For a group G, we denote by AutpGq the group of automorphisms of G. For each k P G, let cpkq : G ÝÑ G cpkqg “ k¨g¨k´1 , denote the conjugation homomorphism of k on G; it preserves every normal subgroup G0 of G. If K and G0 are groups and c : K ÝÑ AutpG0 q

(B.2)

is a homomorphism, the semi-direct product of G0 and K with respect to c is the group ` ˘ G ” G0 ¸c K “ G0 ˆK, pg1 , k1 q¨pg2 , k2 q “ g1 ¨cpk1 qg2 , k1 ¨k2 . The subsets t1uˆK and G0 ˆt1u of G are then a subgroup isomorphic to K and a normal subgroup isomorphic to G0 , respectively. The conjugation action of t1u ˆ K on G0 ˆ t1u under the natural isomorphisms is given by c. Furthermore, the map ˘ L` K ÝÑ G G0 ˆt1u , k ÝÑ rp1, kqs, is a group isomorphism. If G0 is a connected Lie group and K is (at most) countable, then G above is a Lie group and G0 ˆt1u is its identity component. Conversely, if G0 is the identity component of a Lie group G, K is a subgroup of G such that the map K ÝÑ G{G0 ,

k ÝÑ rks,

(B.3)

is a group isomorphism, and (B.2) is the homomorphism induced by the conjugation of G0 by the elements of K, then the map G0 ¸c K ÝÑ G,

pg, kq ÝÑ gk,

is a Lie group isomorphism. An extension of a group K by another group H is a short exact sequence j

r ÝÑ K ÝÑ t1u r r ÝÑ H ÝÑ K t1u

(B.4)

Appendices

435

of groups. An extension of a group homomorphism ι : K ÝÑ K 1 by a group homomorphism r ι0 : H ÝÑ H 1 is a commutative diagram r t1u

/H r ι0

r t1u



/ H1

r /K 

j

r ι

/K r1

/K

/ t1u r

(B.5)

ι j1



/ K1

/ t1u r

of extensions of K by H and of K 1 by H 1 . An extension of K by H as r jq replaced in (B.4) is isomorphic to another such extension with pK, 1 1 r by pK , j q if there exists a commutative diagram as in (B.5) with K 1 “ H, ι “ id, and r ι0 “ id. We denote by ExtpK, Hq the set of equivalence classes of extensions of K by H. r on H Ă K r If H in (B.4) is abelian, the conjugation action of K descends to an action of K on H by group isomorphisms. If H is abelian and c : K ÝÑ AutpHq is any homomorphism, denote by Extc pK, Hq Ă ExtpK, Hq the subset of equivalence classes of extensions as in (B.4) so that ` ˘ r r r r; kq r g @ gr P H Ă K, kPK (B.6) k¨r g ¨r k´1 “ c jpr this condition is well defined on the equivalence classes. Suppose G is a Lie group, G0 is its identity component, and r0 ÝÑ G0 is a connected Lie group covering. In particular, G0 q0 : G is a normal subgroup of G. If k P G and the conjugation homomorr0 qq of π1 pG0 q, phism cpkq of k on G0 preserves the subgroup q0˚ pπ1 pG then it lifts uniquely to a Lie group automorphism ` ˘ ` ˘ r0 , 1 ÝÑ G r0 , 1 . rcq pkq : G 0

We denote by Covq0 pGq the set of equivalence classes of Lie group coverings q of G that restrict to q0 over G0 . The following two statements are established in Section B.3. Lemma B.2. Suppose G is a Lie group, G0 is its identity comr0 ÝÑ G0 is a connected Lie group covering. If ponent, and q0 : G r0 qq of π1 pG0 q is preserved Covq0pGq ‰ H, then the subgroup q0˚ pπ1 pG by the conjugation homomorphism cpkq for every k P G. Proposition B.3. Suppose G is a Lie group, G0 is its identity comr0 ÝÑ G0 is a connected Lie group covering such that ponent, q0 : G

436

Spin/Pin-Structures & Real Enumerative Geometry

r0 qq of π1 pG0 q is preserved by the conjugation the subgroup q0˚ pπ1 pG homomorphism cpkq for every k P G, and K is a subgroup of G so that the map (B.3) is an isomorphism. (a) The map ‰ “ ‰ “ q r ÝÑ G ÝÑ t1u ÝÑ q ´1 p1q ÝÑ q ´1 pKq ÝÑ K ÝÑ t1u q: G 0 (B.7) is a bijection from Covq0pGq to Extrcq0pK, q0´1 p1qq. (b) If ι is a Lie group automorphism of G preserving the subgroup r0 qq of π1 pG0 q and r ι0 is the lift of ι0 ” ι|G0 to a Lie group q0˚ pπ1 pG r automorphism of G0 , then an extension rι of ι : K ÝÑ ιpKq

by r ι0 : q0´1 p1q ÝÑ q0´1 p1q

induces a Lie group isomorphism r ι, lifting ι and extending r ι0 , between the Lie group coverings determined by the associated extensions of K and of ιpKq by q0´1 p1q. B.2

Proof of Lemma B.1

By [36, Theorem 82.1], for every subgroup H Ă π1 pGq there exists r ÝÑ G such that q˚ πpGq r “ H. a connected covering projection q : G Since π1 pGq is abelian and countably generated, the index of H in G r is a second countable topological space. The is countable and thus G r so that smooth structure on G then lifts to a smooth structure on G q becomes a local diffeomorphism and all deck transformations of q are smooth. r be any preimage of the identity 1 P G. Since the images rPG Let 1 r r ˆ Gq r under the homomorphisms determined by of π1 pGq and π1 pG the continuous maps ` ˘´1 r ÝÑ G ÝÑ G, , G g ÝÑ q r r g ˘ ` ˘ ` ˘ ` r G r ÝÑ GˆG ÝÑ G, r g1 ¨q gr2 , Gˆ g1 , gr2 ÝÑ q r r the lifting property for covering projections [36, are q˚ π1 pGq, Lemma 79.1] implies that the inverse ´1 and product ¨ operar and G rˆG r so that the tions on G lift to continuous maps on G

Appendices

437

diagrams r 1q r pG,

´1

/ pG, r 1q r

q

r G, r 1ˆ r 1q r pGˆ

q



pG, 1q

´1

qˆq



/ pG, 1q

¨

/ pG, r 1q r q



pGˆG, 1ˆ1q

¨



/ pG, 1q

commute. Since q is a local diffeomorphism, the two lifts are smooth. r From the following paragraph, they determine a group structure on G r is the identity element. By construction, q commutes with so that 1 the group operations ¨. Since the maps ` ˘ ` ˘ r 1 r 1 r ÝÑ G, r , gr ÝÑ r r 1¨r r g, G, g ¨ 1, r 1q. r Since the lift the identity on pG, 1q, they are the identity on pG, maps ` ˘ ` ˘ r 1 r 1 r ÝÑ G, r , r g, G, g ÝÑ gr¨r g ´1 , gr´1 ¨r r 1q. r lift the constant map on pG, 1q, they are the constant map on pG, Since the maps ` ˘ ` ˘ r Gˆ r G, r 1ˆ r 1 r 1ˆ r 1 r ÝÑ G, r , Gˆ (B.8) ` ˘ ˘ ` ˘ ` g3 , gr1 ¨r g2 ¨r g1 ¨ gr2 ¨r g3 , g1 , gr2 , gr3 ÝÑ r r lift the maps ` ˘ GˆGˆG, 1ˆ1ˆ1 ÝÑ pG, 1q,

`

` ˘ ˘ ` ˘ g1 , g2 , g3 ÝÑ g1¨ g2¨g3 , g1¨g2 ¨g3 ,

which are the same, the two maps in (B.8) are also the same. Thus, rˆG r constructed in the previous paragraph the lifted map ¨ on G r so that 1 r is the identity element and defines a group structure on G ´1 is the inverse operation. the lifted map Let r h P q ´1 p1q. Since the map ` ˘ ` ˘ r 1 r 1 r ÝÑ G, r , r G, g ÝÑ r h¨r g ¨r h´1 , r 1q. r Thus, q ´1 p1q lifts the identity on pG, 1q, it is the identity on pG, r and is in particular is contained in the center of the Lie group G abelian.

438

Spin/Pin-Structures & Real Enumerative Geometry

Let q, q 1 , and ι be as in Lemma B.1(b). By the lifting property for covering projections [36, Lemma 79.1], ι lifts to a continuous map r ι so that the diagram `

r 1 r G,

˘

r ι

`

/ G r1 , 1 r1

q



pG, 1q

ι



˘

q1

/ pG1 , 11 q

commutes if and only if (B.1) holds; if such a lift rι exists, it is unique. If it exists, r ι is a smooth map because q and q 1 are local diffeomorphisms and ι is a smooth map. Since the maps ˘ ` ˘ ` ˘ ` ˘ ` ˘ ` 1 1˘ ` r G, r 1ˆ r ,1 r , r 1 r ÝÑ G g2 , r gr1 , gr2 ÝÑ r ι r g1 ¨r ι gr1 ¨r ι r g2 (B.9) Gˆ lift the maps ` ˘ GˆG, 1ˆ1 ÝÑ pG1 , 11 q,

`

˘ ` ˘ g1 , g2 ÝÑ ι g1 ¨g2 , ιpg1 q¨ιpg2 q,

which are the same, the two maps in (B.9) are also the same. Thus, the map r ι is a Lie group homomorphism. This establishes Lemma B.1(b). r ÝÑ G is a connected cover of a connected Lie group Suppose q : G r 1 r1 P q ´1 p1q. Since π1 pGq is with the identity 1 P G as above and 1, r1 “ ρp1q r for some deck abelian, q is a regular covering and thus 1 1 r with transformation ρ of q. If ¨ and ¨ are Lie group structures on G 1 r and 1 r , respectively, so that q is a group the identity elements 1 homomorphism with respect to both, then ˘ ` ˘ ` r 1 ÝÑ G, r 11 ρ : G, is a Lie group isomorphism with respect to ¨ and ¨1 by the previous paragraph. Thus, the Lie group structure on G (uniquely) determines r so that q is a Lie group homomorphism. a Lie group structure on G This establishes Lemma B.1(a). B.3

Disconnected Lie groups

We next establish Lemma B.2 and Proposition B.3 and then give some examples.

Appendices

439

r ÝÑ G is a Lie group covering which Proof of Lemma B.2. If q : G extends q0 , then ` ˘ ` ˘ r gr P G r0 . gq @ r k P G, (B.10) k¨r g ¨r k´1 “ c qpr kq q0 pr q0 r r0 for every k P G. Thus, cpkq lifts to a Lie group automorphism of G By Lemma B.1(b), this implies that cpkq preserves the subgroup r0 qq of π1 pG0 q.  q0˚ pπ1 pG r ÝÑ G is a Lie group Proof of Proposition B.3. Suppose q : G ´1 r is then a subgroup covering which extends q0 . The subset q pKq Ă G so that t1u ÝÑ q0´1 p1q ÝÑ q ´1 pKq ÝÑ K ÝÑ t1u q

(B.11)

is an extension of K by q0´1 p1q. Since q0´1 p1q is abelian by Lemma B.1(a), the conjugation action of q ´1 pKq on q0´1 p1q descends r1 ÝÑ G is to an action of K. By (B.10), the latter action is rcq0 . If q 1 : G another Lie group covering which extends q0 and ρ is an equivalence from q and q 1 , then ρ induces an equivalence / q ´1 pKq j

/ q ´1 p1q 0

t1u

id





/ q ´1 p1q 0

t1u

/K

ρ

/ q 1´1 pKq

q1



/ t1u

id

/K

/ t1u

(B.12)

between the extensions of K by q0´1 p1q determined by q and q 1 . Thus, the map (B.7) is well defined. Suppose conversely that j

r ÝÑ K ÝÑ t1u t1u ÝÑ q0´1 p1q ÝÑ K is an extension of K by q0´1 p1q such that ` ˘ r r G r0 , r r k P K. kq r h @r h P q0´1 p1q Ă K, k¨r h¨ r k ´1 “ rcq0 jpr The quotient ` r” G r0 ¸rc q: G

q0 ˝j

˘L r „, K

`

(B.13)

(B.14)

˘ ˘ ` k @r h P q0´1 p1q g, r r k „ gr¨ r h, r h´1 ¨ r

is then a Lie group so that the maps “ ‰ “ ‰ r r r ÝÑ G, r ιpr r0 ÝÑ G, g ÝÑ r g, 1 and ι : K kq “ 1, r k (B.15) G

440

Spin/Pin-Structures & Real Enumerative Geometry

r and are a Lie group isomorphism onto the identity component of G an injective group homomorphism, respectively. The map ` ˘ r ÝÑ G, q rr g q¨jpr kq, q: G g, r ks “ q0 pr is a Lie group covering so that its composition with the first map in (B.15) is q0 . Furthermore, the image of ι is q ´1 pKq and the diagram

id





/K

ι

/ q ´1 pKq q

/ q ´1 p1q 0

t1u

j

/K r

/ q ´1 p1q 0

t1u



/ t1u

id

/K

/ t1u

is an equivalence between extensions of K by q0´1 p1q. Thus, the map (B.7) is surjective. r1 ÝÑ G1 be another Lie group covering and Let q 1 : G `

` ˘ r 1 o G,

r0 , 1 ?_ G

q

˘

r ι0

q01

q0



 ? _ pG0 , 1q

pG, 1q o

` 1 ˘  r ,1 / G 0

ι0

 / pG1 , 1q   0

`

r1 , 1 / G

˘

q1

 / pG1 , 1q 3

ι

be a commutative diagram of Lie group homomorphisms. Since ι is a group homomorphism, ` ˘ ` ˘ (B.16) ι0 cpkqg “ c ιpkq ι0 pgq @ g P G0 , k P K. r ÝÑ G r 1 is a Lie group homomorphism lifting ι and extending r ι0 , If r ι: G then the diagram t1u

/ q ´1 p1q 0 r ι0

t1u



/ q 1´1 p1q 0

/ q ´1 pKq 

q

/K

r ι

/ q 1´1 pιpKqq

/ t1u

ι q1



/ ιpKq

/ t1u

(B.17)

is an extension of ι by r ι0 . Suppose conversely that (B.17) is an extension of ι by r ι0 . By (B.16) and Lemma B.1(b), ` ˘ ` ˘ r0 , r r g q @ gr P G k P K. kqr g “c r ιpr kq rι0 pr r ι0 cpr

Appendices

441

This in turn implies that the map ` ˘ r0 , k P q ´1 pKq, r ÝÑ G r1 , r ι gr¨ r k “r ι0 pr g q¨r ιpr kq @ gr P G rι : G

(B.18)

is a Lie group homomorphism lifting ι and extending r ι0 . We now apply the conclusion of the previous paragraph with G “ G1 and q0 “ q01 . If ι is as in Proposition B.3(b), then ι0 ” ι|G0 lifts to a Lie group isomorphism ` ˘ ` ˘ r0 , 1 ÝÑ G r0 , 1 r ι0 : G by Lemma B.1(b). By the previous paragraph, an extension t1u

/ q ´1 p1q 0 r ι0

t1u



/ q ´1 p1q 0

r /K 

j

r ι

/K r1

/K

/ t1u

ι j1



/ ιpKq

/ t1u

(B.19)

ι0 |q´1 p1q induces an isomorphism (B.18), lifting ι and of ι|K by r 0 extending r ι0 , between the Lie group coverings q and q 1 associated with the first and second lines in (B.19). This establishes Proposition B.3(b). In particular, an equivalence between the extensions of K by q0´1 p1q determined by two covers q and q 1 of G extending q0 as in (B.7) induces an equivalence between q and q 1 . Thus, the map (B.7)  is injective. This concludes the proof of Proposition B.3(a). Example B.4. The groups Pin˘ pnq defined directly in Section 2.2 are the quotients of certain semi-direct products Spinpnq ¸ Z22 and Spinpnq ¸ Z4 by natural Z2 -actions. The subgroup q0´1 p1q « Z2 of Spinpnq appearing in (B.13) is the subgroup generated by the element pIn in the notation of Section 2.1. The subgroup K « Z2 of Opnq can be taken to be the subgroup generated by any order 2 element of Opnq´SOpnq, i.e. an order 2 element of Opnq with an odd number of p´1q-eigenvalues. Any two distinct order 2 elements of Opnq with precisely one p´1q-eigenvalue are contained in a subgroup G isomorphic to Op2q and are interchanged by an automorphism ι of Opnq which preserves G. By Proposition B.3(b), ι lifts to an isomorphism r ι

442

Spin/Pin-Structures & Real Enumerative Geometry

˘ between the Lie group coverings Pin˘ K1 pnq, PinK2 pnq determined by the extensions

t1u

id

t1u

/ Pin˘ p1q q

/ Z2





/ Z2

id q1

/ Pin˘ p1q

/ K1

/ t1u

 / K2

/ t1u

of the subgroups K1 , K2 generated by the two elements. The restric˘ tion of r ι to Pin˘ K1 pnq|G is an isomorphism onto PinK2 pnq|G . By Example 2.5, this implies that ˇ ˇ Pin˘ pnqˇ “ Pin˘ pnqˇ K2

G

K1

G

˘ and so Pin˘ K2 pnq “ PinK1 pnq. Thus, the criterion for distinguishing between Pin´ pnq and Pin` pnq above Definition 1.1 and Example 2.5 does not depend on the choice of order 2 element of Opnq with precisely one p´1q-eigenvalue used to generate the subgroup K. However, the first statement in (2.30) implies that the subgroup of Pin` pnq generated by the preimages of an order 2 element of Opnq with precisely three p´1q-eigenvalues is Z4 . Thus, the criterion for distinguishing between Pin` pnq and Pin´ pnq depends on the choice of the conjugacy class of order 2 elements of Opnq. In the notation of Example 2.5,

` ˘L Pin` p2q “ R{2π Z ¸c` Z22 „,

` ˘L Pin´ p2q “ R{2π Z ¸c´ Z4 „,

c` pa, bqpθ`2π Zq “ p´1qb θ`2π Z, ˘ ` ˘ ` θ`2π Z, pa, bq „ θ`π`2π Z, pa`1, bq ,

c´ paqpθ`2π Zq “ p´1qa θ`2π Z, `

˘ ` ˘ θ`2π Z, a „ θ`π`2π Z, a`2 .

Example B.5. In the case of the lift r ιn;m of ιn;m in (2.32), (B.16) becomes ι1n;m pAq “ cpIn;1 qι1n;m pAq ι2n;n´m pcpIn´m;1 qAq “ cpIn;1 qι2n;n´m pAq

@ A P SOpmq, @ A P SOpn´mq.

Appendices

443

These two conditions are equivalent to the two equations in (2.31). In the situation of Remark 2.6, the diagram (B.17) would become t1u

/ Z2 2 r ι0

t1u

r n´m /K rm ˆK 



q

r ι

/K rn

/ Z2

/ Km ˆKn´m

/ t1u

ι q1



/ Kn

/ t1u

r n denoting the group generated by In;1 and its preimwith Kn and K ˘ age in Pin pnq, respectively. However, there is no group homomorphism r ι making this diagram commute. Example B.6. The connected Lie group double cover q0 of Spinp2q « S 1 does not extend to a Lie group double cover q of Pin´ p2q ” R{2πZˆZ2 , ¨ : Pin´ p2qˆPin´ p2q ÝÑ Pin´ p2q, ` ˘ pθ1 , k1 q¨pθ2 , k2 q “ θ1 `p´1qk1 θ2 `k1 k2 π, k1 `k2 . Such an extension would be of the form q : R{2πZˆZ2 ÝÑ R{2πZˆZ2 ,  ( qpθ, kq “ qp2θ, kq, gr0 ” p0, 1q¨p0, 1q P pπ{2, 0q, p3π{2, 0q . The conjugation homomorphism on the double cover of Spinp2q by p0, 1q is the lift of the conjugation homomorphism on Spinp2q by p0, 1q and is thus given by S 1 ÝÑ S 1 ,

θ`2πZ ÝÑ ´θ`2πZ.

g0 cannot be the Since the lifted homomorphism does not fix gr0 , r square of p0, 1q in a Lie group double cover of Pin´ p2q. Since every element of Pin´ p2q´Spinp2q is of order precisely 4, Pin´ p2q contains no subgroup projecting isomorphically to Pin´ p2q{Spinp2q and thus Proposition B.3 does not apply in this case.

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Index of Terms

A

exact triple, 155 orientation, 156

admissible curve, 297 B

G

bordered surface, 6

geometric genus, 317, 325

C

H

closed surface, 6

half-surface, 157

D

I

determinant line bundle, 156

immersion, 297 involution, 156

E

J

elemental component conjugate, 157 real, 157 exponential-like map, 344

J-holomorphic map, 325 real, 328 K

F

Kontsevich’s recursion, 320

flat family of deformations, 171 Fredholm operator determinant, 156

L loop, 6 449

450

Spin/Pin-Structures & Real Enumerative Geometry

N n-connected, 90 nodal surface, 156 node of immersion, 297 type E, 156 type H, 157 normalization, 156 normalization of a curve, 316, 325 O OSpin-structure, 5 equivalent, 5 P Pin group Pin˘ pnq, 4, 36 Pin˘ p1q, 15 Pin˘ p2q, 36 Pin-structure classical, 4 CW trivializations, 6 equivalent, 5 loop trivializations, 6 R rational curve, 317, 325 real bundle pair, 158 C-balanced, 159 dual, 338 pp, qq-forms, 338

real CR-operator, 158, 161 real curve, 325 real symplectic manifold, 323 reduced curve, 316, 325 relative OSpin-structure, 114 relative Pin-structure equivalent, 112 intrinsic, 114 Solomon, 111 relative Spin-structure equivalent, 112 FOOO, 112 intrinsic, 114 S separating fixed locus, 157 simple map, 325, 328 simple marked map, 329 simple node, 317, 326 Spin group Spinpnq, 4, 28 Spinp1q, Spinp2q, 15 Spinp3q, 28 Spin-structure classical, 4 CW trivializations, 6 equivalent, 5 loop trivializations, 6 symmetric surface, 156 decorated, 159 marked, 157 Riemann, 157

Index of Notation

r φ pX, Y q, 324 H 2

rks, rls, 157 rX, Zs, 90 rpX, Y q, pZ, z0 qs, 90

pI1 , 15 rI1;1 , 15 In , rIn , pIn , 28 In;1 , rIn;1 , pIn;1 , 33, 35 In;2 , rIn;2 , pIn;2 , 30, 32 In;4 , rIn;4 , pIn;4 , 30, 32 In;pmq , rIn;pmq , pIn;pmq , 33, 35 In;pm1 ,m2 q , rIn;pm1 ,m2 q , pIn;pm1 ,m2 q , 32 In;n , rIn;n , pIn;n , 35 ι˘ ι˘ n ,r n , 39 ιn;m , ι1n;m , ι2n;m , 27 rιn;m , r ι 1n;m, r ι 2n;m , 29, 37–38 ι 1n;m , r ι 2n;m , 30 r ι n;m , r

c1 pX, ωq, 324 cpBq,rcpBq, 32 δpαq, 297 dX;Y , 324 dY , 393 BY ;Z2 , 323 η, 168 gpΣq, 160 γC , 17 γC;2 , 315 γR , 16 γR;1 , 17 γR;n , 18

ω , 325 λpDq, 156 λpoq, 3 λpV q, 3 λpV, oq, 3 LpY q, 6

p ˚ pX, Y q, 356 p ˚ pX, Y q,H H 2˚ p pX, Y q, 356 H H2φ pX; Zq, 323 451

452

Spin/Pin-Structures & Real Enumerative Geometry

L˚ pY q, 297 LX pY q, 113

qn , 4, 28 qn˘ , 4, 35

μp , 299

s0 pV, oq, 16 SOpnq, 4 SOpV, oq, 4, 44 sp;h puq, 386 sp;l;h puq, 373 Spinpnq, 4 SppV, oq, 7 SpX pV, oq, 115 StpV, oq, 3

o1eo2 , 12 Opnq, 4 o1 o2 , 12 os0 pV, oq, 5–6 OSppV q, 6 OSpX pV q, 114 ØpV q, 43 OpV q, 3 OpV q, 4 oY , 3 ´ p´ 0 pγR;1 q, p1 pγR;1 q, 17, 19 ´ 2 2 p0 pRP q, p´ 1 pRP q, 20 ˘ p0 pV q, 20–21 Pin˘ pnq, 4 P˘ pV q, 6 ˘ PX pV q, 114

q1 , q2 , 15 q1˘ , 15 q2˘ , 36 q3 , 28

τ , 168 τn , 320 τY , 3 tos pαq, 298 tp pαq, 298 V˘ , 6 W1 pV, ϕq, 160 w2 psq, w2 posq, w2 ppq FOOO, 112 intrinsic, 114 os pΣ˚ q, 161 p pΣ˚ q, 161