Equivariant Poincaré Duality on G-Manifolds: Equivariant Gysin Morphism and Equivariant Euler Classes (Lecture Notes in Mathematics) [1st ed. 2021] 3030704394, 9783030704391

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Lecture Notes in Mathematics  2288

Alberto Arabia

Equivariant Poincaré Duality on G-Manifolds Equivariant Gysin Morphism and Equivariant Euler Classes

Lecture Notes in Mathematics Volume 2288

Editors-in-Chief Jean-Michel Morel, CMLA, ENS, Cachan, France Bernard Teissier, IMJ-PRG, Paris, France Series Editors Karin Baur, University of Leeds, Leeds, UK Michel Brion, UGA, Grenoble, France Camillo De Lellis, IAS, Princeton, NJ, USA Alessio Figalli, ETH Zurich, Zurich, Switzerland Annette Huber, Albert Ludwig University, Freiburg, Germany Davar Khoshnevisan, The University of Utah, Salt Lake City, UT, USA Ioannis Kontoyiannis, University of Cambridge, Cambridge, UK Angela Kunoth, University of Cologne, Cologne, Germany Ariane Mézard, IMJ-PRG, Paris, France Mark Podolskij, University of Luxembourg, Esch-sur-Alzette, Luxembourg Sylvia Serfaty, NYU Courant, New York, NY, USA Gabriele Vezzosi, UniFI, Florence, Italy Anna Wienhard, Ruprecht Karl University, Heidelberg, Germany

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Alberto Arabia

Equivariant Poincaré Duality on G-Manifolds Equivariant Gysin Morphism and Equivariant Euler Classes

Alberto Arabia IMJ-PRG, CNRS Paris Diderot University Paris Cedex 13 France

ISSN 0075-8434 ISSN 1617-9692 (electronic) Lecture Notes in Mathematics ISBN 978-3-030-70439-1 ISBN 978-3-030-70440-7 (eBook) https://doi.org/10.1007/978-3-030-70440-7 Mathematics Subject Classification: 55, 55M05, 57R91, 57S15, 55N30, 14F05, 20Cxx © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This is an introduction to equivariant Poincaré duality of oriented G-manifolds, for a compact Lie group G, and to equivariant Gysin morphisms for both proper and non-proper maps, with applications as: the equivariant Gysin exact sequence, the equivariant Lefschetz fixed point theorem, the equivariant Thom isomorphism, and the equivariant Thom and Euler classes. For the most part, we have chosen to focus on smooth manifolds and de Rham (equivariant) cohomology, thereby restricting coefficients to the field of real numbers R. However, we also give a more general and parallel approach, based on the Grothendieck-Verdier’s formalism, which allows us to simultaneously treat equivariant and nonequivariant theories as well as substituting the field of coefficients. The material is organized into three parts: • The first part, which consists of Chaps. 2 and 3, is about nonequivariant de Rham cohomology. Chapter 2 summarizes familiar results and constructions related to Poincaré duality and Gysin morphisms, which we extended, in Chap. 3, to fiber bundles following Grothendieck’s relative point of view. In doing this, we were motivated by the fact that, for a G-manifold M, Poincaré duality of its homotopy quotient MG , relative to the classifying space IBG, is the precise analogue to Gequivariant Poincaré duality of M. This approach thus opens the way to a general definition of equivariant cohomology, not only beyond manifolds but also over arbitrary coefficient rings. • The second part consists of Chaps. 4 to 7 devoted to equivariant de Rham cohomology of manifolds. Chapter 4 reviews the origins of equivariant cohomology, recalls standard definitions and constructions in equivariant de Rham cohomology, and ends with the equivariant de Rham comparison theorems that identify equivariant de Rham and singular cohomologies (4.8). Chapter 5 is devoted to equivariant de Rham Poincaré duality, which we apply in Chap. 6 to define equivariant de Rham Gysin morphisms and Euler classes, which are localized in Chap. 7. • The third part, Chap. 8, explains the approach of equivariant cohomology within the framework of Grothendieck-Verdier duality, whose great advantage is its v

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enormous flexibility, in particular, the fact that the coefficients field k is irrelevant. Chapter 8 thus presents an alternative approach for dealing with equivariant cohomology, equivariant Poincaré duality and equivariant Gysin morphisms for manifolds, allowing the results of previous chapters to be extended to arbitrary coefficient fields. We assume the reader to be familiar with algebraic topology techniques as well as with sheaf cohomology and with derived categories as they appear in Grothendieck-Verdier’s duality theory, for example, as explained in Iversen [58], Kashiwara-Schapira [61, 62], and Weibel [95]. However, for readers with little experience in these subjects we added Appendix A, which gives an overview of Derived Categories and Derived Functors. Beyond recalling the usual terminology and definitions, this appendix describes the common underlying approach for dealing with duality in derived categories of dg-modules over dg-algebras, as it occurs throughout the book. We also assume that the reader knows the basics of equivariant cohomology, both as the ordinary cohomology of the homotopy quotient of G-spaces and as the cohomology of the equivariant de Rham complex in the case of G-manifolds. References for these topics are, for ordinary equivariant cohomology: W. Hsiang [55] and Allday-Puppe [2], and for de Rham equivariant cohomology: Guillemin and Sternberg [50] and Loring Tu [91]. A more specialized reference in equivariant Poincaré duality is the article of Brylinski [24] on G-equivariant Poincaré duality, which, written for stratified spaces and for intersection homology, uses a modified version of the Cartan model relative to a Thom-Mather stratification.When the space is smooth, the model coincides with the usual Cartan model, which is the starting point of this book. Other specialized references are articles by Allday, Franz, and Puppe [2–5], which address G-equivariant Poincaré duality for G := T a torus and coefficients in arbitrary fields using what they call the singular Cartan Model. Equivariant Gysin morphisms, Thom isomorphism, and Euler classes can also be found in Allday et al. [4], Allday-Puppe [2], and Kawakubo [63]. Many people have contributed to the development of equivariant cohomology and equivariant Poincaré duality in significant ways since the 1990s by pursuing other routes than those that we will travel. I apologize in advance to each of them for not having been able to refer to their work as it deserves. The following signs have been used in the book:  to indicate the end of the proof of a theorem, proposition, etc.,  to indicate the end of a sketch of proof,  to indicate the end of a lemma within a proof, to indicate a helpful hint.

Acknowledgements

To Matthias Franz for his remarks on the subject of reflexivity of equivariant cohomology; to Bernhard Keller for enlightening discussions on dg-algebras; to Bruno Klingler both for his helpful remarks and for directing me to valuable references on classification spaces; to the referees whose valuable reviews made this a much better book; to Loïc Merel, Director of the Mathematical Institute of Jussieu (IMJ), who made the institute’s research resources available to me; to Frances Cowell, whose careful proofreading and her wise comments have helped me improve this work; to Ute McCrory, editor at Springer, for her constant encouragement, her gentle prodding and, especially, her remarkable patience, and, of course, to SPRINGER for offering me the opportunity to publish this book. Paris, France January 2021

Alberto Arabia

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Contents

1

2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Equivariant de Rham Poincaré Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Equivariant de Rham Gysin Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Adjunction Properties of Gysin Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Equivariant Cohomology Viewed as a Relative Cohomology Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Equivariant Poincaré Duality over Arbitrary Fields. . . . . . . . . . . . . . . . . . 1.6 Conditions on the Group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Conditions on Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonequivariant Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Category of Cochain Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Fields in Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Vector Spaces and Pairings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 The Category GV(k) of Graded Spaces . . . . . . . . . . . . . . . . . . . . 2.1.4 The Subcategories of Bounded Graded Spaces . . . . . . . . . . . . . 2.1.5 Graded Algebras over Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.6 The Category DGV(k) of Differential Graded Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.7 The Shift Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.8 The Functors Hom•k (−, −) and (− ⊗k −)• . . . . . . . . . . . . . . . . 2.1.9 On the Koszul Sign Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.10 The Functor Hom•k (−, W ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.11 The Duality Functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Categories of Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 The Category of Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 The Category of Manifolds and Proper Maps . . . . . . . . . . . . . . 2.2.4 G-Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 4 5 6 7 8 9 9 9 9 10 10 10 12 13 14 15 16 16 18 18 18 18 19

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2.3

Orientation and Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Orientability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Poincaré Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Poincaré Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 The Fundamental Class of an Oriented Manifold . . . . . . . . . . Poincaré Adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Poincaré Adjoint Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Manifolds and Maps of Finite de Rham Type . . . . . . . . . . . . . . 2.5.3 Ascending Chain Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Existence of Proper Invariant Functions . . . . . . . . . . . . . . . . . . . . 2.5.5 Manifolds with Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.6 Proof of Proposition 2.5.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Gysin Functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 The Right Poincaré Adjunction Map. . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 The Gysin Morphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ............................................ 2.6.3 The Image of DM The Gysin Functor for Proper Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constructions of Gysin Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 The Proper Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.2 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19 19 21 23 23 29 30 30 33 33 34 35 36 37 37 38 42 44 46 46 47

Poincaré Duality Relative to a Base Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Fiber Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 The Categories TopB and ManB . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 The Relative Point of View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Fiber Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5 The Base Change Functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.6 Fiber Products of Fiber Bundles of Manifolds . . . . . . . . . . . . . 3.1.7 Orientable Fiber Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.8 The Categories ManB , FibB and Fibor B ...................... 3.1.9 Proper Subspaces of a Fiber Bundle . . . . . . . . . . . . . . . . . . . . . . . . 3.1.10 Differential Forms with Proper Supports . . . . . . . . . . . . . . . . . . . 3.2 Integration Along Fibers on Fiber Bundles. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 The Case of Trivial Euclidean Bundles . . . . . . . . . . . . . . . . . . . . . 3.2.2 Sheafification of Integration Along Fibers . . . . . . . . . . . . . . . . . . 3.2.3 Thom Class of an Oriented Vector Bundle . . . . . . . . . . . . . . . . . 3.3 Poincaré Duality for Fiber Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Sheafification of the Poincaré Adjunction . . . . . . . . . . . . . . . . . . 3.3.2 Deriving the Sheafified Poincaré Adjunction Functors . . . . 3.3.3 The Poincaré Duality Theorem for Fiber Bundles . . . . . . . . . 3.3.4 Poincaré Duality for Fiber Bundles and Base Change . . . . .

49 49 49 50 51 51 54 56 59 62 62 65 71 72 78 81 83 83 84 85 87

2.4

2.5

2.6

2.7 2.8

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3.4

Poincaré Duality Relative to a Formal Base Space . . . . . . . . . . . . . . . . . . 3.4.1 Formality of Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Poincaré Duality Relative to Classifying Spaces . . . . . . . . . . . Gysin Morphisms for Fiber Bundles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Gysin Morphisms Relative to a Base Space . . . . . . . . . . . . . . . . 3.5.2 Gysin Morphisms for Fiber Bundles and Base Change . . . . Examples of Gysin Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Adjointness of Gysin Morphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Constant Map and Locally Trivial Fibrations. . . . . . . . . . . . . . . 3.6.3 Open Embedding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.4 Proper Base Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.5 Zero Section of a Vector Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.6 Closed Embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Gysin Long Exact Sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 Lefschetz Fixed Point Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90 91 93 95 95 96 98 98 99 100 100 101 103 104 104 106 107

Equivariant Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Significant Dates in Equivariant Cohomology Theory. . . . . . . . . . . . . . . 4.1.1 Cartan’s ENS Seminar (1950) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Borel’s IAS Seminar (1960) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Atiyah-Segal: Equivariant K-Theory (1968) . . . . . . . . . . . . . . . 4.1.4 Quillen: Equivariant Cohomology (1971) . . . . . . . . . . . . . . . . . . 4.1.5 Hsiang’s Book (1975). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.6 Atiyah-Bott and Berline-Vergne: Equivariant de Rham Cohomology (1980) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Category of g-Differential Graded Modules . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Field in Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 The Category of g-Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 g-Differential Graded Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 g-Differential Graded Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Split g-Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Equivariant Cohomology of g-Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 The g-dg-Algebra S(g ∨ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Cartan Complexes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Induced Morphisms on Cartan Complexes . . . . . . . . . . . . . . . . . 4.3.4 Split G-Complexes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Equivariant Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Fields in Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 G-Fundamental Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Interior Products and Lie Derivatives . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Complexes of Equivariant Differential Forms . . . . . . . . . . . . . . 4.4.5 On the Connectedness of G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.6 Splitness of Complexes of Equivariant Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

109 109 109 114 115 115 116

3.5

3.6

3.7

3.8 4

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117 119 119 119 120 121 122 124 124 124 126 128 129 129 129 129 130 131 134

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4.5

Cohomological Properties of Homotopy Quotients . . . . . . . . . . . . . . . . . . 4.5.1 Local Triviality of G-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Slices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Existence of Slices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constructing Classifying Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 The Milnor Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Stiefel Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Borel Construction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 The Homotopy Quotient Functor. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 On the Cohomology of the Homotopy Quotient . . . . . . . . . . . 4.7.3 Orientability of the Homotopy Quotient . . . . . . . . . . . . . . . . . . . . Equivariant de Rham Comparison Theorems . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1 Question Iso 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.2 Question Iso 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.3 Equivariant Cohomology Comparison Theorem . . . . . . . . . . . Cohomology of Classifying Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.1 Canonicity of the Cohomology of Classifying Spaces . . . . . 4.9.2 Formality of Classifying Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . Local Equivariant Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10.1 The Long Exact Sequence of Local Equivariant Cohomology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

139 139 140 141 146 146 147 148 148 148 149 156 157 158 162 163 165 165 166 169

Equivariant Poincaré Duality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Differential Graded Modules over a Graded Algebra . . . . . . . . . . . . . . . . 5.1.1 Graded Modules and Algebras over Graded Algebras . . . . . 5.1.2 The Category of G -Graded Modules . . . . . . . . . . . . . . . . . . . . . . 5.2 The Category of G -Differential Graded Modules . . . . . . . . . . . . . . . . . . 5.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 The Hom•G (−, −) and (−⊗G −)• Bifunctors on DGM(G ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 The Duality Functor on DGM(G ) . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 The Forgetful Functor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.5 On the Exactness of Hom• (−, −) and (−⊗−)• . . . . . . . . . . . . 5.3 Comparing the Categories C(GM(G )) and DGM(G ) . . . . . . . . . . . . 5.3.1 The Tot Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 The Hom•G (−, −) bifunctor on C(GM(G )) . . . . . . . . . . . . . 5.3.3 The (− ⊗G −)• Bifunctor on C(GM(G )) . . . . . . . . . . . . . . . 5.4 Deriving Functors in GM(G ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Augmentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Simple Complex Associated with a Bicomplex . . . . . . . . . . . . 5.4.3 The IR Hom•G (−, −) and (−) ⊗IL Ω G (− ) Bifunctors on GM(G ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 The Ext• and Tor• Bifunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.5 The Duality Functor on D(GM(G )) . . . . . . . . . . . . . . . . . . . . . .

175 175 175 176 179 179

4.6

4.7

4.8

4.9

4.10

5

172

179 180 180 181 182 182 184 186 187 188 188 189 191 192

Contents

xiii

The Duality Functor on D(DGM(G )) . . . . . . . . . . . . . . . . . . . . Spectral Sequences Associated with IR Hom•G (−, G ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equivariant Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Equivariant Integration vs. Integration Along Fibers. . . . . . . Equivariant Poincaré Duality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 The G -Poincaré Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 G-Equivariant Poincaré Duality Theorem . . . . . . . . . . . . . . . . . . 5.6.3 Torsion-Freeness, Freeness and Reflexivity . . . . . . . . . . . . . . . . 5.6.4 T -Equivariant Poincaré Duality Theorem . . . . . . . . . . . . . . . . . .

193 197 197 199 201 201 203 208 209

6

Equivariant Gysin Morphism and Euler Classes. . . . . . . . . . . . . . . . . . . . . . . . . 6.1 G-Equivariant Gysin Morphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Equivariant Finite de Rham Type Coverings. . . . . . . . . . . . . . . . 6.1.2 G-Equivariant Gysin Morphism for General Maps . . . . . . . . 6.1.3 G-Equivariant Gysin Morphism for Proper Maps . . . . . . . . . . 6.1.4 Gysin Morphisms through Spectral Sequences . . . . . . . . . . . . . 6.2 Group Restriction and Equivariant Gysin Morphisms . . . . . . . . . . . . . . . 6.2.1 Group Restriction and Equivariant Cohomology . . . . . . . . . . . 6.2.2 Group Restriction and Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Adjointness of Equivariant Gysin Morphisms . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Adjointness Property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Explicit Constructions of Equivariant Gysin Morphisms . . . . . . . . . . . . 6.4.1 Equivariant Open Embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Equivariant Constant Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Equivariant Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.4 Equivariant Fiber Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.5 Zero Section of an Equivariant Vector Bundle . . . . . . . . . . . . . 6.4.6 Equivariant Gysin Long Exact Sequence . . . . . . . . . . . . . . . . . . . 6.5 Equivariant Euler Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 The Nonequivariant Euler Class. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 G-Equivariant Euler Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3 G-Equivariant Euler Class of Fixed Points . . . . . . . . . . . . . . . . . 6.5.4 T -Equivariant Euler Class of Fixed Points . . . . . . . . . . . . . . . . .

211 211 211 212 214 215 216 216 217 218 218 220 220 220 221 222 222 226 228 228 229 230 230

7

Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 The Localization Functor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Localized Equivariant Poincaré Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Localized Equivariant Gysin Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Torsion in Equivariant Cohomology Modules . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Slices and Orbit Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 The General Slice Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Orbit Type of T -Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Localized Gysin Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

235 235 236 236 237 237 238 238 239 240

5.4.6 5.4.7

5.5

5.6

192

xiv

Contents

7.7 8

The Localization Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 7.7.1 Inversibility of Euler Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

Changing the Coefficients Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Comments about Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Sheafification of Cartan Models over Arbitrary Fields . . . . . . . . . . . . . . 8.2.1 Dictionary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Reformulation of The Poincaré Adjunctions . . . . . . . . . . . . . . . 8.3 Equivariant Poincaré Duality over Arbitrary Fields. . . . . . . . . . . . . . . . . . 8.3.1 The Equivariant Duality Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Formality of IBG over Arbitrary Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 The Integral Cohomology of G/T . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Equivariant Gysin Morphisms over Arbitrary Fields . . . . . . . . . . . . . . . . 8.5.1 Gysin Morphism for General Maps . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 Gysin Morphism for Proper Maps . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 The Localization Formula over Arbitrary Fields . . . . . . . . . . . . . . . . . . . . .

245 245 246 247 250 251 256 256 257 257 261 262 263 263

A Basics on Derived Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Categories of Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.1 The Category of Complexes of an Abelian Category . . . . . . A.1.2 Extending Additive Functors from Ab to C(Ab) . . . . . . . . . . . A.1.3 The Mapping Cone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.4 Homotopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.5 The Homotopy Category K(Ab) . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.6 The Derived Category D(Ab) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.7 The Subcategories C∗ (Ab), K∗ (Ab) and D∗ (Ab) . . . . . . . . . . A.2 Deriving Functors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.1 Extending Functors from Ab to D(Ab) . . . . . . . . . . . . . . . . . . . . . A.2.2 Extending Functors from C(Ab) to K(Ab) . . . . . . . . . . . . . . . . . A.2.3 Extending Functors from K(Ab) to D(Ab) . . . . . . . . . . . . . . . . A.2.4 Acyclic Resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.5 The Duality Functor on D(DGM(G )) . . . . . . . . . . . . . . . . . . . . A.3 DG-Modules over DG-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.1 K-Injective (A, d)-Differential Graded Modules . . . . . . . . . . A.3.2 Formality of DGA’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.3 Formality of DGM’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

265 265 266 267 267 268 270 275 279 279 279 279 280 285 287 292 292 293 295

B Sheaves of Differential Graded Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 Mild Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 The Sheaf of Functions OX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Global Lifting of Germs on OX -modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4 OX -Graded Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

301 301 302 303 305

Contents

B.5

B.6

B.7 B.8 B.9

xv

Localization Functor for OX -GA’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.5.1 The Isomorphism Hom•A (A, −)  Γ (Y ; −) . . . . . . . . . . . . . . B.5.2 Right Adjoint to Hom•A (A, −) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.5.3 The Localization Functor for OX -GA’s . . . . . . . . . . . . . . . . . . . . . Equivalences of Some Derived Functors in D GM(A) . . . . . . . . . . . . . . B.6.1 An Equivalence of Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.6.2 Family of Supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.6.3 The functor Γ in GM(A(X)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . OX -Differential Graded Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.7.1 Localization Functor for OX -DGA’s . . . . . . . . . . . . . . . . . . . . . . . . The Localization Functor for OX -DGA’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equivalences of Derived Functors in DDGM(A, d) . . . . . . . . . . . . . . . . B.9.1 K-Injective Differential Graded Modules . . . . . . . . . . . . . . . . . .

306 306 307 308 310 310 311 312 315 315 316 317 317

C Cartan’s Theorem for g-dg-Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 D Graded Ring of Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 E Hints and Solutions to Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

Chapter 1

Introduction

1.1 Equivariant de Rham Poincaré Duality Given a compact Lie group G, we prove G-equivariant Poincaré duality for an oriented G-manifold M following J.-L. Brylinski’s approach for the equivariant intersection cohomology of G-pseudomanifolds [24]. We will therefore work in the category DGM(G ) of G -differential graded modules (G -dgm), where G denotes the ring S(g ∨ )G of G-invariant real polynomial functions on g := Lie(G) endowed with the grading that doubles polynomial degrees. By setting its differential to zero, G becomes a differential graded ring; which coincides with its cohomology ring, traditionally denoted by HG . The complexes of G-equivariant differential forms G (M) and G,c (M) (4.4.4), which respectively compute the equivariant de Rham cohomologies HG (M) and HG,c (M), belong to the category DGM(G ). When M is an oriented G-manifold of dimension dM , ordinary Poincaré pairing naturally extends to equivariant differential forms as an G -pairing:  ·, · M,G : G (M) × G,c (M) → G ,

 α, β M,G :=

 M

α∧β,

giving rise to the G-equivariant Poincaré adjunctions, the morphisms in DGM(G ):  IDG,M : G (M)[dM ] → Hom•G (G,c (M), G ) (1.1)  : G,c (M)[dM ] → Hom•G (G (M), G ) IDG,M defined respectively by: IDG,M (α) : β →  α, β M,G

and

 IDG,M (β) : α →  α, β M,G ,

for which we have the following equivariant analogue to Poincaré duality theorem. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Arabia, Equivariant Poincaré Duality on G-Manifolds, Lecture Notes in Mathematics 2288, https://doi.org/10.1007/978-3-030-70440-7_1

1

2

1 Introduction

Theorem (5.6.2.1) Let G be a compact Lie group, and let M an oriented Gmanifold of dimension dM . 1. The following left Poincaré adjunction is an injective quasi-isomorphism, ID G,M : G (M)[dM ] −→ Hom•G (G,c (M), G ) .

(1.2)

2. The morphism ID G,M induces the Poincaré duality morphism in G-equivariant cohomology (see 5.4.7.2–(1)) D G,M : HG (M)[dM ] −→ Hom•HG (HG,c (M), HG ) which is an isomorphism if ExtiHG (HG,c (M), HG ) = 0 for all i > 0, for example if HG,c (M) is a free HG -module. 3. If G is connected, then there exist spectral sequences converging to HG (M)[dM ] 

p,q

p

IE 2 (M) = (ExtHG (HG,c (M), HG ))q p,q

p

d +p+q

⇒ HGM

−q

d +p+q

IF 2 (M) = HG ⊗R Hom•R (Hc (M), R) ⇒ HGM

(M) (M)

where, in the first, q refers to the graded vector space grading. 4. If, in addition, M is of finite type, the right Poincaré adjunction ID G,M : G,c (M)[dM ] −→ Hom•G (G (M), G )

(1.3)

is an injective quasi-isomorphism, and mutatis mutandis for (2) and (3).

1.2 Equivariant de Rham Gysin Morphisms As we will show equivariant Poincaré duality is better approached in the derived category D(DGM(G )) of DGM(G ) (5.4.6, A.2.5) by a systematic use of the derived duality functor IR Hom•G (−, G ) : D(DGM(G ))  D(DGM(G )) .

(1.4)

Indeed, although in the statements of theorem 5.6.2.1 we still use nonderived Hom• , possible since both G (MG ) and G,c (MG ) are free G -graded modules (cf . proof of 5.6.2.1-(1)), the right framework to deal with Gysin morphisms is derived category, the reason being that the general definition of these requires inverting quasi-isomorphisms. Let f : M → N be a G-equivariant map between oriented G-manifolds of dimensions dM and dN respectively.

1.2 Equivariant de Rham Gysin Morphisms

3

When f is proper, the pullback f ∗ : G,c (N ) → G,c (M) induces, through the duality functor (1.4), a morphism in D(DGM(G )), which we denote: f∨ : IR Hom•G (G,c (M), G ) → IR Hom•G (G,c (N ), G ) . We therefore have in D(DGM(G )) the diagram: IR Hom• G

G,c

G)

f∨

IR Hom• G

ID G,M G (M)[dM ]

G,c

G)

(1.5)

ID G,N f∗

G (N)[dN ]

by which, we define the Gysin morphism f∗ for the proper f as f∗ := ID G,N −1 ◦ f∨ ◦ ID G,M : G (M)[dM ] → G (N )[dN ] , thanks to equivariant Poincaré duality (5.6.2.1–(1)). Now if g : N → N  is also G-equivariant and proper between oriented Gmanifolds, we immediately get (g ◦ f )∗ := ID G,N  −1 ◦ (g ◦ f )∨ ◦ ID G,M = (ID G,N  −1 ◦ g∨ ◦ ID G,N ) ◦ (ID G,N −1 ◦ f∨ ◦ ID G,M ) = g∗ ◦ f∗ . Whence, the Gysin functor for proper maps (−)∗ : G-Manpr  D(DGM(G )) ,

M  G (M)[dM ] ,

f  f∗ ,

where G-Manpr is the category of G-manifolds and G-equivariant proper maps. When f is not proper, we can proceed in the same way if dim(Hc (N )) < ∞, since in that case the morphism ID G,N (1.3) is also an isomorphism (5.6.2.1–(3)). We can then consider the pullback f ∗ : G (N ) → G (M) with no condition on supports and replace ID G,M with ID G,M in diagram (1.5). The definition of the Gysin morphism f! for arbitrary f is then f! := (ID G,N )−1 ◦ (f ∗ )∨ ◦ ID G,M : G,c (M)[dM ] → G,c (N )[dN ] . Whence, the Gysin functor for arbitrary maps (−)! : G-Man  D(DGM(G )) ,

M  G,c (M)[dM ] ,

f  f! ,

where G-Man is the category of G-manifolds and G-equivariant maps.

4

1 Introduction

1.3 Adjunction Properties of Gysin Morphisms By taking cohomology, Gysin morphisms induce morphisms of graded HG -modules which we denote by the same notations: 

f∗ = HG (M)[dM ] → HG (N )[dN ] , f! = HG,c (M)[dM ] → HG,c (N )[dN ] .

The commutativity of diagram (1.5) then amounts to the equalities 

 α, f ∗ (β) M,G =  f∗ (α), β N,G ,  f ∗ (α), β M,G =  α, f! (β) N,G .

(1.6)

for all α ∈ HG (M) and all β ∈ HG,c (N ), showing that Gysin morphisms are Poincaré adjoints to pullbacks. This is often the departure point for introducing Gysin morphisms, but to solve the adjunction equalities (1.6) in the category DGM(G ) generally demands additional constraints in the map f , constraints that are no more needed in D(DGM(G )). In other words, the Gysin morphisms f∗ and f! cannot always be defined in DGM(G ). In the classical approach, given a map f : M → N , one has explicit definitions of Gysin morphism for compact supports in the category DGM(G ) for both the projection p : M × N → N and the graph embedding Gr(f ) : M → M × N . These allow defining f! := p! ◦ Gr(f )! . But although this is a necessary condition to functoriality, it does not justify it, something which would still have to be proven. Working in derived category D(DGM(G )) simplifies things. • It allows an a priori justification for the functoriality of Gysin morphisms. • Given a closed inclusion of oriented manifolds N ⊆ M, we can simply dualize the exact triangle in D(DGM(G )) G,c (M  N ) → G,c (M) → G,c (N ) → , applying the duality functor (1.4), immediately getting, thanks to equivariant Poincaré duality, the Gysin exact triangle in D(DGM(G )): G (N )[dN ] → G (M)[dM ] → G (M  N )[dM ] → , whence, the Gysin exact sequence of equivariant cohomology: → HGi (N ) → HGi+dM −dN (M) → HGi+dM −dN (M  N ) → .

1.4 Equivariant Cohomology Viewed as a Relative Cohomology Theory

5

• In the same easy way, if E → B is a vector bundle with fibers of dimension n and oriented base space, the zero section map ι : B → E, being proper, automatically induces the Gysin morphism ι∗ : G (B) → G (E)[n] in D(DGM(G )). We deduce the equivariant Thom morphism of HG -graded modules ι∗ : HGi (B)  HGi+n (E), the Thom equivariant class then being ι∗ (1) ∈ HGn (E). In a more refined approach of Poincaré duality of E relative to the base space B developed in Chap. 3, we see in the same easy way that the Thom morphism i+n establishes an isomorphism ι∗ : HGi (B)  HG,B (E), where HG,B (−) denotes the equivariant cohomology with supports in B. • The Equivariant Euler class for a closed embedding i : N ⊆ M of oriented manifolds, is simply EuG (N, M) := i ∗ i∗ (1) where i ∗ i∗ : HG (N ) → HG (N ) is the push-pull operator.

1.4 Equivariant Cohomology Viewed as a Relative Cohomology Theory Chapter 3 is a transition chapter between nonequivariant and equivariant de Rham cohomologies, in which we make explicit, in the category of fiber bundles (E, B, π, M), the Grothendieck-Verdier formalism of cohomology and of Poincaré duality of the total space E relative to the base space B. The classic setting, thereafter referred to as absolute, corresponds to the case where B := {•}. The change from the absolute to the relative point of view amounts to the following replacements: absolute cohomology ↔ relative cohomology R := ({•}) ↔ (B)  R B D(DGM(R)) ↔ D(DGM((B), d)) ∼ D(B; R) (M) ↔ π∗ (E)  IR π∗ R E c (M) ↔ π! (E)  IR π! R E •

HomR (−, R) ↔ IR Hom •( B ,d) (−,  B )  IR Hom •R B (−, R B ) where (−) denotes the sheaf of differential forms, and D(B; R) is the derived category of sheaves of R-vector spaces on B. The relative Poincaré duality, also known as the Poincaré-Verdier Duality, states the existence of an isomorphism in the derived category of sheaves over B. Theorem (3.3.3.1) Let (E, B, π, M) be an oriented fiber bundle with fiber M a manifold of dimension dM .

6

1 Introduction

1. The following morphisms induced by the left Poincaré adjunction, π∗

E [dM ]

ID B,M q.i.

IR Hom •

B ,d)

(π!

E

B)

q.i. Grothendieck-Verdier Duality

IR Hom •R B (π!

q.i.

(1.7) E , RB )

are isomorphisms in D(B; R). 4. If M is of finite type, the previous statements remain true if we swap terms π!  E ↔ π∗  E and cv (E) ↔ (E). The right Poincaré adjunction: π!

E [dM ]

ID B,M q.i.

IR Hom •

B ,d)

(π∗

E

B)

(1.8)

is therefore a quasi-isomorphism in D(B; R) too. This theorem is the basis of all versions of Poincaré Duality. Indeed, when B := {•}, we have the classical (absolute) Poincaré Duality; when B is the classifying space IBG and that we consider the fiber bundle (MG , IBG, π, M), where π : MG := IEG ×G M → IBG is the Borel construction for an oriented G-manifold M, the cohomology of MG relative to IBG is precisely the singular equivariant cohomology of M with coefficients in R. The equivariant comparison de Rham theorems, proved in Sect. 4.8, establish the existence of canonical isomorphisms HG (M)  H (MG ; R)

and

HG,c (M)  Hcv (MG ; R) ,

where H (MG ; R) := H (IBG; π∗  MG ) and Hcv (MG ; R) := H (IBG; π!  MG ),

1.5 Equivariant Poincaré Duality over Arbitrary Fields As we recalled in the Preface, Grothendieck-Verdier’s formalism is especially valuable because of its generality: not only does it allow topological spaces to be much more general than simple manifolds, but also coefficients rings can be arbitrary. Nevertheless, in this book we limited generality to fiber bundles (E, B, π, M) where the fiber M is a manifold and the base space B is a mild topological space (cf . Sect. 1.7 in this chapter). We also limited the coefficients ring to be a field k, albeit of arbitrary characteristic. Poincaré-Verdier Duality is then relative to B with coefficients in k. In Chap. 8, we extend Theorem 3.3.3.1 to this more general framework, thus extending equivariant Poincaré duality and Gysin morphisms to arbitrary coefficients fields.

1.6 Conditions on the Group G

7

1.6 Conditions on the Group G In the literature on G-equivariant cohomology, the most frequent assumption about group G is that it is a compact connected Lie group whose action on manifolds is differentiable. Compactness and differentiability hypotheses seem unavoidable when one needs, as we do, G-averaging operators on G-manifolds, which allow us to prove the following fundamental (and constantly used) facts for G-manifolds. • • • •

Existence of proper invariant functions (2.5.4.2). Existence of G-invariant Riemannian metrics. Existence of G-Slices (4.5.3.1, 7.5.1); Existence of G-invariant tubular neighborhoods around G-stable submanifolds, essential to the proof of Localization formulas (7.7) • Existence of quotient manifolds for free actions. • Equivariant de Rham theorems, which, for G-manifolds M, N , identify the cohomology of the Cartan Models G (M), G,c (M) and G,N (M) with the corresponding cohomologies for the homotopy quotients MG , NG (4.8). Connectedness of G is less necessary and can (and will) frequently be avoided. This hypothesis is usually invoked to warrant

• homotopic triviality of the action of G on G-spaces. But connectedness can be unnecessarily restrictive, for example, when, G is connected and we are interested in the K-equivariant cohomology HK (M) of a G-manifold, where K is any closed subgroup of G, connected or not (see 4.4.6.3). • the action of G preserves the orientation of a G-manifold, which is not always true when G is not connected. But here, rather than excluding nonconnectedness, we call oriented G-manifold a G-manifold in which the action of every g ∈ G preserves orientation (see 5.5.1) • simple connectedness of the classifying space IBG. This is a useful property when dealing with local systems on IBG since, in that case, local systems are trivial and the description of Leray spectral sequences converging to HG (M) and HG,c (M) is greatly simplified. We can however still avoid connectedness of G altogether and get around nontrivial local systems by including in our considerations the finite group G/G0 where G0 denotes the connected component of the identity element e ∈ G, and by replacing HG (M) and HG,c (M) respectively with HG0 (M)G/G0 and HG0 ,c (M)G/G0 . (We will then need to add the hypothesis that the cardinality of G/G0 be prime to char(k) when introducing equivariant cohomology over fields of positive characteristic.) • use of Cartan-Weil theory as described in Cartan’s Seminar (4.1.1). Indeed, the theory concerns the category of g-differential graded algebras, where g := Lie(G0 ). But, again, instead of restricting to connected G, we extend CartanWeil theory in such a way that it incorporates the action of the finite group G/G0 , which we do in 4.4.5, Proposition 4.4.5.2, beyond which we need no longer assume G to be connected to apply this theory (see 4.4.4.2).

8

1 Introduction

Convention For these reasons, in this book, and otherwise explicitly stated, the group G is only assumed to be a compact Lie group whose action on manifolds is differentiable.

1.7 Conditions on Topological Spaces All topological spaces are assumed to be Hausdorff , locally contractible, paracompact and perfectly normal.1 We call such spaces mild spaces (cf . Appendix B.1). The most immediate examples of which are: manifolds and inductive limits of such, as are the classifying spaces IBG, and, more generally, open subspaces of CWcomplexes. Working with mild spaces is especially pleasant in so far as Alexander-Spanier, ˇ Singular, Cech, de Rham (for manifolds) and Sheaf cohomology are all canonically isomorphic to each other.2 This flexibility allows us to approach Poincaré Duality from different points of view and, in particular, through Grothendieck-Verdier formalism. Notice that, in contrast to frequent practice, we do not require the property of being locally compact in the definition of mild spaces. The reason being that we need to apply Poincaré-Verdier Duality to fiber bundles (E, B, π, M) where, while the fiber space M is a manifold, the base space B can fail to be so, in which case being a mild space is a sufficient condition for the theory to work.

1 A topological space X

is ‘normal’ if for every pair of disjoint closed subspaces F1 , F2 , there exist disjoint neighborhoods Vi ⊇ Fi , and it is ‘perfectly normal’ if, in addition, there exist continuous functions f : X → R≥0 verifying f V1 = 1 and f V2 = 0. 2 See Bredon [20], ch. III, Comparison with other cohomology theories.

Chapter 2

Nonequivariant Background

2.1 Category of Cochain Complexes We begin recalling standard terminology and notations.

2.1.1 Fields in Use We denote by k an arbitrary field, and by Vec(k) the category of k-vector spaces and k-linear maps. Later, as we consider de Rham cohomology on manifolds, the field k will be the field of real numbers R.

2.1.2 Vector Spaces and Pairings Expressions such as vector space, linear, bilinear,. . . always refer to the field k, unless otherwise stated. The dual of the vector space V is denoted by V ∨ := Homk (V , k). A bilinear map b : V ×W → k, also called a pairing, gives rise to the linear maps γb : V → W ∨ , γb (v)(w) := b(v, w), and ρb : W → V ∨ , ρb (w)(v) := b(v, w), respectively called the left and right adjunctions associated with b. One says that b is a nondegenerate pairing whenever the adjunctions are injective, and that b is a perfect pairing whenever they are bijective. For example, the canonical pairing V ∨ × V → k, (λ, v) → λ(v), is always nondegenerate and is perfect if and only if V is finite dimensional.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Arabia, Equivariant Poincaré Duality on G-Manifolds, Lecture Notes in Mathematics 2288, https://doi.org/10.1007/978-3-030-70440-7_2

9

10

2 Nonequivariant Background

2.1.3 The Category GV(k) of Graded Spaces A graded (vector) space is a family V := {V m }m∈Z of vector spaces indexed by the set of integers. Given graded spaces V and W , a graded homomorphism α : V → W of degree d =: deg(α) is a family α := {αm : V m → W m+d }m∈Z of linear maps. The composition of graded homomorphisms is defined degree by degree, i.e. β ◦ α = {βm+d ◦ αm }m∈Z , and we have deg(α ◦ β) = deg(α) + deg(β). We denote by Homgrdk (V , W ) the vector space of graded homomorphisms of degree d. When d = 0, we simply write Homgrk (V , W ) for Homgr0k (V , W ). The graded space of graded homomorphisms from V to W is defined as   Homgr∗k (V , W ) := Homgrdk (V , W ) d∈Z .

(2.1)

The category GV(k) of graded spaces, also denoted by Vec(k)Z , is the category whose objects are graded spaces and whose morphisms are graded homomorphisms of degree 0. Given V , W ∈ GV(k), we denote by MorGV(k) (V , W ) the set of morphisms from V to W . Note that the three notations MorGV(k) (−, −), Homgrk (−, −) and Homgr0k (−, −) denote the same set. Comment 2.1.3.1 It is worth mentioning that since a graded space a  is simply m} m and family of vector spaces V := {V , it underlies the direct sum V m∈Z m∈Z  the direct product m∈Z V m . Nevertheless, it would be a mistake to identify graded spaces with any kind of sum or product. In categories of sheaves, forexample, the distinction between a graded sheaf {Fi }i∈Z and a direct-sum sheaf i∈Z Fi is   crucial since the inclusion i∈Z Γ (X; Fi ) ⊆ Γ (X; i∈Z Fi ) is generally strict (cf . fn. (6 ), p. 305).

2.1.4 The Subcategories of Bounded Graded Spaces A graded space V := {V m }m∈Z is said to be bounded, bounded below (by ), bounded above (by ), if V m = 0 respectively for |m|  0, m  0 (m < ), m  0 (m > ). We denote by GVb (k), GV+ (k), GV≥ (k), GV− (k) and GV≤ (k) the full subcategories of GV(k) whose objects are respectively the bounded, bounded below and bounded above graded spaces. The notation GV∗ (k) will be used in statements concerning all of these categories.

2.1.5 Graded Algebras over Fields A graded k-algebra is a graded space A ∈ GV(k) together with a multiplication operation, i.e. a family of k-bilinear maps { · : Aa × Ab → Aa+b }a,b∈Z , such that the triple (A, 0, +, · ) verifies the axioms of a k-algebra.

2.1 Category of Cochain Complexes

11

The graded algebra is said to be • positively graded: if Am = 0 for all m < 0, • evenly graded: if Am = 0, for all odd m, • anticommutative:1 if a1 · a2 := (−1)d1 d2 a2 · a1 , for all a1 ∈ Ad1 , a2 ∈ Ad2 . A morphism of graded (resp. unital) algebras α : (A, 0, +, · ) → (B, 0, +, · ) is a morphism of graded spaces α ∈ Homgrk (A, B) which is compatible with multiplication, i.e. α(x · y) = α(x) · α(y) (resp. α(1A ) = 1B ). Examples 2.1.5.1 1. For any V ∈ GV(k), the space (Endgr∗k (V ), 0, +, id, ◦) of graded endomorphisms (cf . Sect. 2.1.3–(2.1)) is a noncommutative graded algebra. 2. The algebra G := S(g ∨ )g of g-invariant polynomial functions over a real Lie algebra g equipped with the grading that doubles the polynomial grading is an evenly positive graded commutative R-algebra. 3. Given a manifold M, the algebras of differential forms ((M), d) and (c (M), d) are positively graded anticommutative R-algebras. Notice that c (M) does not have an identity element if M is not compact. Exercise 2.1.5.2 Let GV(k) be the category of graded spaces where the sets of morphisms are the graded spaces MorGV(k) (V , W ) := Homgr∗k (V , W ) , where the composition rule is the coordinate-wise composition of linear maps. Denote by GVb (k) the full subcategory  of GV(k) of bounded  graded spaces. Given V := {V m }m∈Z , let ⊕V := m∈Z V m and V := m∈Z V m . 1. Show the correspondence  ⊕ : GV(k)  Vec(k) ,

V  ⊕V   α : V → W  ⊕α : ⊕V → ⊕W

where ⊕α(vm )m∈Z := m∈Z αm (vm ), is a faithful functor,2 and conclude that GV(k) and GV(k) can be viewed as subcategories of Vec(k). 2. Same as (1), replacing ⊕ by .

1 This

is the terminology used in the Colloque de Topologie sur les Espaces Fibrés de Bruxelles (1950) [32], notably in Cartan’s talks [25, 26]. Although the terminology is still widely used, it is being progressively replaced by graded commutative, especially in the theory of dg-algebras, see Stacks Project [87]. 2 A functor F : C → C is said to be faithful (resp. full) if, for all pair of objects X, Y ∈ Ob(C ), 1 2 1 the map FX,Y : MorC1 (X, Y ) → MorC2 (F (X), F (Y )) is injective (resp. surjective).

12

2 Nonequivariant Background

3. Let V , W ∈ GM(k). Given a k-linear map λ : ⊕V → ⊕W define, for all k, d ∈ Z, the map λk,d ∈ Homk (V k , W k+d ) as the composition Vk

ιk

m∈Z V

m

λ

m∈Z W

m pk+d

W k+d

λk,d

where ιk denotes the canonical injection of the k-th coordinate space, and pk+d the canonical projection onto the (k+d)-th coordinate space. The family λd := {λm,d }m∈Z belongs to Homgrdk (V , W ), hence the map:  V ,W Homk ⊕V , ⊕W −−−→ Homgr∗k (V , W ) ,

λ → {λd }d∈Z .

– Show that V ,W is a natural injective homomorphism. – Same question replacing ⊕ with . ( , p. 329) 4. Give the conditions of surjectivity for V ,W . ( , p. 329) a. Show that GVb (k) is equivalent to the category whose objects are the finite direct sums of k-vector spaces, and whose morphisms are the sets of all klinear maps. b. Show that GV(k) is not equivalent by ⊕ (resp. ) to the category of countable direct sums (resp. products) of k-vector spaces and k-linear maps.

2.1.6 The Category DGV(k) of Differential Graded Vector Spaces A differential graded (vector) space (V , d) (also called complex of vector spaces), is a graded space V together with a graded endomorphism d ∈ Endgr1k (V ) such that d 2 = 0, called the differential. • Differential graded spaces are usually represented as cochain complexes: (V , d) := · · ·

dm−2

V m−1

dm−1

Vm

dm

V m+1

dm+1

··· .

2.1 Category of Cochain Complexes

13

• A morphism of complexes α : (V , d) → (V  , d  ) is a graded homomorphism of degree 0 such that α ◦ d = d  ◦ α. (V , d) := · · ·

dm−2

dm−1

αm−1

α

(V , d ) := · · ·

V m−1

dm−2

V

m−1

Vm

dm

V

dm+1

···

dm+1

···

αm+1

αm dm−1

V m+1

m

dm

V

m+1

• The complexes and their morphisms constitute the category DGV(k) of differential graded vector spaces. Also denoted as C(Vec(k)) and called the category of complexes of k-vector spaces. • The cohomology of a complex (V , d) is the graded space denoted by h(V , d), or simply h(V ), whose i-th term is hi (V , d) := ker(di )/ im(di−1 ). • A morphism of complexes α : (V , d) → (V  , d  ) induces a morphism between the graded spaces of cohomologies denoted by h(α) : h(V , d) → h(V  , d  ) . The morphism α is called a quasi-isomorphism, a quasi-injection or a quasisurjection, if h(α) is an isomorphism, an injection or a surjection respectively. • The correspondence which associates V  h(V ) and α  h(α) constitutes the (covariant) cohomology functor which we denote by h : DGV(k)  GV(k) . Exercise 2.1.6.1 Let Ab and Ab be abelian categories. Show that every additive functor F : Ab → Ab is exact if and only if Ab is split. 3 Apply to Vec(k) and GV(k). Explain the exactness of bifunctors 2.1.8.1–(2.8). ( , p. 329)

2.1.7 The Shift Functors Let s ∈ Z. S-1. Given V ∈ Vec(k), we denote by V [s] the graded space with V [s]−s := V and V [s]m := 0 for s + m = 0. Given a k-linear map α : V → V  , we denote by α[s] : V [s] → V  [s] the morphism of graded spaces with α[s]−s := α and α[s]m := 0 for m = −s. The correspondences V  V [s] and V  (V [s], 0),

an abelian category Ab, a morphism f : X → Y is said to be split, if there are isomorphisms X ∼ K ⊕ L and L ⊕ M ∼ Y through which f reads (k, l) → (l, 0). The category Ab is called split, if all its morphisms are split.

3 In

14

2 Nonequivariant Background

together with α  α[s], define respectively the shift functors [s] : Vec(k) → GV(k)

and

[s] : Vec(k) → DGV(k) .

(2.2)

S-2. Given V := {V m }m∈Z ∈ GV(k), we denote by V [s] the graded space with V [s]m := V s+m . Given a morphism of graded spaces α : V → W , we denote by α[s] : V [s] → W [s] the morphism with α[s]m := αs+m . The correspondence V  V [s] and α  α[s] defines the shift functor [s] : GV(k)  GV(k) .

(2.3)

S-3. Given (V , d) ∈ DGV(k), we set  (V , d)[s] := V [s], (−1)s d[s] . The correspondence (V , d)  (V , d)[s], and α  α[s] is the shift functor [s] : DGV(k)  DGV(k) .

(2.4)

All these functors are covariant additive and exact.4

2.1.8 The Functors Hom•k (−, −) and (− ⊗k −)• Given two complexes V := (V , d) and V  := (V  , d  ) in DGV(k), we recall the definition of the complexes 

Hom•k (V , V  ), D



and

 (V ⊗k V  )• , Δ .

(2.5)

As graded spaces they are5  m    Homm k (V , V ) := Homgrk (V , V [m]) = Homgrk (V , V ) , (2.6) m ∈ Z →  (V ⊗k V  )m := b+a=m V a ⊗k V b .

4

In an abelian category Ab the sets of morphisms MorAb (X, Y ) are abelian groups. A covariant functor between abelian categories F : Ab → Ab is then said to be additive if, for all X, Y ∈ Ob(Ab), the map FX,Y : MorAb (X, Y ) → MorAb (F (X), F (Y )) is a group homomorphism. Likewise for contravariant functors. The functor is then said to be exact, left exact or right exact, if it transforms every short exact sequence of Ab in respectively an exact, left exact or right exact sequence of Ab , and this regardless of whether F is covariant or contravariant. 5 These notations are standard and will be used in several different categories, in the present case, we get two equivalent notations Hom•k (−, −) = Homgr∗k (−, −).

2.1 Category of Cochain Complexes

15

The differentials are6 

Dm (α) :=d  ◦ α − (−1)m α ◦ d ,

Δm (v ⊗ v  ) :=d(v) ⊗ v  + (−1)a v ⊗ d  (v  ) ,

(2.7)

where v ⊗ v  ∈ V a ⊗ V b and a + b = m. Definition 2.1.8.1 The previous constructions are natural w.r.t. each entry, and define the following two bifunctors:  Hom• ( , ) : DGV(k) × DGV(k)  DGV(k) , k − − (− ⊗k −)• : DGV(k) × DGV(k)  DGV(k) .

(2.8)

The functor ‘Hom• ’ is contravariant and exact in the first entry, and covariant and exact in the second entry. The functor ‘⊗’ is covariant and exact in each entry. Exercise 2.1.8.2 Show that the natural morphism of GV(k)  : V  ⊗k Hom•k (V , k) → Hom•k (V , V  ) defined, for all homogeneous λ ∈ Hom•k (V , k) and v  ∈ V  , by  (v  ⊗ λ) := v → λ(v) v  , is a morphism of DGV(k). ( , p. 331) Exercise 2.1.8.3 Check that the following complexes coincide as graded spaces but not as complexes even though they are naturally isomorphic. ( , p. 331) Hom•k (V [s], W [t])  Hom•k (V , W )[t − s] , V [s] ⊗ W [t]  (V ⊗ W )[s + t] .

2.1.9 On the Koszul Sign Rule This is the name given to the mnemonic determining signs when manipulating differential graded objects,7 as we did in the previous paragraphs. Essentially, it says that when transposing two consecutive homogeneous objects x and y respectively [x][y] must be added, e.g. xy = (−1)[x][y] yx. of degrees [x] and [y], the sign (−1)

6 See 7 See

Stacks Project [88] §15.67 Hom complexes, and [87] §12. Tensor product, p. 15. Stacks Project [86] §68 Sign rules, p. 167, for a commented full list of these rules.

16

2 Nonequivariant Background

In the case of the differential D in (2.7), if α ∈ Hom•k (V , V  ) and v ∈ V are homogeneous, and if we write α(v) ↔ α v, we see that the rule gives d  (α v) = Dα v + (−1)[α][D] α dv , 

where the sign (−1)[α][d ] corresponds to the transposition Dα ↔ αd.

2.1.10 The Functor Hom•k (−, W ) Let (W , d) be a complex. Given a morphism of complexes α : (V  , d) → (V , d), the map m  α ∗ : Homm k (V , W ) → Homk (V , W ) ,

β → β ◦ α ,

is well-defined for all m ∈ Z and commutes with differentials: D(α ∗ (β)) = d ◦ (β ◦ α) − (−1)[β◦α] (β ◦ α) ◦ d = (d ◦ β) ◦ α − (−1)[β] (β ◦ d) ◦ α = α ∗ (D(β)) . The correspondence   Hom•k (α, W ) : Hom•k (V , W ), D → Hom•k (V  , W ), D , which associates (V , d)  Hom•k (V , W ) and α  Hom•k (α, W ) := α ∗ , is then easily seen to be an additive functor: Hom•k (−, W ) : DGV(k)  DGV(k) , which is contravariant and exact.

2.1.11 The Duality Functor This is the functor (−)∨ := Hom•k (−, k[0]) : DGV(k)  DGV(k) .

(2.9)

2.1 Category of Cochain Complexes

17

The dual complex associated with a complex (V , d) is then defined as (V , d)∨ := (V ∨ , D) ,

(2.10)

where (V ∨ )m := (V −m )∨ and Dm = (−1)m+1 td−(m+1) .8 Remark 2.1.11.1 Take care that the natural embedding of graded vector spaces φ : V → V ∨∨ ,

φ(v)(λ) := λ(v) ,

gives only an embedding of complexes (V , −d) ⊆ (V , d)∨∨ , where the sign of the differential has changed ! Indeed, applying the definition (2.7), we have  D φ(v) (λ) = − (−1)[v] φ(v)(D(λ)) = (−1)1+[v] (−1)1+[λ] φ(v)(λ ◦ d) = (−1)[v]+[λ] λ(dv) = φ(−dv)(λ) , since [λ] = [v] + 1 is the only nonzero case to check. The canonical isomorphism  : (V , d) → (V , −d) ,

m = (−1) m idV m

(2.11)

is then necessary to obtain an embedding φ ◦  : (V , d) → (V , d)∨∨ which does not alter differentials. Proposition 2.1.11.2 1. There exists a canonical isomorphism between the cohomology of the dual and the dual of the cohomology, i.e.   ∨ h (V , d)∨ −−→ h(V , d) . 

2. A morphism of complexes α : (V , d) → (V  , d  ) is a quasi-isomorphism if and only if so is α ∨ . Proof (1) Immediate since the duality functor is additive and that Vec(k) is a split category (cf 2.1.6.1).  . Exercise  (2)  Results from (1)  which entails the equivalences α is q-iso ⇔ h(α) is iso ⇔ h(α)∨ is iso ⇔ α ∨ is q-iso .  

we denote by t(d−(m+1) ) the adjoint map to d−m−1 : V −m−1 → V −m , hence the map Dm := td−(m+1) : (V ∨ )m → (V ∨ )m+1 .

8 Here,

18

2 Nonequivariant Background

2.2 Categories of Manifolds 2.2.1 Manifolds The names manifold, map of manifolds and function will be shortcuts respectively for real Hausdorff paracompact differentiable manifold,9 for differentiable map of class C∞ , in short smooth maps, and for maps with values in R. A manifold is said to be equidimensional if its connected components have the same dimension, in which case ‘dM ’ denotes this common dimension. Expressions such as ‘M is of dimension d ’, presupposes that M is equidimensional.

2.2.2 The Category of Manifolds By Man (resp. Manor ) we denote the category of manifolds (resp. oriented) and (smooth) maps. Over Man one has the de Rham contravariant functor (−) : Man  DGV(R) ,

M  (M, dM )

(2.12)

and the de Rham cohomology contravariant functor H (−) := h((−)) : Man  ModN (R) ,

M  H (M) .

(2.13)

2.2.3 The Category of Manifolds and Proper Maps or By Manpr (resp. Manor pr ) we denote the subcategory of Man (resp. Man ) of all 10 manifolds and all proper maps. ( , p. 332). Over Manpr one has, in addition to the functors (2.12) and (2.13), the compactly supported de Rham contravariant functor:

c (−) : Manpr  DGV(R) ,

9A

M  c (M, dM )

(2.14)

connected manifold M which is paracompact, is automatically countable at infinity, second countable, and perfectly normal, which means that two disjoint closed subspaces F1 and F2 admit disjoint neighborhoods, Vi ⊇ Fi , and that there exist continuous functions f : M → R verifying f V1 = 1 and f V2 = 0. 10 A continuous map between locally compact spaces f : M → N is said to be proper if f −1 (F ) is compact for every compact subspace F ⊆ N , or, equivalently, if f is closed with compact fibers (exercise !) (See also 3.1.9.3–(3).)

2.3 Orientation and Integration

19

and the compactly supported de Rham cohomology contravariant functor Hc (−) : Manpr  ModN (R) ,

M  Hc (M, dM ) .

(2.15)

The inclusion c (−) ⊆ (−) induces a morphism of functors Hc (−) → H (−).

2.2.4 G-Manifolds Let G denote a Lie group. A manifold endowed with a smooth (left) action of G is called a G-manifold. A map f : M → N between G-manifolds is said to be Gequivariant if it is compatible with the action of G, i.e. if f (g(m)) = g(f (m)), for all g ∈ G and m ∈ M. The G-manifolds and the G-equivariant maps constitute the category of G-manifolds which we denote by G- Man. The categories G- Manor , or G- Manpr and G- Manor pr are the equivariant versions of the categories Man , Manpr or and Manpr .

2.3 Orientation and Integration11 2.3.1 Orientability A manifold M of dimension dM is said to be orientable if it admits an atlas A all of whose transition maps T locally preserve the canonical orientation of RdM , in other terms, are such that the Jacobians J (T ) are positive functions. Such an atlas A is called an oriented atlas of M. Proposition 2.3.1.1 1. A manifold M of dimension dM is orientable if and only if there exists a nowhere vanishing differential form of highest degree ω ∈ dM (M). 2. A simply connected manifold is orientable. Proof (1) If A = {ϕa : Ua → RdM }a∈A is an oriented atlas of M, then, for any partition of unity Φ := {φa : Ua → R≥0 }a∈A , the dM -form ω(A, ) ω(A, Φ) :=

11 See

 a

φa ϕa∗ (dx1 ∧ · · · ∧ xdM ) ,

also Bott-Tu [18] §3 Orientation and Integration, p. 27.

(2.16)

20

2 Nonequivariant Background

is nowhere vanishing. Indeed, the pushforward of ω(A, ) through the chart (ϕa  : Ua  → RdM ) is the differential form ϕa  ∗ (ω(A, Φ)) =

 a

∗ Ta,a  (φa dx1 ∧ · · · ∧ xdM )

(2.17)

where Ta,a  : ϕa  (Ua  a ) → ϕa (Uaa  ) is the transition map. But then ∗ Ta,a  (dx1 ∧ · · · ∧ xdM ) = ga,a  dx1 ∧ · · · ∧ xdM

with ga,a  a strictly positive function on ϕa  (Ua  a ) ⊆ RdM and the sum (2.17) is nowhere vanishing on Ua  . Conversely, for any connected open subset U ⊆ M and any diffeomorphism ϕ : U → ϕ(U ) ⊆ RdM there exists a uniquely defined function f : ϕ(U ) → R such that ϕ∗ (ω) = fω dx1 ∧ · · · ∧ dxdM . When ω is nowhere vanishing, we have either fω > 0 or fω < 0. We then say that ϕ is ω-oriented if fω > 0. It is clear that if ϕ := (ϕ1 , ϕ2 , . . . , ϕdM ) is not ω-oriented, then, by switching the first two coordinates, the diffeomorphism ϕ˜ := (ϕ2 , ϕ1 , . . . , ϕdM ) is ω-oriented. As a consequence, the charts (ϕa , Ua ) in any atlas A of M can be easily modified to make all the charts ω-oriented, in which case the atlas becomes oriented. (2) Let T M → → M denote the tangent vector bundle of M. Statement (1) says that M is orientable if and only if the determinant bundle det T M := dM T M admits a nowhere vanishing section. But since det T M is a line bundle, the bundle det T M  {0} retracts to an S0 -bundle above M, which is trivial if M is simply connected, after a well-known result in the theory of coverings.  

2.3.1.1

Orientations

Two atlases A1 , A2 are said to define the same orientation, and we write A1 ∼ A2 , if the atlas A1  A2 , defined by the collection of all charts of A1 and A2 , is also an oriented atlas. The relation ‘∼’ is an equivalence relation on the set of all oriented atlases. Let M be connected. If ω1 , ω2 ∈ dM (M) are nowhere vanishing, we have ω1 = f12 ω2 for a unique nowhere vanishing function f12 : M → R, which will therefore satisfy either f12 > 0 or f12 < 0. We say that ω1 and ω2 define the same orientation, and we write ω1 ∼ ω2 , if f12 > 0. The relation ‘∼’ is an equivalence relation on the set of nowhere vanishing differential forms of highest degree. It is thus clear that, in the proof of proposition 2.3.1.1, the correspondence A  ω(A, Φ) verifies A1 ∼ A2 if and only if ω(A1 , Φ1 ) ∼ ω(A2 , Φ2 ). As a consequence, if M is connected and orientable, there are exactly two classes of oriented atlases for M. To choose an orientation of M then means to choose one these classes, or, equivalently, to choose a nowhere vanishing dM -form on M. We denote such a choice by [M] and we say that (M, [M]) is oriented by [M].

2.3 Orientation and Integration

21

When M is not connected, it is orientable if each of its connected components is orientable. In that case, to choose an orientation for M means to choose an orientation for each of its connected components.

2.3.2 Integration Let (M, [M]) be an oriented manifold of dimension dM . Let A = {ϕa : Ua → RdM }a∈A be an oriented atlas corresponding to the orientation [M] and let Φ := {φa : Ua → R≥0 }a∈A a partition of unity. Lemma 2.3.2.1 Given β ∈ dc M (M), there is a unique f ∈ 0c (M) such that β = f ω(A, Φ) (cf. (2.16)). Then, the sum    −1 ∗ β := (ϕ ) (φ β) = (fa ◦ϕa−1 ) dx1 ∧· · ·∧xdM , (2.18) a a d d [M]

a

R

M

R

a

M

where fa := φa f has compact support in the chart (ϕa : Ua → Rn ) ∈ A, is a finite sum independent of the choice of A and Φ. In particular, if |β| ⊆ Ua , then  [M]

β=

 RdM

(ϕa−1 )∗ (β) .

Proof We need only consider the case where A is the full atlas A := {ϕa : Ua → ϕa (Ua ) ⊆ RdM }a∈A corresponding to [M].12 We begin proving that if |β| ⊆ Ua 0 for some a 0 ∈ A, then   −1 ∗ (ϕ ) (β) = (ϕa−1 )∗ (φa β) , (2.19) a 0 d d R

M

a

R

M

for any partition of unity Φ := {φa : Ua → R≥0 }. Indeed, in that case, for all a ∈ A, we have |φa β| ⊆ Ua 0 a , in which case  RdM

(ϕa−1 )∗ (φa β) = 0

 RdM

(ϕa−1 )∗ (φa β)

by change of variables since the transition map map Taa 0 preserves the orientation. The right-hand term in (2.19) can then be rewritten as:  a

RdM

(ϕa−1 )∗ (φa β) = =

 a RdM

RdM

(ϕa−1 )∗ 0

proving the claim for |β| ⊆ Ua 0 . 12 See

(ϕa−1 )∗ (φa β) 0

also Bott-Tu [18] Proposition 3.3, p. 30.



 φ β = a a

RdM

(ϕa−1 )∗ (β) 0

22

2 Nonequivariant Background

We can now approach the central question of the lemma. Let β ∈ c (M) be arbitrary. If Φ  := {φa : Ua → R≥0 }a∈A is a second partition of unity, then  a

RdM

(ϕa−1 )∗ (φa β) =

 RdM

a,a 

=1

(ϕa−1 )∗ (φa φa  β)



RdM

a,a 

∗  (ϕa−1  ) (φa φa  β) =

where (=1 ) is justified by the fact that |φa φa  β| ⊆ Uaa  .

 a

RdM

∗  (ϕa−1  ) (φa  β)

 

The sum (2.18) is called the integral of β relative to [M], and will be simply denoted by M β when the orientation [M] is understood. Proposition 2.3.2.2 Let M be an oriented manifold of dimension dM . Extend the integration map M : dc M (M) → R by linearity to the whole complex c (M) by setting M β := 0 for all β ∈ ic (M) and all i < dM . The resulting map 

: c (M)[dM ] → R[0] ,

M

(2.20)

is then a morphism of complexes, which is invariant under the action of orientation preserving diffeomorphisms of M. Proof Let n := dM . We need only prove that  M

d n−1 c (M) = 0 .

Let A = {ϕa : Ua → Rn } be an oriented atlas of M corresponding to the orientation of M, and let Φ := {φa : Ua → R≥0 } be a partition of unity. For all β ∈ n−1 c (M), we have β = a φa β, in which case dβ =

 a

d(φa β) ,

where the support of d(φa β) is compact and contained in Ua . By the definition of the integration operation and Lemma 2.3.2.1, we then have  M

dβ =

 a

Rn

dβa ,

n with βa := (ϕa−1 )∗ (φa β) ∈ n−1 c (R ). Hence, to prove that M dβ = 0, we can restrict ourselves to M := Rn . There, we have a system of global coordinates x := n 0 {x1 , . . . , xn } and β ∈ n−1 c (R ) is a linear combination over c (M) of the (n − 1)forms dxi1 ∧ · · · ∧ dxin−1 so that to verify the equality Rn dβ = 0, we need only

2.4 Poincaré Duality

23

check it for β := f (x) dx2 ∧ · · · ∧ dxn with f ∈ 0c (M), in which case dβ = (∂x1 f ) (x) dx1 ∧ dx2 ∧ · · · ∧ dxn . But then, 

dβ = n



R

R

···

  R

R

 (∂x1 f )(x1 , x2 , . . . , xn ) dx1 dx2 . . . dxn = 0 ,

by Fubini’s theorem, and because, for all fixed x2 , . . . , xn ∈ R,  R

(∂x1 f )(x1 , x2 , . . . , xn ) dx1 = lim

t→+∞

 f (t, x2 , . . . , xn ) − f (−t, x2 , . . . , xn ) = 0  

since f has compact support.

2.4 Poincaré Duality 2.4.1 Poincaré Pairing Let M be an oriented manifold of dimension dM not necessarily connected. The composition of the exterior product ∧ : dM −i (M) × ic (M) → dc M (M) , with integration

M

(α, β) → α ∧ β ,

(2.21)

: dc M (M) → R, gives rise to a pairing (2.1.2)

·, ·M : dM −i (M) × ic (M) → R ,

(α, β) →

 M

α∧β,

(2.22)

[α] ∪ [β] .

(2.23)

inducing the Poincaré pairing in cohomology ·, ·M : H dM −i (M) × Hci (M) → R ,

([α], [β]) →

 M

Proposition 2.4.1.1 (and Definitions) The Poincaré pairing (2.22) is nondegenerate and the map ID M : (M)[dM ] → c (M)∨

(2.24)



 ID M (α) := β → α∧β ,

(2.25)

defined by M

24

2 Nonequivariant Background

is an injective morphism of complexes. It will be called the left Poincaré adjunction associated with the pairing (2.22). It induces in cohomology the map DM : H (M)[dM ] → Hc (M)∨

(2.26)

which will be called the left Poincaré adjunction in cohomology Proof Following 2.1.7–(S-3) and 2.1.11, we have   ID M (−1)dM d α (β) = (−1)dM d α ∧ β M

=



M

(−1)dM d(α ∧ β) + (−1)dM +[α]+1

 = (−1)[β] ID M (α) d β

 M

α∧dβ

(2.27)

= (−1)[β] (−1)1+dM +[α] (DID M (α))(β) = (DID M (α))(β) ,

where M d(α∧β) = 0, after Proposition 2.3.2.2, which proves that the left Poincaré adjunction (2.24) is a morphism of complexes. Injectivity of ID M Let α ∈ i (M) and assume α(x) = 0 for some x ∈ M. We can ϕa then choose a chart M ⊇ Ua −−→ Wa ⊆ Rm (m := dM ) such that ∼

α

Ua

 = ϕa∗ fa dx1 ∧ · · · ∧ dxi + · · · ,

for some fa ∈ 0 (Wa ) such that fa > 0.13 In that case, for any ga ∈ c (Wa ), we can consider the (m−i)-differential form  β := ϕa∗ ga dxi+1 ∧ · · · ∧ dxm ∈ cm−i (M) , so that  α ∧ β = ϕa∗ fa ga dx1 ∧ · · · ∧ dxm ∈ m c (M) . In this way, if in addition ga ≥ 0 and ga = 0, then 

13 In

α, β

 M

:=

 Rm

fa ga dx1 ∧ · · · ∧ dxm > 0 .

fact, it can easily be seen that one can always assume fa = 1.

2.4 Poincaré Duality

25

The same argument obviously works for α with compact support. The Poincaré pairing (2.22) is therefore nondegenerate and ID M is injective.   Exercise 2.4.1.2 Show that the Poincaré pairing (2.22) is perfect if and only if M is a finite set. Show that if dM > 0 then none of the adjunctions induced by the pairing ·, ·M is bijective. ( , p. 332) Theorem 2.4.1.3 (Poincaré Duality) Let M be an oriented manifold of dimension dM not necessarily connected. We assume neither dimR H (M) < +∞ nor dimR Hc (M) < +∞. 1. The left Poincaré adjunction ID M : (M)[dM ] → c (M)∨ is an injective quasiisomorphism. It induces the Poincaré duality isomorphism DM : H (M)[dM ] −→ Hc (M)∨ . 

(2.28)

2. The Poincaré pairing in cohomology ·, ·M : H (M) × Hc (M) → R

(2.29)

is nondegenerate, and is perfect (2.1.2) if and only if dim(H (M)) < +∞. Proof (1) Let O(M) denote the category of open subspaces of M where morphisms are inclusion maps.14 Consider the functors • • • •

(−) : O(M)  DGV(R), the de Rham complex; c (−) : O(M)  DGV(R), the compactly supported de Rham complex; ID(−) : (−)[dM ] → c (−)∨ , the left Poincaré adjunction; D(−) : H (−)[dM ] → Hc (−)∨ , the left Poincaré adjunction in cohomology.

Denote by • OPD (M) ⊆ O(M), subcategory on which D(−) is an isomorphism. Lemma 1 If I := {Ui }i∈I is an increasing family of open subspaces in OPD , its union ∪ I = ∪i Ui also belongs to OP D .

14 The

proof is close to that given in Bott-Tu [18] §5 The Mayer-Vietoris and Poincaré Duality on an Orientable Manifold, pp. 42–.

26

2 Nonequivariant Background

Proof of Lemma 1 Consider the following diagram where Uj ⊆ Ui ∈ I

(2.30)

and where the vertical arrows are the natural restriction maps. The morphism lim I D(U ) is an isomorphism since the third and fourth lines ←− are isomorphisms by hypothesis. The vertical morphisms ξ and ξ  are also isomorphisms. Indeed, it is well-known that the natural maps lim S (U ; R) → S∗ (∪ I; R) −→ I ∗ lim ∗ (U ) → ∗c (∪ I) −→ I c

(2.31)

(the first denotes the complex of singular chains) are isomorphisms of complexes. Applying the duality functor to (2.31), we get the two isomorphisms lim H (∪ I; R)∨ → H∗ (U ; R)∨ ←−I ∗ ξ  : lim I Hc∗ (∪ I)∨ → Hc∗ (U )∨ ←−

(2.32)

since (−)∨ is exact and transforms inductive limits into projective limits. The first line in (2.32) concerns duals of singular homology which, thanks to de Rham theorem, coincide with corresponding de Rham cohomology, so that ξ is also an isomorphism. The morphism D(∪ I) = ξ −1 ◦ lim I D(U ) ◦ ξ is therefore an ←− isomorphism.  Lemma 2 If U, V ∈ OPD , then (U ∪ V ) ∈ OPD if and only if (U ∩ V ) ∈ OPD . Proof of Lemma 2 Consider the two familiar Mayer-Vietoris short sequences ⎧ p q ⎨ 0 → (U ∪ V ) −−→ (U ) ⊕ (V ) −−→ (U ∩ V ) → 0 ⎩

p

q

0 → c (U ∩ V ) −−→ c (U ) ⊕ (V ) −−→ c (U ∪ V ) → 0

(2.33)

2.4 Poincaré Duality

where 

p(ω) := (ω

27

U,ω V )



p (ω) := (ω, −ω)

and and

q(ω,  ) := ω

U ∩V

−

U ∩V

(2.34)



q (ω,  ) := ω + 

While it is easy to check that both sequences are left exact, the surjectivity of q and q  needs justification, which we can do thanks to the existence of a smooth partition of unity in U ∪ V subordinate to the cover {U, V }. We will therefore consider two smooth positive functions φU , φV : U ∪ V → R satisfying the support conditions |φU | ⊆ U and |φV | ⊆ V and such that φU + φV = 1. In that case, • for ω ∈ (U ∩ V ), the differential form φV ω extends by zero to U , and φU ω to V . Hence, q(φV ω, −φU ω) = ω, and the surjectivity of q. • for all ω ∈ c (U ∩ V ) we have φU ω) ∈ c (U ) and φV ω ∈ c (V ). Hence, q  (φU ω, φV ω) = ω, and the surjectivity of q  . These justifications are useful to describe the connecting morphisms c and c in the following long exact sequences of cohomology associated with (2.33): [1] [1]

c

H (U ∪ V )

H (U ) ⊕ H (V )

H (U ∩ V )

Hcn− (U ∪ V )

Hcn− (U ) ⊕ Hcn− (V )

Hcn− (U ∩ V )

[1] c [1]

(2.35)

Indeed, a simple computation shows that we have: c(α) = dφV ∧ α = −dφU ∧ α

and

c (β) = dφU ∧ β = −dφV ∧ β .

(2.36)

If we now connect the sequences (2.35) with the adjunction morphisms, we get [1]

H (U ∪ V )[dM ] D(U ∪V )

[1]

Hc (U ∪ V )

(I) ∨

H (U ∩ V )[dM ]

H (U )[dM ] ⊕ H (V )[dM ] D(U )

Hc (U ) ⊕ Hc (V ) ∨

(II)

D(V ) ∨

c [ 1]

D(U ∩V )

Hc (U ∩ V )

(2.37) ∨

c [1]

28

2 Nonequivariant Background

where the choices in (2.34) immediately give the commutativity of the sub-diagrams (I) and (II). For the commutativity of the connecting sub-diagram H (U ∩ V )[dM ]

c [1]

H

+1 (U

D(U ∩V )

∪ V )[dM ] D(U ∪V )

HcdM − (U ∩ V )∨

c [1]

d − +1)

Hc M

(U ∪ V )

the equalities (2.36) lead to D(U ∪ V )(c(α))(β) =



(D(U ∩ V )(α)(c (β)) =

M

d φV ∧ α ∧ β



M

−α ∧ d φV ∧ β = (−1) +1 D(U ∪ V )(c(α))(β) ,

which suggest modifying c in (−1) +1 c if we want (2.37) to be a morphism of long exact sequences, which we do. Applying the Five Lemma then finishes the proof.  The difficulty in using Lemma 2, is that while the open subspaces U and V can be very simple, the intersection U ∩ V can be problematic, with no reason why if U and V belong to OPD , the same should be true for U ∩ V . It is at this point that the idea of Leray of good covers comes on stage. A good cover of M is an open cover U = {Ui } of M such that all finite intersections Ui1 ∩ . . . ∩ Uik are either vacuous or homeomorphic to RdM .15,16 Using good covers in connection with Poincaré duality is quite simple. Let U = {B1 , B2 , . . . } be a good cover of M and define for n ∈ N: Un := B1 ∪ B2 ∪ · · · ∪ Bn .

15 The

existence of good covers on manifolds (also called simple covers or Leray covers) is proved by a Riemannian geometry argument, see Bott-Tu [18] §5 Theorem 5.1, p. 42. Given any open cover V of M, there always exists a good cover U subordinate to V . Since M is countable to infinity, the cover U can always be chosen to be locally finite (hence countable). 16 As reported by Christian Houzel, in A Short History in Kashiwara-Schapira [61], p. 7, in a conversation with Henri Cartan and André Weil in 1945, Leray explained his idea of good covers, following which Weil proved later the de Rham theorems and Poincaré duality in the modern approach used today, see Weil [96]: Sur les théorèmes de de Rham, p. 17, and Lettre à Henri Cartan, p. 45. See also the interesting historical review in Bott’s introduction to Bott-Tu [18], especially p. 7.

2.4 Poincaré Duality

29

Then • Each Bi belongs to OPD . Indeed, since Bi  RdM , we are lead to verify that the Poincaré pairing  ·, · Rn : H (RdM ) × Hc (RdM ) → R is perfect. This results from Poincaré lemmas, both for arbitrary and compact supports, which show that cohomologies are concentrated respectively in degrees 0 and dM . We then simply need to check that M : Hc (RdM ) → R is an isomorphism of vector spaces which is obvious.17 • For all n, the open subspace Un belongs to OP D . Indeed, by induction on n, we can assume Un−1 ∈ ODP . Then, by Lemma 2, Un ∈ OP D if and only if (Un−1 ∩ Bn ) ∈ OP D . But here, the intersection is gentle since (Un−1 ∩ Bn ) = (B1 ∩ Bn ) ∪ · · · ∪ (Bn−1 ∩ Bn )

(2.38)

is a good cover with fewer that n terms. We can therefore apply the inductive hypothesis and conclude that Un = (Un−1 ∩ Bn ) ∈ OP D . • M ∈ OPD . Indeed, M is the union of the increasing family of open subspaces {Un } ⊆ OPD and we apply Lemma 1, which ends the proof of (1). ∨ : HcdM −i (M)∨∨ → H i (M)∨ is bijective and, (2) For each i, the morphism DM,i

composed with the canonical embedding i : HcdM −i → (HcdM −i )∨∨ (2.11), gives the right adjunction ρ·,· : Hc (M)[dM ] → H (M)∨ . The finite dimensionality condition is then equivalent to the bijectivity of i , hence of ρ·,· .  

2.4.2 The Fundamental Class of an Oriented Manifold Let M be an oriented manifold of dimension dM not necessarily connected. An immediate corollary of the Poincaré duality theorem 2.4.1.3 is that the left Poincaré adjunction gives a canonical isomorphism DM : H 0 (M) −→ HcdM (M)∨ , 

where the image of 1 ∈ H 0 (M), the integral operator [M] defined in 2.3, is a nonzero linear form over HcdM (C) for each connected component C ∈ 0 (M).

17 See

Bott-Tu [18]. Corollary 4.1.1 (Poincaré Lemma), p. 35; Corollary 4.7.1 (Poincaré Lemma for Compact Supports), p. 39.

30

2 Nonequivariant Background

When M is connected, we have 1 = dim H 0 (M) = dim HcdM (M), and there exists a unique cohomological class ζ[M] ∈ HcdM (M) such that  [M]

ζ[M] = 1 .

This class is called the fundamental class of the oriented manifold (M, [M]). It will be denoted simply by ζM when the orientation [M] is understood. Exercise 2.4.2.1 Let M be an orientable manifold of dimension dM , not necessary connected. ( , p. 333) 1. Show that to choose an orientation of M is equivalent to choosing a nonzero class in HcdM (C) for each connected component C of M.

2. Show that if M is oriented and |0 (M)| < +∞, then DM (1) = C∈0 (M) ζC . 3. Can statement (2) be true when |0 (M)| is not finite ?

2.5 Poincaré Adjunctions 2.5.1 Poincaré Adjoint Pairs Let M and N be oriented manifolds of dimensions dM and dN respectively. A pair (L, R) of morphisms of complexes L : (N) → (M)[L] and

R : c (M) → c (N )[R]

is called a Poincaré adjoint pair for (M, N ), if one has [L] − [R] = dM − dN



and M

L(α) ∧ β =

 N

α ∧ R(β)

(2.39)

for all α ∈ (N ) and β ∈ c (M). Analogously, in the cohomological framework, a pair (L, R) of graded morphisms L : H (N) → H (M)[L] and

R : Hc (M) → Hc (N )[R]

is called a Poincaré adjoint pair for (M, N ), if one has [L] − [R] = dM − dN



and M

for all [α] ∈ H (N) and [β] ∈ Hc (M).

L([α]) ∪ [β] =

 N

[α] ∪ R([β])

(2.40)

2.5 Poincaré Adjunctions

31

Proposition 2.5.1.1 (and more Definitions) 1. If (L, R1 ) and (L, R2 ) are Poincaré adjoint pairs, then R1 = R2 , and we say that R := R1 is the Poincaré right adjoint of L. 2. If (L1 , R) and (L2 , R) are Poincaré adjoint pairs, then L1 = L2 , and we say that L := L1 is the Poincaré left adjoint of R. 3. If (L1 , R1 ) and (L2 , R2 ) are Poincaré adjoint pairs respectively for (M, N ) and (N, L), then (L1 ◦ L2 , R2 ◦ R1 ) is a Poincaré adjoint pair for (M, L). 4. Define the right Poincaré adjunction (cf. Sect. 2.6.1) ID M : c (M)[dM ] → (M)∨ , by 

 ID M (β) = α → α∧β . M

If (L, R) is a Poincaré adjoint pair, then ID M ◦ L = R ∨ ◦ ID N

and

ID N ◦ R = L∨ ◦ ID M ,

i.e. the following diagrams are commutative ID M

(M)[dM ] −−−−→ c (M)∨ q.i.   ⏐ ∨ ⏐ L⏐ ⏐R ID N

(N )[dN ] −−−−−→ c (N )∨ q.i.

ID M

c (M)[dM ] −−−−→ (M)∨ ⏐ ⏐ ⏐ ∨ ⏐ R L ID N

c (N )[dN ] −−−−−→(N )∨

5. If (L, R) is a Poincaré adjoint pair of morphisms of complexes, then (H (L), Hc (R)) is an adjoint pair in cohomology, so that we have DM ◦ H (L) = Hc (R)∨ ◦ DN ,

  DN ◦ Hc (R) = H (L)∨ ◦ DM .

In particular, H (L) and Hc (R) are adjoint operators via Poincaré duality. 6. If (L, R) is a Poincaré adjoint pair and L (resp. R) is a quasi-isomorphism, then R (resp. L) is a quasi-isomorphism too. Proof (1) If R1 and R2 are adjoints of the same L, we have, 

α, R1 (β) − R2 (β)

 N

= 0,

(∀α ∈ (N )) (∀β ∈ c (M)) ,

in which case R1 (β) − R2 (β) = 0 since Poincaré pairing is nondegenerate on differential forms (cf . Proposition 2.4.1.1).

32

2 Nonequivariant Background

(2) Same proof as (1). If L1 and L2 are adjoints of the same R, we have, 

L1 (α) − L2 (α), β

 M

= 0,

(∀α ∈ (N )) (∀β ∈ c (M)) ,

in which case L1 (α) − L2 (α) = 0 since Poincaré pairing is nondegenerate. (3) Follows from the equalities: 

L1 ◦ L2 (−), (−)

 M

    = L2 (−), R1 (−) N = (−), R2 ◦ R1 (−) L

(4) Straightforward verifications:     ID(L(α))(β) = L(α), β M = α, R(β) N = R ∨ (ID(α))(β)     ID  (R(β))(α) = α, R(β) N = L(α), β M = L∨ (ID  (β))(α) (5) Left to the reader. (6) After the commutative diagram DM

H (M)[dM ] −−−−→ Hc (M)∨    ⏐ ⏐ H (L)⏐ ⏐ H (R)∨ DN

H (N)[dN ] −−−−→ Hc (N )∨ 

   H (L) is an iso ⇔ H (R)∨ is an iso ⇔ H (R) is an iso since the R-duality functor is exact.

 

Comments 2.5.1.2 We shall see that, given f : M → N , the pullback morphism f ∗ : (N ) → (M) may or may not admit a right Poincaré adjoint at the level of complexes, although such adjoint will always exist in derived category, hence in cohomology. In cohomology, this right adjoint is the Gysin morphism (for arbitrary maps) f! : Hc (M) → Hc (N ). The pair (H (f ∗ ), f! ) is then a Poincaré adjoint pair in cohomology. When f is a proper map, the pullback f ∗ : c (N ) → c (M) is well-defined and one may look for a left Poincaré adjoint to f ∗ , i.e. one may look for some morphism of complexes L : (M)[dM ] → (N)[dN ] such that  N

L(α) ∧ β =

 M

α ∧ f ∗ (β) .

As in the previous case, such L may or may not exist for complexes, but it always will in derived category, hence in cohomology.

2.5 Poincaré Adjunctions

33

In cohomology, this left adjoint is the Gysin morphism for proper maps f∗ : H (M)[dM ] → H (N)[dN ]. The pair (f∗ , Hc (f ∗ )) is a Poincaré adjoint pair in cohomology.

2.5.2 Manifolds and Maps of Finite de Rham Type Definition 2.5.2.1 A manifold M is said to be of finite (de Rham) type, if its de Rham cohomology H (M) is finite dimensional. A map between manifolds f : M → N is said to be of finite (de Rham) type if N is the union of a countable ascending chain U := {U0 ⊆ U1 ⊆ · · · } of open subspaces of finite type such that each subspace f −1 (Um ) ⊆ M is of finite type. Proposition 2.5.2.2 If M is a manifold, orientable or not, which admits a finite good cover (e.g. M compact), then dim H (M) < +∞ and dim Hc (M) < +∞. Proof By induction on the cardinality of a good cover {B1 , . . . , Bn } of M, the open subspaces U := B1 ∪ · · · ∪ Bn−1 and U ∩ Bn = (B1 ∩ Bn ) ∪ · · · ∪ (Bn−1 ∩ Bn ) verify the statement. Use of Mayer-Vietoris long exact sequence sequences for (U, Bn ): [1]

[1]

H (M)

H (U ) ⊕ (Bn )

H (U ∩ Bn )

Hc (U ∩ Bn )

Hc (U ) ⊕ Hc (Bn )

Hc (M)

then ends the proof.

[1]

[1]

 

Remark 2.5.2.3 An oriented manifold is of finite type if and only if its Poincaré pairing in cohomology is perfect (2.4.1.3–(2)), in which case the compactly supported cohomology is also finite dimensional. Exercises 2.5.2.4 1. Show that a locally trivial fibration f : M → N with finite type fiber (3.1), is a finite type map. ( , p. 334) 2. Let M be a finite type manifold. Show that dim(Hc (M)) < +∞ if and only if the orientation manifold M˜ of M is also of finite type. ( , p. 334)

2.5.3 Ascending Chain Property Although general manifolds need not be of finite type, they are always the inductive limit of such. More precisely, any manifold M is the union of an ascending chain {U0 ⊆ U1 ⊆ · · · } of open subsets of finite type of M.

34

2 Nonequivariant Background

This weaker finiteness property, sufficient for our needs, results from the existence of countable good covers (fn. (15 ), p. 28). Indeed, if V := {V0 , V1 , . . . } is such a cover of M, the open subsets Un := V0 ∪ · · · Vn then verify dim H (Un ) < +∞ (as well as dim Hc (Un ) < +∞). When a manifold is endowed with the action of a Lie group G, we will also need each Un to be G-stable. Proposition 2.5.3.1 Let G be a compact Lie Group. A G-manifold M is the union of a countable ascending chain U := {U0 ⊆ U1 ⊆ · · · } of G-stable open subsets of M of finite type (as well as dim Hc (Un ) < +∞). The following sections summarize certain facts needed in the proof of this proposition, which we postpone to 2.5.6.

2.5.4 Existence of Proper Invariant Functions The aim of this section is to show, for G compact, that there always exist positive proper G-invariant functions on a G-manifold (cf . (fn. (10 ), p. 18)). Recall that, by our convention in 2.2.1, manifolds are paracompact spaces. Fix a countable, locally finite cover U := {Un }n∈N of M, where each Un is a relatively compact open subset of M. Next, fix a smooth partition of unity {ϕn }n∈N subordinate to U . This means in particular that for each n ∈ N, the equality ϕn (x) = 0 holds whenever x ∈ Un . Then one has, for every N ∈ N, 1=

 n>N

∀x ∈ U0 ∪ · · · ∪ UN .

ϕn (x) ,

(2.41)

Now, for every x ∈ M, the infinite sum φ(x) :=

 n∈N

n · ϕn (x)

is finite and smooth on M, as it is a locally finite sum of smooth functions. Lemma 2.5.4.1 The function φ : M → R>0 is proper and differentiable. Proof By property (2.41) one has, for all x ∈ U0 ∪ · · · ∪ UN , φ(x) ≥

 n>N

n · φn (x) > N

 n>N

 φn (x) = N .

(2.42)

Now, to see that  is proper, note that if F ⊆ R is compact, then F ⊆ [−N, N] for some N ∈ N and φ −1 (F ) ⊆ U0 ∪· · ·∪UN by (2.42). But the closure U0 ∪ · · · ∪ UN is a compact subset of M because each U i is assumed compact. As a closed subset of a compact set, φ −1 (F ) is also compact.  

2.5 Poincaré Adjunctions

35

As a corollary of the previous lemma, we can now prove the existence of positive proper invariant functions. Proposition 2.5.4.2 A manifold M endowed with a smooth action of a compact Lie group G admits proper G-invariant positive functions  : M → R>0 . Proof Let φ : M → R>0 denote a proper positive function (see 2.5.4.1), and set: (x) :=

 G

φ(g · x) dg ,

where dg is a G-invariant differential form of top degree on the compact Lie group G, such that 1 = G dg.18 The correspondence x → (x) is clearly a well-defined nonnegative unbounded G-invariant function of M into R. Now, for each N ∈ N, the set MN := G · φ −1 ([−N, N ]) is compact and G-stable, and if y ∈ MN , then φ(g · y) > N for all g ∈ G, so that (y) =

 G

φ(g · y) dg > N .

(2.43)

The properness of  then follows as for Lemma 2.5.4.1, i.e. if F is a compact subset of R, then F ⊆ [−N, N ] for some N ∈ N, and −1 (F ) ⊆ MN by (2.43). The subspace −1 (F ) is then compact since it is closed in the compact set MN .  

2.5.5 Manifolds with Boundary The following elementary finiteness property of compact manifolds with boundary will be needed in the proof of Proposition 2.5.3.1. Proposition 2.5.5.1 Let U be the interior of a compact manifold with boundary. Then, dim H (U ) < +∞ and dim Hc (U ) < +∞. Proof Let N be a compact manifold with boundary ∂N . Denote by U := N  ∂N its interior. Gluing N with itself along its boundary, one gets the ‘double’ N := N ∂N N , which is a compact manifold without boundary. We know from (2.5.2.2) that H (N) and H (∂N) are finite dimensional. The finiteness of dim Hc (U ) then results from the exactness of the long sequence of compactly supported cohomology associated with the closed embedding ∂N ⊆ N (see 3.7.1–(1)): · · · −→ Hci (U ) ⊕ Hci (U ) −→ H i (N ) −→ H i (∂N ) −→ · · · .

18 See

Tu [91] §13.2 Integration Over a Compact Connected Lie Group, p. 105.

36

2 Nonequivariant Background

Next, let N be an open collar neighborhood of each copy of N within N. The open subspace (∂N ) := N ∩ N ⊆ N is homotopy-equivalent to ∂N , so that its cohomology is finite dimensional. The finiteness of Hc (N ) then results from the Mayer-Vietoris sequence associated with the pair (N , N ): · · · −→ H i (N ) −→ H i (N ) ⊕ H i (N ) −→ H i ((∂N ) ) −→ · · · . We can then conclude, as before, that dim H (N ) < +∞, and, since N is homotopy-equivalent to U , that dim H (U ) < +∞ too.  

2.5.6 Proof of Proposition 2.5.3.1 The connected components of a manifold  M are always open and closed submanifolds of M. In particular, if M = i∈I Ci denotes the decomposition of M in connected components, then the indexing set I is finite or countable, and the restriction of a proper function  : M → R to each Ci remains proper. If all the connected components of M are compact, we may index them by natural numbers C0 , C1 , . . . and define Un := C0 ∪ C1 ∪ · · · ∪ Cn . Each Un is then open in M and is also a compact manifold, hence it is of finite type. The ascending chain {U0 ⊆ U1 ⊆ · · · } meets the requirements of Proposition 2.5.3.1. If M contains a noncompact connected component C, fix any proper positive Ginvariant function  : M → R, which is possible due to 2.5.4.2, and note that (C) is necessarily unbounded, since otherwise C ⊆ −1 ([0, T ]) for some T ∈ R, and C would be compact as  is proper over C. Moreover, there exists N ∈ N such that (M) ⊇ (C) ⊇ (]N, +∞[), since (C) is unbounded and connected. Now, by Sard’s theorem, the interior of the set of critical values of  : M → R is empty so that there exists an unbounded increasing sequence of positive real numbers {N < t0 < · · · < tn < · · · }n∈N which are regular values of . (see Fig. 2.1) Each subset Mn := −1 (tn ) is then a submanifold of codimension 1 in M and, moreover, it is compact and G-stable since  is proper and G-invariant. Similarly, the sets Un := −1 (]−∞, tn [) and Wn := −1 (]tn , +∞[), clearly nonempty, are G-stable open subsets of M. We then easily check that U n = Un  Mn and W n = Mn  Wn are in fact Gmanifolds with boundary Mn embedded in M. Furthermore, U n is compact as we have U n := −1 (]−∞, tn ]) = −1 ([0, tn ]) since  is positive. By Proposition 2.5.5.1, the G-stable open subspace Un verifies dim H (U ) < +∞ and dim Hc (U ) < +∞. Hence, the increasing chain {U0 ⊆ U1 ⊆ · · · } meets the requirements of Proposition 2.5.3.1. 

2.6 The Gysin Functor

37

R

M Φ Wn

Wn

Wn

tn n

n

n

n

Un

Un M

0 Fig. 2.1 Proof’s figure

2.6 The Gysin Functor 2.6.1 The Right Poincaré Adjunction Map In Sect. 2.4, we introduced the left Poincaré adjunction, ID M : (M)[dM ]−−→ c (M)∨ .

(2.44)

q.i.

By duality, this map yields ID ∨M : c (M)∨∨ → (M)[dM ]∨ which is also a quasiisomorphism and, composed with the embedding c (M) ⊆ c (M)∨∨ , gives rise to the injection and quasi-injection (2.4.1.1, 2.1.6, 2.1.11.1) c (M)[dM ], d

c (M)

∨∨[d

M ], −D

ID ∨ q.i.

∨

, −D

ID M

The resulting morphism of complexes is the right Poincaré adjunction:   ID M : c (M)[dM ], d) −→ (M)∨ , −D

(2.45)

It is given by (cf . 2.5.1.1) 

 ID M (β) := α → α∧β , M

(2.46)

38

2 Nonequivariant Background

inducing the right Poincaré adjunction in cohomology  DM : Hc (M)[dM ] −→ H (M)∨

(2.47)

Exercises 2.6.1.1 1. Check, as for ID in (2.27) (p. 24), that formula (2.46) for ID  defines a morphism of differential graded modules. ( , p. 335) 2. The natural inclusion map ι : (c (M)[dM ], d) ⊆ ((M), d)[dM ] is not a morphism of complexes, for which a sign needs to be introduced, for example, 

(c (M)[dM ], d) −−→ ((M), d)[dM ]

β → (−1)[β]dM β .

Show then that the following diagram, where Ξ (λ) := (−1)|λ|+dM λ ◦ ι, is a commutative diagram of complexes ( , p. 335) [dM ]

c (M)[dM ], d)

ID M

ID M

−D)∨

Ξ

c (M)

∨

, D) .

Proposition 2.6.1.2 Let M be an oriented manifold. The right Poincaré adjunction   ID M : c (M)[dM ], d) −→ (M)∨ , −D is an injection and a quasi-injection. Furthermore, it is a quasi-isomorphism if and only if M is of finite type.  

Proof Same as 2.4.1.3–(2).

2.6.2 The Gysin Morphism The last statement shows that for oriented manifolds of finite type, the compactly supported cohomology canonically coincides with the dual of closed support cohomology so that if N is such kind of manifold, then the diagram  DM

∨ Hc (M)[dM ] −−−−−−→ H (M) ⏐ ⏐ ⏐ f!  ⊕  H (f ∗ )∨  DN

Hc (N )[dN ] −−−−−−→ H (N )∨ 

(2.48)

2.6 The Gysin Functor

39

can be commutatively closed in a unique way by a morphism of graded spaces f! : Hc (M)[dM ] → Hc (N)[dN ]

(2.49)

It follows that the correspondence which assigns M  M! := Hc (M)[dM ] and f  f! , is covariant and functorial.  is still an injection When the manifold N in (2.48) is not of finite type, DN but it is no longer surjective so that it is not obvious that the diagram can be closed. Statement (2) in the next theorem establishes that this is in fact the case. It is therefore always possible to define the morphism f! : M! → N! , which we call the Gysin morphism for compact supports associated with f . The resulting correspondence  (−)! : Man  GV(R) or

M  M! := Hc (M)[dM ] f  f!

is thus a well-defined covariant functor on the whole category Manor , called the Gysin functor.19 Theorem 2.6.2.1 (and Definitions) 1. Let M be oriented and endow its open subsets with induced orientations. For any inclusion of open subsets j : V ⊆ W , denote by j! : c (V ) → c (W ) the map that assigns to β ∈ c (V ) its extension by zero to W , also called the pushforward of β. Then, the following diagrams ID V

c (V )[dM ] −−−→ (V )∨ ⏐ ⏐ ⏐ ∗∨ ⏐ j∗   (j ) ID W

c (W )[dM ] −−−→ (W )∨

DV

Hc (V )[dM ] −−−→ H (V )∨ ⏐ ⏐ ⏐ ⏐ Hc (j∗ )  H (j ∗ )∨  DW

Hc (W )[dM ] −−−→ H (W )∨

are commutative, i.e. (j ∗ , j! ) is a Poincaré adjoint pair (cf. Proposition 2.5.1.1). 2. For any map f : M → N between oriented manifolds, we have the diagram  DM

Hc (M)[dM ] −−−−−−→ H (M)∨ ⏐ ⏐ ⏐ f!   H (f ∗ )∨  DN

Hc (N )[dN ] −−−−−−→ H (N )∨

19 See

Sect. 8.1 for a justification of the notation.

(2.50)

40

2 Nonequivariant Background

  ) ⊆ Im(D  ), so that there exists a unique morphism of where H (f ∗ )∨ Im(DM N graded spaces f! : Hc (M)[dM ] −→ Hc (N )[dN ]

(2.51)

called the Gysin morphism for compact supports associated with f, such that the diagram (2.50) is commutative, i.e. (H (f ∗ ), f! ) is a Poincaré adjoint pair in cohomology, which means that, for any [α] ∈ H (N ) and [β] ∈ Hc (M), the equation in X,  M

f ∗ ([α]) ∪ [β] =

 N

[α] ∪ X ,

(2.52)

admits a unique solution in Hc (N ), namely X = f! [β]. Furthermore, f! in (2.51) is a morphism of H (N )-modules, i.e. the equality, called the projection formula,  f! f ∗ ([α]) ∪ [β] = [α] ∪ f! ([β])

(2.53)

holds for all [α] ∈ H (N) and [β] ∈ Hc (M). 3. The correspondence  (−)! : Manor  GV(R)

M  M! := Hc (M)[dM ]

with

f  f!

is a covariant functor, called the Gysin functor. 4. If M and N are oriented of finite type, then f ∗ : H (N ) → H (M) is an isomorphism if and only if the Gysin morphism f! : Hc (M)[dM ] → Hc (N )[dN ] is also an isomorphism. Proof (1) The commutativity results from the equality 

α V

V



∧β =

W

α ∧ j! β

for α ∈ (W ) and β ∈ c (V ), which is clear since the support of α ∧ j! β is contained in V . (2) We need to verify that, given [β] ∈ Hc (M), there exists [β  ] ∈ Hc (N ) such that the linear form [α] ∈ H (N) →

 M

f ∗ [α] ∪ [β]

2.6 The Gysin Functor

41

Fig. 2.2 Diagram (D)

coincides with [α] ∈ H (N) →

 N

[α] ∪ [β  ]

(3, 4) Thanks to Proposition 2.5.3.1, there exists an open subset W ∈ N of finite type such that f −1 W contains the support of β, denoted β := β f −1 W . We then have the commutative diagram (D) (Fig. 2.2) where the subdiagrams (I) are commutative after (2) and the commutativity of (II) is simply the functoriality of pullback morphisms. Following the arrows, we see that  (f ∗ )∨ ◦ DM ([β]) = (i ∗ )∨ ◦ (f ∗ )∨ ◦ Df −1 W ([β])   = (j ∗ )∨ ◦ DW ([β  ]) = DN ◦ Hc (j! )([β  ])

where [β  ] ∈ Hc (W )[dN ] verifies  ([β  ]) = (f ∗ )∨ ◦ Df −1 W ([β]) DW  is surjective as W is of finite type ! which is possible since DW The statement about Eq. (2.52) is clear and formally implies the projection formula since

 N

  [ω] ∪ f! f ∗ [α] ∪ [β] = f ∗ [ω] ∪ f ∗ [α] ∪ [β] =

M M

f ∗ ([ω] ∪ [α]) ∪ [β] =

 N

[ω] ∪ [α] ∪ f! [β] .

Finally, (3) is trivial since D  is bijective over its image, and (4) is clear.

 

42

2 Nonequivariant Background

Remark 2.6.2.2 It is important to note that the main ingredients in the proof are (1) the Poincaré pairings, (2) Poincaré duality and (3) the ascending chain property (2.5.3). In later sections we will show that these three ingredients exist also in the equivariant setting so that the last theorem and its proof will extend verbatim to G-manifolds and G-equivariant cohomology. Exercise 2.6.2.3 Let f : M → N be a map of oriented manifolds. Show that the left Poincaré adjoint of the Gysin morphism f! : Hc (M)[dM ] → Hc (N )[dN ] is the pullback morphism f ∗ : H (N) → H (M). ( , p. 336)  2.6.3 The Image of DM

The next proposition will be used when extending the Gysin functor to the  in terms of ascending equivariant context. It gives a description of the image of DM chains of open finite type subspaces of M, which was the main reason for proving that such covers always exist (see 2.5.3.1). Proposition 2.6.3.1 Let U be a filtrant open cover20 of a manifold M. 1. Let j : V ⊆ W denote an inclusion of open subsets of M. The extension by zero morphism j! : c (V ) ⊆ c (W ), that assigns to β ∈ c (V ) the differential form j! (β) ∈ c (W ), equal to β over V and 0 otherwise, is a morphism of complexes inducing, in cohomology, the morphism of graded spaces Hc (j! ) : Hc (V ) → Hc (W ) We also have the morphism of complexes j ∗ : (W ) → (V ) that restricts a differential form of W to V , and the corresponding morphism of graded spaces H (j ∗ ) : H (W ) → H (V ) . These constructions, applied to the elements of U , give rise to the inductive systems {c (U )}U ∈U and {Hc (U )}U ∈U , and to the projective systems {(U )}U ∈U and {H (U )}U ∈U , whence the canonical maps ν : lim c (U ) → c (M) and −→ U ∈U

μ : (M) → lim (U ) ←− U ∈U

and

H (ν) : lim Hc (U ) → Hc (M) , −→ U ∈U

H (μ) : H (M) → lim H (U ) . ←− U ∈U

All these maps are bijective. recall that U = {Ui }i∈I is said to be filtrant whenever for all U1 , U2 ∈ U there exists U3 ∈ U such that (U1 ∪ U2 ) ⊆ U3 .

20 We

2.6 The Gysin Functor

43

2. Suppose M is oriented, then the map   ID U : c (M), d [dM ] −→ lim U ∈U (U )∨ , −D −→ 

 β → α −→ α∧β

(2.54)

M

is a well-defined morphism of complexes inducing in cohomology the map  : Hc (M)[dM ] → lim U ∈U H (U )∨ DU −→

3. Suppose further that each U ∈ U is of finite type. Then ID U is a quasiisomorphism, and one has  ) = lim U ∈U H (U )∨ ⊆ H (M)∨ Im(DM −→

(2.55)

 ∨ canonically identifies with D ; more precisely, Moreover, the adjunction DU M the following diagram is commutative:  ∨ DU

lim H (U )[dM ] = ( lim H (U )∨ )∨ [dM ] −−−−−−→ Hc (M)∨ ←− −→   U ∈U U ∈U ⏐  ⏐  ⏐  DM

H (M)[dM ] −−−−−−−−−−−−−−−−−−−−−−−−→ Hc (M)∨ Proof (1) The map ν : lim U ∈U ∗c (U ) → ∗c (M) is injective since it is the limit of a −→ filtrant inductive system of injective maps. The image of ν is the union of ∗c (U ) for the same reason. Now, if ω ∈ ∗c (M), then its support, being compact, is ∗ contained in some U ∈ U so that ω is the∗pushforward of ω U ∈ c (U ). This ∗ justifies the equality c (M) = U ∈U c (U ) and proves that ν is surjective. Standard arguments on the homology of filtrant inductive systems of complexes prove that H (ν) is bijective. The map μ : (M) → lim U ∈U (U ) is injective since a differential form ←− is null if and only if it is locally so. To see that μ also surjective, let {αU ∈ (U )}U ∈U be a given projective system of differential forms, then note that for any x ∈ M, the element α(x) ˜ := αU (x) is independent of the choice of U # x. Indeed, for x ∈ U1 ∈ U and x ∈ U2 ∈ U , we can choose U3 ∈ U such that U1 ∪ U2 ⊆ U3 , since U is filtrant, in which case αU1 (x) = αU3 (x) = αU2 (x). The differentiability of α˜ is obvious since this is a local property. Finally, for all U ∈ U we have α˜ U = αU by construction, which ends the proof of the surjectivity of μ.

44

2 Nonequivariant Background

We need now only justify that H (μ) is bijective. This is immediate when M is orientable, since H (μ) is then just the Poincaré dual of Hc (ν) which has already been shown to be bijective. Otherwise, when M is not orientable, we lift U to the orientation manifold M˜ associated with M through the canonical Z/2Z-covering p : M˜ → → M, setting therefore U˜ :={U˜ := −1 ˜ ˜ → lim p (U )|U ∈U }. As M is orientable, the map H (M) H (U˜ ) is ←− U ∈U now bijective, and because this map is also compatible with the reversingorientation action of Z/2Z, it induces a bijection between invariants sub˜ Z/2Z −−→ lim spaces H (M) H (U˜ )Z/2Z , and one concludes since H (U ) =  ←− U ∈U Z/2Z ˜ H (U ) . (2) Endow each U ∈ U with the orientation induced by M. Taking the inductive limit of the maps ID U : Hc (U )[dM ] → H (U )∨ and applying (1) one sees immediately that ID U = lim U ∈U ID U . −→ (3) By 2.6.1.2 the maps ID U : Hc (U )[dM ] → H (U )∨ are quasi-isomorphisms for each U ∈ U , hence ID U = lim U ∈U ID U is also a quasi-isomorphism since −→ U is filtrant. The rest of the statement is then clear by duality.  

2.7 The Gysin Functor for Proper Maps In this section, the Gysin morphism for compact supports f! : Hc (M)[dM ] → Hc (N)[dN ] are extended to arbitrary closed supports f∗ : H (M)[dM ] → H (N )[dN ] when f : M → N is a proper map. We will see that this case is much simpler than the general one as it results immediately from Poincaré duality. When f : M → N is proper, the pullback f ∗ : (N ) → (M) respects compact supports inducing a morphism of complexes f ∗ : c (N ) → c (M), which gives rise to the covariant functor from Manpr to Vec(R) M  Hc (M)∨ ,

f  Hc (f ∗ )∨ .

When M is oriented, the right Poincaré adjunction ID M (Sect. 2.6.1) can be extended from c (M) to (M) by 

 β ∧α , ID M (α) = β → M

∀α ∈ (M),

∀β ∈ c (M) ,

2.7 The Gysin Functor for Proper Maps

45

so that the diagram of morphisms of complexes ID M

(M)[dM ] −−−−→ c (M)∨ q.i.   ⏐ ⏐ ⊆⏐ ⏐ ID M

c (M)[dM ] −−−−→ (M)∨ is commutative with its upper row a quasi-isomorphism, since it is simply the right Poincaré adjunction map ID M up to a sign ±1 related to the anticommutativity of the wedge product β ∧ α = (−1)[α][β] α ∧ β. Definition 2.7.1 If f : M → N is a proper map between oriented manifolds, the Gysin morphism associated with f is the map f∗ : H (M)[dM ] → H (N )[dN ]

(2.56)

making commutative the diagram  DM

H (M)[dM ] −−−−−−→ Hc (M)∨  ⏐ ⏐ ⏐ ⏐ f∗   Hc (f ∗ )∨  DN

H (N)[dN ] −−−−−−→ Hc (N )∨ 

Theorem 2.7.2 (and more Definitions) 1. Let f : M → N be a proper map between oriented manifolds. Then, for β ∈ Hc (N) and α ∈ H (M), the equation in X,  M

f ∗ ([β]) ∪ [α] =

 N

[β] ∪ X ,

(2.57)

admits a unique solution in H (N), namely X = f∗ [α]. Furthermore, (a) f∗ is a morphism of Hc (N )-modules, i.e. the following equality, called the projection formula for proper maps,  f∗ f ∗ ([β]) ∪ [α] = [β] ∪ f∗ [α]

(2.58)

holds for all [β] ∈ Hc (N ), [α] ∈ H (M). (b) The pullback f ∗ : Hc (N ) → Hc (M) is an isomorphism if and only if the Gysin morphism f∗ : H (M)[dM ] → H (N)[dN ] is an isomorphism.

46

2 Nonequivariant Background

2. The following correspondence is a covariant functor:  (−)∗ : Manor pr  GV(R)

with

M M∗ := H (M)[dM ] f f∗

We refer to it as the Gysin functor for proper maps 3. The natural map φ(−) : Hc (−)[d− ] → H (−)[d− ] (Sect. 2.2.3) is a homomorphism of Gysin functors (−)! → (−)∗ on the category Manor pr , i.e. the following diagrams are natural and commutative. φ(M)

Hc (M)[dM ] −−−→ H (M)[dM ] ⏐ ⏐ ⏐ ⏐ f!   f∗

(2.59)

φ(N )

Hc (N )[dN ] −−−→ H (N )[dN ] Proof (1,2) Same as 2.6.2.1. (3) Immediate after definitions.

 

2.8 Constructions of Gysin Morphisms We summarize the steps in the construction of the Gysin morphisms.

2.8.1 The Proper Case Let f : M → N be a proper map of oriented manifolds. To α ∈ (M) we assign the linear form on c (N ) defined by ID f (α) : β → M f ∗ β ∧ α. In this way we obtain the diagram [dM ]

f∗ ID f

⊕

[dN ] ID N (quasi-iso) c (N)

∨

which may be closed in cohomology, since ID N is a quasi-isomorphism. Note that the closing arrow f∗ , the Gysin morphism for proper maps, in general exists only at the cohomology level.

2.8 Constructions of Gysin Morphisms

47

2.8.2 The General Case Let f : M → N be a map of oriented manifolds. To β ∈ c (M) we assign the linear form on (N ) defined by ID f (β) : α → M f ∗ α ∧ β. In this way we obtain the diagram c (M)[dM ]

c (N)[dN

f!

quasi-iso, when N

⊕

ID f

ID N

is of finite type

∨

(2.60)

which may be closed in cohomology (as in the proper case), when N is of finite type, since then ID N is a quasi-isomorphism (2.6.1.2). When N is not of finite type, we fix a filtrant cover U of N made up of open finite type subspaces of N (see 2.5.3.1), and replace ID N with ID U . In this way, we get (see 2.6.3.1–(2,3)), the following diagram: c (M)[dM ]

c (N)[dN ]

f!

⊕

ID f,

lim U ∈

ID

=

(quasi-iso) ∨

⊆

c (N)[dN ] ID N ∨

where ID f,U is defined as follows. For β ∈ c (M) denote by |β| its support and by Uβ ⊆ U the system consisting of U ∈ U s.t. |β| ⊆ f −1 U . One has a natural map lim U (U )∨ → lim U (U )∨ (which in fact is −→ −→ β  bijective). Now, for every U ∈ Uβ the linear map M f ∗ (−) ∧ β : (U ) → R, is well-defined and compatible with restriction, so that it defines an element of lim (U )∨ , and then of lim U (U )∨ . This element is ID f,U (β) by definition. −→ Uβ −→ The closing arrow f! , the Gysin morphism associated with a general map f , is  −1 ◦ H (ID then defined in cohomology as the composition DU f,U ). Remark 2.8.2.1 In all cases, the Gysin morphism appears as the composition of a morphism in the category C(Vec(R)) with the ‘inverse’ of a quasi-isomorphism. While this is generally impossible in C(Vec(R)), it is possible in the derived category of complexes D(Vec(R)), since its fundamental property is that: a morphism in derived category is an isomorphism if and only if it induces an isomorphism in cohomology (cf . Sects. A.1.6 and A.1.6.3). Gysin morphisms are hence naturally defined in derived categories.

Chapter 3

Poincaré Duality Relative to a Base Space

In this chapter we extend the concepts of Orientability, Differential Form with Compact Support, Integration and Poincaré Adjunctions from manifolds to fiber bundles with the aim of extending the definition of Poincaré Duality to make duality be relative to a base space. This is an important step towards the equivariant cohomology of a G-space X, as this is the name given to the cohomology of the total space of the fiber bundle XG := IEG ×G X → → IBG where IBG is a classifying space for the Lie group G (cf . Sect. 4.7).

3.1 Fiber Bundles A map π : E → B between topological spaces is a locally trivial fibration of fiber F , if there is an open cover U := {Ui }i∈I of B such that for each i ∈ I there exists a homeomorphism Φi : π −1 (Ui ) → Ui × F such that p ◦ Φi = π , π −1 (U ) π

U

Φ ∼

U ×F

⊕

p

where p(x, y) = x.

U

3.1.1 Terminology F-1. The subspaces Ui are said to be trivializing for π . The trivialization Φi will also be denoted (π(−), ϕi (−)) : π −1 (Ui ) → Ui × F , with ϕi : π −1 (Ui ) → F .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Arabia, Equivariant Poincaré Duality on G-Manifolds, Lecture Notes in Mathematics 2288, https://doi.org/10.1007/978-3-030-70440-7_3

49

50

3 Poincaré Duality Relative to a Base Space

F-2. For b ∈ B, we denote by Fb := π −1 (b) the fiber of π at b, which is, after the definition, a closed subspace homeomorphic to F . F-3. The quadruple (E, B, π, F ) is called a fiber bundle of total space E, base space B, projection map π and fiber space F , F-4. A morphism of fiber bundles from (E, B, π, F ) to (E  , B  , π  , F  ), is a pair of maps f : E  → E and f : B  → B such that E π◦f = f ◦π ,

E

f

⊕

π

B

π

(3.1)

B

f

The map f therefore transforms the fiber Fb  into the fiber Fb , with b := f (b ). F-5. The definition of fiber bundles of manifolds and their morphisms are the same replacing spaces by manifolds and maps by differentiable maps. Exercise 3.1.1.1 Let (E, B, π, F ) be a fiber bundle of manifolds. Given a connected component E  of E, let B  := π(E  ) and denote by π  : E  → B  the restriction of π to E  . Show that B  is a connected component of B and that π  is a locally trivial fibration of fiber F  , union of connected components of F of dimension dE  − dB  . Hence, (E  , B  , π  , F  ) is a fiber bundle of equidimensional manifolds. The inclusion maps E  ⊆ E, B  ⊆ B are open and define a morphism of fiber bundles from (E  , B  , π  , F  ) to (E, B, π, F ). ( , p. 336)

3.1.2 The Categories TopB and ManB The definition of a fibre bundle is the mathematical formulation of the familiar idea of a space E which is obtained by glueing together a family {Fb }b∈B of copies of a given fiber space F , parametrized by the elements of a base space B. When the fibers Fb are no longer the same, we have the more general concept of a space X above B, which, while in essence a simple map π : X → B, may be better understood as the glueing of the family of fibers {Fb := π −1 (b)}b∈B . Given topological space B, we denote by TopB the category whose objects are spaces above B, i.e. topological spaces X together with a continuous map π : X → B. A morphism in TopB from π  : X → B to π : X → B is then a continuous map f : X → X such that π◦f = π ,

X

f

X π

π

B A morphism thus appears as a continuous glueing of a family of continuous maps between fibers {fb : π −1 (b) → π −1 (b)}b∈B .

3.1 Fiber Bundles

51

When data consist of differentiable manifolds and differentiable maps, we define in the same way the category ManB of manifolds above B. What makes interesting these categories are the constraints imposed by the presence of common base space B. Not only does it restrict the sets of morphisms between spaces, but, most importantly for us, the target of the cohomology functor becomes the category of H (B)-modules (which is a key point in equivariant cohomology).

3.1.3 The Relative Point of View The idea of fixing a base space B turned out to be a very fruitful heuristic in the hands of Grothendieck who named it the relative point of view, as opposed to the absolute point of view where B := {•} and which, therefore, do not impose any constraint to morphisms. A fundamental transformation in this heuristic, which corresponds to changing of point of view, is the base change functor B  ×B (−) : TopB  TopB  , based in the operation of fiber product, which we now review.

3.1.4 Fiber Product Given two morphisms πi : Xi → Z with equal target in a category C, a fiber product1 of (π1 , π2 ) is an object in C, denoted by X1 ×(π1 ,π2 ) X2 , together with two morphisms pi : X1 ×(π1 ,π2 ) X2 → Xi such that π1 ◦ p1 = π2 ◦ p2 p1

X1 ×(π1 ,π2 ) X2 p2

1 Also

X1

⊕ X2

π1

Z, π2

called fibre product, fibered product or Cartesian square.

52

3 Poincaré Duality Relative to a Base Space

representing the functor MorC (−, (π1 , π2 )) : C → Set, which associates with W ∈ C, the set of morphisms wi : W → Xi such that π1 ◦ w1 = π2 ◦ w2 . w1

W

(∃w) ˜

p1

X1 ×(π1 ,π2 ) X2 w2

X1

π1

⊕

Z

p2

(3.2)

π2

X2

Recall that what is meant by this is that the correspondence MorC (−, X1 ×(π1 ,π2 ) X2 ) → MorC (−, (π1 , π2 )) ,

w˜ → (p1 ◦ w, ˜ p2 ◦ w) ˜ ,

is an isomorphism of functors. The existence and uniqueness of w˜ : W → X1 ×(π1 ,π2 ) X2 verifying pi ◦ w˜ = wi is usually referred to as the universal property of the fiber product. It implies that fiber products, when they exist, are unique up to canonical isomorphism. The category C is said to have fiber products if, for every pair of morphisms πi : Xi → Z, the fiber product X1 ×(π1 ,π2 ) X2 exists. Notice that if Z is a final object in C, for example a singleton in a category of spaces, then the natural map X1 ×Z X2 → X1 × X2 is an isomorphism. The concept of fiber product thus naturally extends that of product. Convention The fiber product X1 ×(π1 ,π2 ) X2 will be denoted simply by X1 ×Z X2 when the morphisms πi are understood and no confusion can arise. Exercise 3.1.4.1 Let C be a category with fiber products. ( , p. 336) 1. For π ∈ MorC (X, W ), let X×W W := X×(π,idW ) W and show that the morphism (idX , π ) : X → X ×W W is an isomorphism. 2. Define a natural isomorphism X1 ×Z X2 → (X1 ×W W ) ×Z X2 . The following proposition summarizes basic properties of the fiber product. Proposition 3.1.4.2 (And Definitions) 1. The category of topological spaces (resp. locally compact) has fiber products. More precisely, given continuous maps πi : Xi → Z, a fiber product of (π1 , π2 ) is given by the closed subspace of X1 × X2 :    X1 ×Z X2 := (x1 , x2 ) ∈ X1 × X2  π1 (x1 ) = π2 (x2 )

(3.3)

together with the maps pi : X1 ×Z X2 (x1 , x2 )

Xi xi

X1 ×Z X2

p1

p2

X2

X1 π1

π2

Z

(3.4)

3.1 Fiber Bundles

53

Furthermore, the restriction to fibers: p1 : p2−1 (x2 ) → π1−1 (π2 (x2 )) and

p2 : p1−1 (x1 ) → π2−1 (π1 (x1 ))

are homeomorphisms for all xi ∈ Xi . Terminology A commutative diagram of spaces is said to be Cartesian if it is isomorphic to the diagram of a fiber product (3.4). A square box is then usually drawn inside the diagram indicate this. 2. The commutative diagram (I) of continuous maps is Cartesian if and only if it is locally Cartesian relative to Z, i.e. for every z ∈ Z there exists an open neighborhood V ⊆ Z such that the diagram (II), with U := f1−1 (π1−1 (V )) = f2−1 (π2−1 (V )), is Cartesian. f1

(I)

X1

⊕

Y f2

X2

π1

Z π2

f1

(II)

π1−1(V )

π1

U f2

V

(3.5)

π2

π2−1(V )

3. The fiber product X1 ×(π1 ,π2 ) X2 together with the map π˜ := π ◦ p1 = π ◦ p2 , is a product of π1 and π2 in TopB . Furthermore, if (E1 , B, π1 , F1 ) and (E2 , B, π2 , F2 ) are fiber bundles, then π˜ : E1 ×B E2 → B is a fiber bundle (E1 ×B E2 , B, π˜ , F1 × F2 ). Proof (1) By the universal property of products, a pair of maps wi : W → Xi factors as the composition of (w1 , w2 ) : W → X1 × X2 and the canonical projections pi : X1 × X2 → Xi . If moreover the maps wi verify π1 ◦ w1 = π2 ◦ w2 , then the image of (w1 , w2 ) is the closed subspace X1 ×(π1 ,π2 ) X2 (locally compact when the Xi ’s are so) and the first part of (1) is proved. The converse is obvious. The statement about the restrictions of the pi ’s to fibers follows by a straightforward verification, after the definition (3.3). (2) If (I) is a fiber product, then every pair of maps (w1 , w2 ) : W → πi−1 (V ) such that π1 ◦ w1 = π2 ◦ w2 , uniquely factors through Y , hence through U , proving that (II) in (3.5) is Cartesian. Conversely, by the universal property of fiber product, if (I) is commutative, we have a canonical continuous map ξ : Y → X1 ×(π1 ,π2 ) X2 . This map is bijective since we are assuming (I) locally Cartesian and that, after (1), the underlying set of a fiber product of topological spaces is the fiber product of the underlying sets. The map ξ is then a homeomorphism since its restrictions to the open subspaces U := f1−1 (π1−1 (V )), which cover Y , are homeomorphisms. (3) In the category TopB , to give a morphism from pW : W → B to the pair πi : Xi → B, means to give a pair of maps wi : W → Xi such that π1 ◦ wi = pW ,

54

3 Poincaré Duality Relative to a Base Space

hence, to give the map (w1 , w2 ) : W → X1 ×B X2 , after (3.2), which is a morphism in TopB since π˜ ◦ (w1 , w2 ) = pW . For the second part of (3), we note that, given an open subspace U ⊆ B and setting Ui := πi−1 (U ), the natural map U1 ×U U2 → X1 ×B X2 is, after the definition (3.3), an open embedding. Therefore, if U is trivializing for both fiber bundles (Ei , B, πi , Fi ), we have (πi : Ui → B) ∼ (p1 : U × Fi → B) in TopB , whence (U × F1 ) ×U (U × F2 ) ∼ U1 ×B U2 → X1 ×B X2 , but, (U × F1 ) ×U (U × F2 ) ∼ U × (F1 × F2 ) (cf . Exercise 3.1.4.1).

 

Exercise 3.1.4.3 Let H ⊆ G be a group inclusion. Given a G-equivariant continuous map f : X → Y between G-spaces, denote by νH,G,X : X/H → X/G and νH,G,Y : Y /H → Y /G the orbit maps and let fH : X/H → Y /H and fG : X/G → Y /G be the induced maps. Show that the following commutative diagram is Cartesian if and only if the restrictions fH : G · x → G · fH (x) are bijective for all x ∈ X/H , ( , p. 337) X/H

νH,G,X

⊕

fH

Y/H

νH,G,Y

X/G fG

Y/G .

3.1.5 The Base Change Functor Given a continuous map h : B  → B, consider the correspondence h−1 (−) : TopB  TopB   which associates with (π : X → B), the map h−1 (π ) : X ×B B  → B  in the Cartesian diagram X ×B B

h

h−1 (π)

B

X π

h

B

where

h(x, y) := x , h−1 (π)(x, y) := y ,

3.1 Fiber Bundles

55 X

and which associates with

X

f

π

∈ MorTopB (π , π), the morphism

π

B

X ×B B h−1 (π )

(f,id)

B

X ×B B h−1 (π)

∈ MorTopB (h−1 (π ), h−1 (π)) .

Proposition 3.1.5.1 1. The correspondence h−1 (−) : TopB  TopB  is a covariant functor. Furthermore, if (π : X → B) ∈ TopB is a locally trivial fibration of fiber F , then h−1 (π ) is a locally trivial fibration of fiber F . 2. Given a continuous map h : B  → B  , we have an isomorphism of functors h −1 (−) ◦ h−1 (−) → (h ◦ h )−1 (−) . Terminology The functor h−1 (−) : TopB  TopB  is called the pullback, or the base change, functor induced by h. Proof (1) The fact that h−1 (−) is a covariant functor, easily follows from the fact that, after 3.1.4.2-(3), base change results by composing two functorial operations, first the product in TopB by (h : B  → B), which already gives the covariant endofunctor (−) ×B B  : TopB  TopB , and second the factorization of the projection map π(−) ◦ p1 through p2 : (−) ×B B  → B  : (−) ×B B

p2

π(−) ◦p1

h

B

B

To see that h−1 (−) preserves locally trivial fibrations of given fiber F , we proceed as for 3.1.4.2-(3). If U ⊆ B is an open trivializing subspace for π : X → B, then (π : π −1 (U ) → B) is isomorphic to (p1 : U × F → B) in TopB , and we have the open embedding (U × F ) ×U h−1 (U ) ∼ p−1 (U ) ×B h−1 (U ) → X ×B B  , where (U × F ) ×U h−1 (U )  F × h−1 (U ) (cf . Exercise 3.1.4.1). (2) Concatenating the Cartesian diagrams generated by h−1 (−) and h−1 (−) : X ×B B ×B B h

h

1 (h−1 (π))

B

X ×B B

h

X

h−1 (π) h

B

π h

B

56

3 Poincaré Duality Relative to a Base Space

we immediately get a commutative diagram X ×B B ×B B h

X ×B B

⊕

1 (h−1 (π))

(h◦h )−1 (π)

B

B

which is natural with respect to (π : X → B) and where the horizontal arrow is a homeomorphism (cf . Exercise 3.1.4.1).  

3.1.6 Fiber Products of Fiber Bundles of Manifolds Fiber products may not always exist in some categories of topological spaces, for example in the category of manifolds (exercise ! ( , p. 337)). But they do exist when one of the maps pi : Xi → Z is a locally trivial fibration. Proposition 3.1.6.1 1. Let (E, B, π, F ) be a fiber bundle of manifolds and let B  be a manifold. For every map f : B  → B, the fiber product E  := E ×B B  exists in the category Man of manifolds and coincides as a topological space with the fiber product for locally compact spaces (3.3). – The map π  : E  → B  , defined as π  (x1 , x2 ) := x2 , is a locally trivial fibration of manifolds with fiber F . – The diagram π◦f = f ◦π ,

E

f

E

B

f

B

π

π

(3.6)

where f : E  → E denotes the map f (x1 , x2 ) := x1 , is a Cartesian diagram. Terminology The fiber bundle (E  , B  , π  , F ) is called the pullback bundle of (E, B, π, F ) by f and is denoted by f −1 (E, B, π, F ), or simply f −1 (E). 2. A commutative diagram of differentiable maps E π

B

f

⊕ f

E π

B

where π is a locally trivial fibration, is a Cartesian diagram if and only if it is locally Cartesian relative to B (cf. Proposition 3.1.4.2–(2)).

3.1 Fiber Bundles

57

3. Given two fiber bundles of manifolds (Ei , B, πi , Fi ) with base B, the product (E1 ×B E2 , B, π˜ , F1 × F2 ) exists in the category ManB of manifolds above B (see 3.1.8). It coincides as a topological space with the product in TopB . Proof (1) The differential structure of the topological space E  := B  ×B E (3.3), will be given by an atlas of differentiable manifolds. Let   E := Φi := (π, ϕi ) : π −1 (Ui )  Ui × F i∈I ,

(3.7)

be an open cover of trivializations for the fiber bundle of manifolds π : E → B (see Exercise 3.1.9.3-(4)). The transition maps of E are the diffeomorphisms Tii  : Uii  × F → Uii  × F ,

Tii  (x, y) = (x, ϕii  (x, y)) ,

(3.8)

where ϕii  (x, −) : F → F is a diffeomorphism depending on x ∈ Ui,i  differentiably, which is defined by the constraint ϕii  (x, ϕi (y)) = ϕi  (y) ,

∀y ∈ π −1 (Uii  ) .

(3.9)

Let B  := {Vi }i∈I be the open cover of B  , where Vi := f −1 (Ui ). We have π −1 (Vi ) = Vi ×Ui π −1 (Ui ) , by the universal property of fiber products for topological spaces. Now, a trivialization (π, ϕi ) : π −1 (Ui )  Ui ×F in E, immediately gives us a trivialization (π  , ϕi ) : π −1 (Vi )  Vi × F , by setting (cf . Exercise 3.1.4.1) π −1 (Vj ) = Vi ×Ui π −1 (Ui ) −−→ Vi ×Ui (Ui × F ) = Vi × F 

(π



, ϕi )([x, y])

(3.10)

:= (x, ϕi (y))

where [x, y] denotes an element in the fiber Vj ×Ui π −1 (Ui ).   product   The family of trivializations E := (π , ϕi ) : π −1 (Vi ) → Vi × F i∈I is then an open cover of the space E  with transition homeomorphisms, the maps Tii  (x  , y) = (x  , ϕii  (f (x  ), y)) ,

∀x  ∈ Vii  ,

(3.11)

which are clearly étale, hence diffeomorphic. The cover E  therefore defines a differentiable structure on the topological fiber product E  := B  ×B E.

58

3 Poincaré Duality Relative to a Base Space

The map f is differentiable since it appears through the trivializations (π  , ϕi ) and (π, ϕi ) as the differentiable map (f , idF ) π

1 (V

)

f

π −1 (U )

(π ,ϕ )

V

(π,ϕ )

F

(f ,idF )

U

F

In the same way, for any manifold M and maps g : M → E and g  : M → B  , such that π ◦ g = g  ◦ f , the induced continuous map g˜ : M → E  := B  ×B E, appears through the trivialization (π  , ϕi ) as the differentiable map m → (g  (m), ϕi (g(m))) . The manifold E  := B  ×B E with f : E  → E and π  : E  → B  is therefore a fiber product for π : E → B and f : B  → B, in the category of manifolds. (2) Same proof as 3.1.4.2-(2). (3) As in the proof of 3.1.5.1-(1), the fact that the product in ManB is obtained by the base change (−) ×B E  : ManB  ManE  , which we proved it exists, followed by the extension of the base through the (differentiable) map π  : E  → B, immediately shows that the product exists in ManB . It only remains to show that π˜ : E ×B E  → B is a locally trivial fibration of manifolds of fiber F1 × F2 . This follows as in 3.1.4.2-(3), by considering an atlas U := {Uj }j∈J of B which is simultaneously trivializing for both fiber bundles (Ei , B, πi , Fi ) with respective trivializations, for i = 1, 2, (πi , ϕi,j ) : πi−1 (Uj ) → Uj × Fi . We then have the trivialization of π˜ −1 (Uj ) (cf . Exercise 3.1.4.1) π1−1 (Uj ) ×B π2−1 (Uj ) → (Uj × F1 ) ×Uj (Uj × F1 ) = Uj × F1 × F2 given by the map Φj := (π˜ , (ϕ1,j ◦ p1 , ϕ2,j ◦ p2 )) : π˜ −1 (Uj ) → Uj × F1 × F2 . The transition map from Φj to Φj , Tjj : Ujj × (F1 × F2 ) → Ujj × (F1 × F2 ) ,

(3.12)

Tjj (x, (y1 , y2 )) = (x, (ϕ1,jj (x, y1 ), ϕ2,jj (x, y2 ))) ,

(3.13)

is therefore

which is étale, hence diffeomorphic, since the maps ϕi,jj are so (cf . 3.8).

 

3.1 Fiber Bundles

59

3.1.7 Orientable Fiber Bundles A connected fiber bundle of manifolds (E, B, π, F ) is said to be orientable if there exists ωπ ∈ dF (E) such that, for all b ∈ B, the restriction ωπ

Fb

∈ dF (Fb ) ,

(3.14)

is nowhere vanishing. A necessary, but not sufficient, condition for a fiber bundle to be orientable is that its fiber F be orientable (cf . Proposition 2.3.1.1). A nonconnected fiber bundle is said to be orientable if its connected components are so. Proposition 3.1.7.1 (and Definition) A fiber bundle of manifolds (E, B, π, F ), with orientable fiber F , is an orientable fiber bundle if and only if there exists an open cover U := {Ui }i∈I of B by trivializing subspaces, and a trivialization Φi := (π, ϕi ) : π −1 (Ui ) → Ui × F , for each i ∈ I, such that in the transition maps from Φi  to Φi (see (3.9)) Tii  : Uii  × F → Uii  × F ,

Tii  (x, y) = (x, ϕii  (x, y)) ,

the maps ϕii  (x, −) : F → F preserve the orientation, for all x ∈ Uii  . In that case, the family of trivializations {(π, ϕi ) : π −1 (Ui ) → Ui × F }i∈I is said to be oriented (cf. 2.3.1). A nowhere vanishing differential form ωF ∈ dF (F ) then determines the differential form ωπ ∈ dF (E) in (3.14) by taking a partition of unity {φi }i∈I subordinate to U and setting ωπ := The restriction to fibers ωπ

Fb

 i

φi ϕi∗ (ωF ) ∈ m (E) .

∈ dF (Fb ) are then nowhere vanishing.

Proof We assume, after Exercise 3.1.1.1, that E, B and F are equidimensional respectively of dimensions n, b and m. Assume the orientability condition (3.14). Fix an orientation [F ] for F . Let U := {Ui }i∈I be a cover of B by connected trivializing open subspaces. Fix i ∈ I, and fix a trivialization (π, ϕi ) : π −1 (Ui ) → Ui × F , and denote its inverse by Ψi : Ui × F → π −1 (Ui ). For each connected component Fc of F , we have the family of embeddings indexed by x ∈ Ui ιc,x

Ψi

Fc −−→ Ui × F −−→ π −1 (Ui ) , where ιc,x (y) = (x, y). Since Ui is connected, there exists c = ±1 such that [Fc ] = c [ι∗c,x Ψi∗ (ωπ )] ,

∀x ∈ Ui .

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3 Poincaré Duality Relative to a Base Space

When c = −1, the maps ιc,x : Fc → {x} × Fc inverse orientation  for all x ∈ Ui , and we modify (π, ϕi ) by composing ϕi : π −1 (Ui ) → F = c Fc with ιc,x0 , for any arbitrary choice of x0 ∈ Ui . We proceed in the same way for each connected component Fc and we denote by (π, ϕi ) : π −1 (Ui ) → Ui × F the resulting modified trivialization, with inverse denoted by Ψi . Then, by construction, [F ] = [ι∗x Ψi∗ (ωπ )] ,

∀x ∈ Ui ,

where ιx : F → Ui × F is now simply ιx (y) = (x, y). We have thus a family of trivializations {(π, ϕi ) : π −1 (Ui ) → Ui ×F }i∈I whose transition maps respect well the orientations of fibers as announced. Conversely, let ωF ∈ m (F ) nowhere vanishing and such that [ωF ] = [F ]. Then, given an open cover U := {Ui }i∈I of B by trivializing subspaces, and for each i ∈ I, a trivialization (π, ϕi ) : π −1 (Ui ) → Ui × F , we define ωi := ϕi∗ (ωF ) ∈ m (π −1 (Ui )) . Now, if {φi }i∈I is a partition of unity subordinate to U , the sum ωπ :=

 i

φi ωi ∈ m (E)

is well-defined, and is such that, for x ∈ B, we have  φi (x) ϕii  (x, −)∗ (ωF ) ωπ Fx = i

Fx

(3.15)

for any i  ∈ I such that x ∈ Ui  . In particular, if, as assumed, the maps ϕii  (x, −) preserve the orientation of fibers for x ∈ Uii  (hence for x such that φi (x) = 0), then 3.15 is a nowhere vanishing differential form in m (Fx ) as required.   Corollary 3.1.7.2 1. Let f : B  → B be a map of manifolds. The pullback f −1 (E) of an orientable fiber bundle (E, B, π, F ) (cf. Proposition 3.1.6.1-(1)) is orientable. 2. The product in ManB (cf. Sect. 3.1.8) of orientable fiber bundles is orientable. 3. A fiber bundle of manifolds (E, B, π, F ) is orientable in the following cases. (a) E = B × F ;

(b) E and B are orientable;

(c) B is simply connected.

Proof (1) In describing the manifold structure of the pullback f −1 (E) in the proof of 3.1.6.1-(1), we fixed an open cover of trivializations for π : E → B,   E := (π, ϕi ) : π −1 (Ui )  Ui × F i∈I ,

3.1 Fiber Bundles

61

with transition maps Tii  : Uii  × F → Uii  × F , Tii  (x, y) = (x, ϕii  (x, y)) ,

∀x ∈ Uii  .

We then defined in (3.10) an open cover of trivializations for E  := f −1 (E),   E  := (π  , ϕi ) : π −1 (Vi ) → Vi × F i∈I , where Vi := f −1 (Ui ), whose transition maps have the form Tii  (x  , y) = (x  , ϕii  (f (x  ), y)) ,

∀x  ∈ Vii  ,

where the ϕii  are the same for E and E  . In particular, applying Proposition 3.1.7.1, the family E is oriented if and only if the family E  is so, which proves (1). (2) The proof is the same as for (1) so that we will be sketchy. We use the trivializing cover for the product of fiber bundles already described in the proof of 3.1.6.1-(3), where the transition maps are (3.13): Tjj (x, (y1 , y2 )) = (x, (ϕ1,jj (x, y1 ), ϕ2,jj (x, y2 ))) ,

(3) (3a) (3b)

(3c)

where the ϕi,jj are the same that appear in the transition maps of trivializing covers of the fiber bundles (Ei , B, πi , Fi ). Hence, again by Proposition 3.1.7.1, an oriented trivialization open cover for the product π˜ : E ×B E  → B exists. It suffices to assume B connected. Is obvious and is also consequence of (1) for the constant map h : B → {•}. Let ωE ∈ n (E), ωB ∈ b (B) and ωF ∈ m (F ) be nowhere vanishing. Then, for any trivializing U ⊆ B and any trivialization Φ : π −1 (U )  U × F , there is a unique choice of sign U ∈ {1, −1} such that ωE and Φ ∗ (ωB ∧ (U ωF )) define the same orientation on π −1 (U ) (2.3.1.1). The differential forms ωΦ,U := Φ ∗ (1 ⊗ U ωF ) are then m-forms defining the same orientation on every fiber Fx for x ∈ U , independently of the trivialization Φ of π −1 (U ). Hence the orientability of the fiber bundle after Proposition 3.1.7.1. The proof of (3b) shows that the orientations of the fibers Fb define a twosheeted covering E of B. When E and B are both oriented, there is a canonical way to choose one of the two sheets over each trivializing open subspace U ⊆ B, entailing the triviality of E. When we do not know whether E is orientable, but we do know that B is simply connected, we can still assert that B is orientable and that E is trivial (cf . Proposition 2.3.1.1-(2)). We can therefore still choose the signs U in a compatible way, and doing so, endow E of an orientation, in which case statement (3b) applies.  

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3 Poincaré Duality Relative to a Base Space

Comment 3.1.7.3 The definition of orientability 3.1.7 is adapted to fiber bundles of manifolds (E, B, π, F ), but in many important cases, while the fiber F is a manifold, the base B is not. For example, in the homotopy quotients IEG ×G F → → IBG where IBG is a CW-complex of infinite dimension. In those cases, where B is at least locally contractible, an alternative definition of orientability can be based on the idea in the proof of 3.1.7.2-(3) of considering the covering E of B defined by the orientations of the connected component of the fibers Fb , as b runs over B. One then says that (E, B, π, F ) is orientable if E is a trivial covering, in which case to choose an orientation means to choose one sheet of E for each connected component of E. With this definition of orientability, Propositions 3.1.7.1 and 3.1.7.2-(1,2) remain true and 3.1.7.2-(3) is verified by the classifying space IBG. Exercises 3.1.7.4 1. Show that in an orientable fiber bundle of manifolds (E, B, π, F ), the orientability of E and B are equivalent properties. ( , p. 338) 2. Give an example of a nonorientable fiber bundle of manifolds (E, B, π, F ) where B and F are orientable. ( , p. 339)

3.1.8 The Categories ManB , FibB and Fibor B As for topological spaces in 3.1.2, given a manifold B, we define the category of manifolds above B, denoted by ManB . Its objects are the differentiable maps of manifolds π : M → B, and the morphisms from π : M  → B to π : M → B are the differentiable maps f : M  → M such that π ◦ f = π  . We denote by Fibor B and FibB the full subcategories of ManB respectively of fiber bundles and oriented fiber bundles of base B. Exercise 3.1.8.1 Check that by Propositions 3.1.6.1, 3.1.7.1 and 3.1.7.2, the analogue to Proposition 3.1.4.2 is verified, with the same proof, when replacing the category TopB by the categories ManB , FibB and Fibor B.

3.1.9 Proper Subspaces of a Fiber Bundle We introduce the concept of properness in the category TopB as the counterpart of compactness in the category Top. As we will see in 3.1.10, differential forms with proper supports on the total space of fiber bundles of manifolds constitute the right analogue to differential forms with compact supports on manifolds. Definition 3.1.9.1 Let π : X → B be a continuous map. A subspace P ⊆ X is said to be π -proper, or simply proper when π is understood, if π : P → Z is a

3.1 Fiber Bundles

63

proper map, i.e. if for every compact subspace K ⊆ B, the subspace π −1 (K) ⊆ X is compact. Given a commutative diagram of continuous maps X

X

g

⊕

π

B

(3.16)

π

B

g

we say that g −1 preserves properness, if for every π -proper subset P ⊆ X, the subspace g −1 (P ) ⊆ X is π  -proper. Proposition 3.1.9.2 1. In a fiber product X1 ×B X2

p1

X1

p2

π1

X2

π2

(3.17)

B

the inverse image p1−1 preserves properness. In particular, if π1 is a proper map, so is p2 , and likewise exchanging 1↔2. 2. Given a commutative diagram X

g

X

⊕

f

B

g

f

B

consider the factorization g

X

ξ

X ×B B

p1

p2

f

B

B

X f

g

B

where ξ : X → X ×B B  is the induced map from X to the fiber product of f : X → B and g : B  → B, whence g = p1 ◦ ξ and p2 ◦ ξ = f  . Then, a. the map g −1 preserves properness if and only if ξ is a proper map; b. if g is proper, then g −1 preserves properness if and only if g is proper.

64

3 Poincaré Duality Relative to a Base Space

Proof (1) We must show that p1−1 (P ) ∩ p2−1 (K) is compact for K ⊆ X2 compact. By the definition of fiber product 3.1.4.2-(1), we have   p1−1 (P ) ∩ p2−1 (K) = K ∩ π2−1 (π1 (P )) ×B π1−1 (π2 (K)) ∩ P ,

(3.18)

where K ∩ π2−1 (π1 (P )) is always compact. When, in addition, π1 : P → B is a proper map, then π1−1 (π2 (K)) ∩ P is also compact, and the right-hand side of (3.18) is compact since it is closed in a product of compact spaces. (2) Lemma. Given a commutative diagram X

ξ

W q

f

B ,

the map ξ is proper if and only if ξ −1 preserves properness. Proof of Lemma For every P ⊆ W and every compact K ∈ B  , we have ξ −1 (P ∩ q −1 (K)) = ξ −1 (P ) ∩ f −1 (K) .

(3.19)

When ξ : X → W and q : P → B  are proper, the left-hand side of (3.19) is compact, so that ξ −1 (P ) is f  -proper. Conversely, if K ⊆ W is compact, it is q-proper and ξ −1 (K) is f  -proper, so that ξ −1 (K) ∩ f −1 (q(K)) is compact. But, ξ −1 (K) ∩ f −1 (q(K)) = ξ −1 (K). Hence ξ is proper.  (2a) Assume ξ proper. If P ⊆ X is f -proper, then p1−1 (P ) is p2 -proper, by (1), and ξ −1 (p1−1 (P )) = g −1 (P ) is f  -proper after the preliminary result. Conversely, if K ⊆ X ×B B  is compact, then ξ −1 (K) is compact too, since it is closed in g −1 (p1 (K))∩f −1 (p2 (K)) which is compact because p1 (K), being compact, is f -proper and that g −1 preserves properness. (2b) If g is proper, then p1 is proper, by (1). Hence, g is proper if and only if ξ is proper, and, by (2a), if and only if g −1 preserves properness.   Exercises 3.1.9.3 1. In the definition of proper subspaces 3.1.9.1, show that if g is a proper map, then g −1 preserves properness, but that the converse is not true. ( , p. 339) 2. Show that in 3.1.9.2-(1), if π1 is open, then p2 is open, but that the same is not true for ‘closed’ in lieu of ‘open’. ( , p. 339) 3. Call π1 : X1 → B universally closed if for all π2 : X2 → B, the map p2 : X1 ×B X2 → X2 is closed. Show that in the category of locally compact spaces a map is proper if and only if it is universally closed. ( , p. 339)

3.1 Fiber Bundles

65

4. In the definition of a locally trivial fibrations π : E → B of fiber F (3.1), a trivialization Φ : π −1 (U ) → U × F is of the form Φ(−) = (π(−), ϕ(−)). Show that (π(−), ϕ(−)) is a trivialization if and only if the following diagram is Cartesian ϕ

π−1(U ) π

⊕

U

F {•}

or, equivalently, the restrictions of ϕ to fibers are homeomorphisms and that ϕ −1 preserves properness, or even, in the category of manifolds, if and only if the restrictions of ϕ to fibers are diffeomorphisms. ( , p. 340)

3.1.10 Differential Forms with Proper Supports Given a map of manifolds π : E → B, a differential form β ∈ (E) is said to be of proper (or compact vertical) support, if |β| is π -proper (3.1.9.1), i.e. For every compact subspace K ⊆ B, the subspace π −1 (K) ∩ |β| is compact. We denote by cv (E) ⊆ (E) the subset of such differential forms. Notice that if B := {•} then cv (E) = c (E). The forthcoming propositions 3.1.10.2 and 3.2.1.3 show that the most relevant properties of c (F ) extend naturally to cv (E). Exercise 3.1.10.1 Given a map of manifolds π : E → B, for φ ∈ 0 (B) and β ∈ (E), let φ β := (φ ◦ π )β. Show that β ∈ cv (E) if and only if φ β ∈ c (E) for all φ ∈ 0c (E). ( , p. 340) Proposition 3.1.10.2 Let (E, B, π, F ) be a fiber bundle of manifolds. 1. The set cv (E) is a differential graded ideal of (E), in other words, cv (E) is a subcomplex of c (E) such that (E) ∧ cv (E) ⊆ cv (E) . In particular, cv (E) is a graded differential module over (B) through the pullback morphism π ∗ : (B) → (E). 2. If a morphism of fiber bundles, E

f

⊕

π

B

f

E π

B

66

3 Poincaré Duality Relative to a Base Space

is either Cartesian or is such that f is proper, then f ∗ (cv (E)) ⊆ cv (E  ), thus inducing a diagram of pullback morphisms of complexes f∗

)

π∗

π

cv (E)

f∗

cv (E

)

(3.20)

⊆

⊆

f∗

)

where, for all α ∈ (E) and β ∈ cv (E) (resp. β ∈ (E)), we have f ∗ (α ∧ β) = f ∗ (α) ∧ f ∗ (β) .

(3.21)

Proof (1) Given βi ∈ cv (E) and α ∈ (E), we have the following obvious relations on supports |dβ| ⊆ |β| ,

|β1 + tβ2 | ⊆ |β1 | ∪ |β2 | , ∀t ∈ R ,

|α ∧ β| ⊆ |β| ,

which immediately imply that cv (E) is a differential graded ideal over (E). (2) The inclusion f ∗ (cv (E)) ⊆ cv (E  ) results applying Propositions 3.1.9.2(1,2b). The equality (3.21) is then obvious since f ∗ : (E) → (E  ) is a homomorphism of algebras.  

3.1.10.1

Poincaré Lemmas for cv (E) and (E)

We extend the classic Poincaré Lemmas from manifolds to fiber bundles. Proposition 3.1.10.3 1. Given two homotopic morphisms of fiber bundles (f1 , f 1 ) and (f0 , f 0 ), E π

B

fi

⊕ fi

E π

B

either Cartesian or such that the fi ’s are proper maps, the pullbacks fi∗ : cv (E) → cv (E  )

and

are homotopic morphisms of complexes.

fi∗ : (E) → (E  ) ,

3.1 Fiber Bundles

67

2. Poincaré Lemmas. If (E, B, π, F ) is the projection π : Rm × F → → Rm , m π(x, y) := x, then the pullbacks induced by p2 : R × F → → F: p2∗ : c (F ) → cv (E)

and

p2∗ : (F ) → (E) ,

are homotopic morphisms of complexes. Proof Let (h, h) be a homotopy for the (fi , f i )’s R×E R

E

h

⊕

π

B

π

h

B

Denote by t ∈ R the variable parametrizing the homotopy. For every ω ∈ i (E), we have a unique decomposition h∗ (ω) = α + dt ∧ β , with α(t, y) ∈ i (E  ) and β(t, y) ∈ i−1 (E  ), for all t ∈ R. The map ηi : i (E) → i−1 (E) ,

by

ηi (ω) :=

 1

(3.22)

β dt , 0

verifies η ◦ d + d ◦ η = (f1∗ − f2∗ ) : (E) → (E  ) .  Indeed, we have h∗ (dω) = d(h∗ (ω)) = dE  (α) + dt ∧ ∂t α − dE  (β) , so that  1 ⎧ ∂t α − dE  (β) dt ⎨ η(dE  (ω)) = 0

 1  1 ⎩ dE  (η(ω)) = dE  β dt = dE  (β) dt 0

0

Whence, (η ◦ d + d ◦ η)(ω)(y) =

 1 0

∂t α dt = α(1, y) − α(0, y)

= h∗ (ω)(t, y)

t=1

− h∗ (ω)(t, y)

t=0 ,

and the homotopy-equivalence of fi∗ : (E) → (E  ) is proved. When ω ∈ cv (E), we have β ∈ cv (R × E  ) by 3.1.10.2-(2), in which case, for every φ ∈ 0c (B  ), the support |φ β| is compact in R × E  . Hence, there exists

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3 Poincaré Duality Relative to a Base Space

1 K ⊆ E  such that |φ β| ⊆ R × K, and |φ 0 β dt| ⊆ K. We can therefore conclude 1 that 0 βdt ∈ cv (E  ) (Exercise 3.1.10.1). The same formula (3.22) thus gives a map η : cv (E) → cv (E)[−1], and the same arguments show that it gives a homotopy for the morphisms fi∗ : cv (E) → cv (E  ). (2) The family of maps f t : Rm → Rm , ft (v) := t v for t ∈ R, induces a family of homotopic morphisms (ft , f t ) : Rm × F → Rm × F . The induced morphism of complexes f1∗ = id is then homotopic to f0∗ by (1). On the other hand, since (f0 , f 0 ) factors as the composition: f0

Rm

p2

×F

ι0

F

π

Rm × F

π c0

Rm

π ι0

0

Rm

the pullbacks p2∗ : (F ) → (Rm × F ) and p2∗ : c (F ) → cv (Rm × F ) admit as homotopy left inverse the morphism ι∗0 . But this same ι∗0 is clearly also a right inverse. Hence the proof of Poincaré Lemmas.  

3.1.10.2

Sheafification of cv (E) and (E) on the Base Space B

For every open inclusion ιU : U ⊆ B, denote by EU := (π −1 (U ), U, π, F ) the pullback ι−1 U (E, B, π, F ). Then, thanks to the functoriality of the base change functors (3.1.5.1), the correspondences which associate with an open subspace U ⊆ B, the complexes U  cv (EU )

and

U  (EU ) ,

and which associate with the inclusion maps ιV U : V ⊆ U , the pullbacks ι∗V U : cv (EU ) → cv (EV )

and

ι∗V U : (EU ) → (EV ) ,

define presheaves of  B -differential graded modules, respectively denote by π!  E and π∗  E

(3.23)

Proposition 3.1.10.4 1. The presheaves π!  E and π∗  E are sheaves. 2. The graded sheaves of cohomology they define: H(π! ) := H(π!  E )

and

H(π∗ ) := H(π∗  E ) ,

are locally trivial sheaves of fibers respectively Hc (F ) and H (F ).

(3.24)

3.1 Fiber Bundles

69

Furthermore, if E is connected, then the sheaf HdF (π! ) admits nowhere vanishing global sections if and only if (E, B, π, F ) is an orientable fiber bundle. In particular, a fiber bundle (E, B, π, F ) with connected fiber F is orientable if and only if the sheaf HdF (π! ) is isomorphic to the trivial sheaf R B . Proof (1) The correspondence U  (π −1 (U )) is the direct image presheaf π∗  E which is well-known to be a sheaf. For the correspondence U  cv (π −1 (U )) we  need to justify the sheaf condition, i.e. that for every open cover U = i∈I Ui , the sequence 0 → cv (π −1 (U )) →

 i∈I

cv (π −1 (Ui )) →

 i,i  ∈I

cv (π −1 (Uii  )) , (3.25)

is well-defined and exact. For this, we need to justify two facts about a continuous map p : X → U . – First. If f is proper, then the restriction p : p−1 (V ) → V is proper for every open subspace V ⊆ U , which is clear after the definition of proper maps. Then, taking for X the supports of differential forms, this first fact shows that (3.25) is well-defined.  – Second. For every open cover U = i∈I Ui , if p : p−1 (Ui ) → Ui is proper for all i ∈ I, then f : X → U is proper. For this, we need only recall  that, given K ⊆ U compact, there exists a finite decomposition K = j ∈J Kj , where Kj is compact and contained in some Ui . But then,  p−1 (K) = j ∈J p−1 (Kj ), where p−1 (Kj ) is compact in p−1 (Ui ), hence in X. Now, as π!  E is contained in the sheaf π∗  E , to prove that (3.25) is exact, we need only check exactness at its central term. Given a family of differential forms {βi ∈ cv (π −1 (Ui ))}i∈I such that βi Uii = βi  Uii , for all i, i  ∈ I, we already know that there exists a unique β ∈ (π −1 (U )) such that β Ui = βi , and, again, taking for X supports of differential forms, the second fact shows that β has proper support. (2) Local triviality in (3.24) is immediate by Poincaré Lemmas 3.1.10.3-(2). By definition, a connected fiber bundle is orientable if and only if there exists ωπ ∈ dF (E) such that ωπ Fb is nowhere vanishing for all b ∈ B. This form ωπ therefore defines a global section of HdF (π! ) since it determines a fundamental class ζCb ∈ HcdM (Cb ) on each connected component Cb of the fiber Fb (cf . Sect. 2.4.2). For the converse, one notes that a nowhere vanishing global section σ of HdF (π! ), although it determines a fundamental class ζCb for at least one component Cb of Fb , it could vanish on the others. However, this cannot occur since we assumed E connected. Indeed, in that case any component Cb of Fb can be joined to Cb by some path γ in E, and it is clear, by the local triviality

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3 Poincaré Duality Relative to a Base Space

of π , that if we follow the section σ along the path π ◦ γ , the fundamental class ζCb moves on a fundamental class of Cb , hence proving that the section σ determines an orientation of the fiber bundle.   Corollary 3.1.10.5 Given two manifolds B and F , the following inclusions are quasi-isomorphisms :  B ⊗ c (F ) ⊆ π!  B×F

and

(B) ⊗ c (F ) ⊆ cv (B × F ) and

 B ⊗ (F ) ⊆ π∗  B×F , (B) ⊗ (F ) ⊆ (B × F ) .

Proof Recall that given two topological spaces X and Y , and sheaves of vector spaces F ∈ Sh(X; k) and G ∈ Sh(Y ; k), one defines the external tensor product F  G ∈ Sh(X × Y ; k) as the sheaf associated with the presheaf U × V  F (U ) ⊗k G (V ) . In the case of the product B × F , we have a natural inclusion of complexes of sheaves  B   F ⊆  B×F giving rise to the inclusions of complexes  B ⊗ c (F ) ⊆ π!  B×F

and

 B ⊗ (F ) ⊆ π∗  B×F ,

(3.26)

inducing the morphisms in the cohomology sheaves: p

R B ⊗ Hc (F ) → Hp (π! )

and

R B ⊗ H p (F ) → Hp (π∗ )

(3.27)

which are isomorphisms since this is so at stalks level after 3.1.10.4-(2). The terms of the complexes (3.26), being  0B -modules, are Γ (B; −)-acyclic, so that the natural map Γ (B; −) → IR Γ (B; −) induces quasi-isomorphisms when applied to the complexes (3.26) (Theorem B.6.3.4-(3)).2 The morphisms 

IR Γ (B;  B ⊗ c (F )) → IR Γ (B; π!  B×F ) IR Γ (B;  B ⊗ (F )) → IR Γ (B; π∗  B×F )

2 See

(3.28)

also Bredon [20], ch. II-§9 Φ -soft and Φ -fine sheaves, p. 65 , in particular examples 9.4 and 9.17 which state that the sheaf of rings  0 (M) of differentiable functions on a manifold M is Φ-soft for any paracompactifying family of supports Φ. This implies that every  0 (M)-module is Φ-soft, hence acyclic for the functor ΓΦ (M; −) of global sections with supports in Φ, which includes the functor Γ (M; −) and Γ c (M; −) (cf . Sect. B.6.2). Other references for these questions are Godement [46] ch. II-§3.7. Faisceaux fins, p. 156 , and Iversen [58] ch. III-§2. Soft sheaves. p. 149.

3.2 Integration Along Fibers on Fiber Bundles

71

then generate, by Godement’s simultaneous resolutions (cf . proof of 3.2.2.1-(2)), spectral sequences which, in the second page, give rise to the natural maps 

p,q

IR p Γ (B; R B ⊗ Hc (F ))) → IR p Γ (B; Hq (π! ))

p,q

IR p Γ (B; R B ⊗ H q (F ))) → IR p Γ (B; Hq (π∗ ))

(IE 2 ) (IE 2 )

q

where one recognizes IR p Γ (B; −) acting on the isomorphisms (3.27). The spectral sequences (IE r ) are therefore isomorphic for r ≥ 2 and the morphisms (3.28) are quasi-isomorphisms, which proves the corollary.  

3.2 Integration Along Fibers on Fiber Bundles In 2.3 we defined, for an oriented manifold M of dimension dM , the integration map  M

: c (M)[dM ] → R[0] ,

which is both a morphism of complexes (Proposition 2.3.2.2), and the main ingredient of another morphism of complexes: the left Poincaré adjunction map (Proposition 2.4.1.1) ID M : (M)[dM ] −→ HomR (c (M), R) ,

(3.29)

which is the basis of the Poincaré Duality theorem. These facts extend to any oriented fiber bundle of manifolds (E, B, π, M) with equidimensional fiber.3 For this, we will define an operation of integration along the fiber M  M

: cv (E)[dM ] → (B) ,

and establish it is a morphism of (B)-differential graded modules. We will then extend the left Poincaré adjunction map (3.29) to a morphism of complexes ID B,M : (E)[dM ] −→ Hom(B) (cv (E), (B)) , which, in turn, will be the basis of the Poincaré Duality theorem relative to B. But before dealing with the general case, we take a closer look at the special case of trivial Euclidean fiber bundles.

3 Most

of the time we shall assume the total space to be connected only for the purpose of guaranteeing equidimensionality of fibers (cf . Exercise 3.1.1.1).

72

3 Poincaré Duality Relative to a Base Space

3.2.1 The Case of Trivial Euclidean Bundles By this terminology we mean a fiber bundle (U × V , U, p1 , V ), with U ⊆ Rb and V ⊆ Rm open subspaces, and where p1 : U × V → U is the map (x, y) → x. What makes trivial Euclidean fiber bundles particularly handy is that, in them, we have the global systems of coordinates given by the Euclidean coordinates, in U the set x := {x1 , . . . , xb }, and in V the set y := {y1 , . . . , ym }. For I ⊆ [[1, b]], let dI := dxi1 ∧ · · · ∧ dxir , if I := {i1 < · · · < ir } is nonempty, and let d∅ = 1. Proceed likewise for J ⊆ [[1, m]]. Then, a differential form ω ∈ (U × V ) can be written in a unique way as a sum ω=

 I,J

fI J (x, y) dI ∧ dJ ,

(3.30)

with fI J ∈ 0 (U × V ). In particular, ω ∈ cv (U × V ) ↔ fI J ∈ 0cv (U × V ) ,

∀I ⊆ [[1, b]] , J ⊆ [[1, m]] .

We define  V

: cv (U × V ) → (U ) ,

(3.31)

by the following conditions: 

• V

is R-linear;

• if J = [[1, m]], then • if J = [[1, m]], then

 V V

fI J (x, y) dI ∧ dJ = 0 ; fI J (x, y) dI ∧ dJ = (−1)m

 V

 fI J (x, y) d[[1,m]] dI .

Proposition 3.2.1.1 (And Definitions) 1. Let U ⊆ Rb and V ⊆ Rm be open. The map  V

: cv (U × V )[m] → (U ) ,

(3.32)

is a morphism of left (U )-differential graded modules.  Furthermore, for any map f : U  → U , where U  ⊆ Rb is open, we have a Cartesian diagram U ×V

f

p1

U

U ×V p1

f

U

f (x, y) = (f (x), y) ,

(3.33)

3.2 Integration Along Fibers on Fiber Bundles

73

and a corresponding commutative diagram of morphisms of complexes cv (U

× V )[m]

f∗

cv (U

× V )[m] V

V f∗

).

2. The map ID U,V :

× V )[m]

cv (U

Hom

×

,

(3.34)

defined by 

 α∧β , ID U,V (α) := β →

(3.35)

V

is an injective morphism of left (U )-differential graded modules, called the left Poincaré adjunction map relative to U . Proof

(1) The map V is, by definition, a morphism of left (Rb )-graded modules, so that to prove that it is a morphism of left (U )-graded modules, it remains only to verify that V commutes with the action of g ∈ p∗ (0 (U )), which is also clear since these functions are independent of the integration variables. For the compatibility with differentials, we have, on the one hand 

d V

f (x, y) dI ∧ dJ = (−1)|I |+|J | d

 V

 f (x, y) dJ dI 

 ∂ f (x, y) dJ dxi ∧ dI i=1 ∂xi V  b  ∂f =1 (−1)|I |+|J | (x, y) dJ dxi ∧ dI i=1 V ∂xi

= (−1)|I |+|J |

b

(3.36)

where (=1 ) is justified since, for x fixed, the support of f (x −) is compact. And we have, on the other hand,    b ∂f d(f (x, y)) dI ∧ dJ = (x, y) dxi ∧ dI ∧ dJ i=1 ∂xi V V  m ∂f (3.37) + (x, y) dyj ∧ dI ∧ dJ j =1 ∂yj  b  ∂f =2 (−1)1+|I |+|J | (x, y) dJ dxi ∧ dI i=1 V ∂xi

74

3 Poincaré Duality Relative to a Base Space

where (=2 ) is due to the vanishing of the last term in the previous line, in the only case where it could not vanish, i.e. for J = [[1, m]]  {j }. Indeed, in that case, since f (x, −) has compact support in V , we can write, for all fixed x,  V



 ∂f  ∂f (x, y) dyj ∧ dI ∧ dJ = (−1)|I |+|J | n−1 (x, y) dyj dJ ∧ dI , ∂yj R R ∂yj

by Fubini’s theorem. But then  R

∂f (x, y) dyj = f (x, y) ∂yj

yj =+∞

− f (x, y)

yj =−∞

=0

since, again, f ∈ 0cv (U × V ). Putting together (3.36) and (3.37), we get d ◦



= (−1)m

V

 V

◦ d.

The statement in (1) concerning the Cartesian diagram (3.33) is now quite straightforward and results from elementary checks, left to the reader. (2) The compatibility of ID U,V with differentials results almost as for Proposition 2.4.1.1, except that now we have also to consider the differential in (U ). We have    ID U,V (−1)m d α (β) = (−1)m d(α ∧ β) + (−1)m+[α]+1 α∧dβ V

V

  =1 d ID U,V (α)(β) − (−1)[β]−1 ID U,V (α) d β = (DID U,V (α))(β) ,

where, for the first term after (=1 ), we applied (1). – Injectivity of ID U,V results from the description (3.30) of α ∈  (U ×V ): α=

 I,J

fI J (x, y) dI ∧ dJ ,

where fI J ∈ 0 (U × V ). Indeed, for every I0 ⊆ [[1, b]] and J0 ⊆ [[1, m]], denote I0 := [[1, b]]  I0 and J0 := [[1, m]]  J0 . Then, for g ∈ 0c (U × V ), we have  α ∧ g(y) dI0 ∧ dJ0 = ±fI0 J0 (x, y) g(x, y) d[[1,b]] ∧ d[[1,m]] , and

   fI0 J0 (x, y) g(x, y) d[[1,b]] . ID U,V (α) g(x, y) d[[1,b]] ∧ d[[1,m]] = ± V

(3.38)

3.2 Integration Along Fibers on Fiber Bundles

75

But then, if fI0 J0 = 0, we can choose g with compact support small enough to have both: fI0 J0 g = 0 and fI0 J0 g ≥ 0. In which case the right-hand side of (3.38) is clearly nonzero, showing that ID U,V (α) = 0, as announced.  

3.2.1.1

The Case of General Fiber Bundles

We can now define for any connected and oriented fiber bundle of manifolds (E, B, π, M) (cf . fn. (3 ), p. 71), the operation of integration along fibers  M

: cv (E)[dM ] → (B) .

For this, consider an atlas U := {Ui }i∈I for B, where the Ui ’s are trivializing, and a corresponding partition of unity {φi }i∈I . Then, following Proposition 3.1.7.1, fix an oriented family of trivializations {(π, ϕi ) : π −1 (Ui ) → Ui × M}i∈I . Let V := {Vj }j∈J be an atlas for M, let {φj }j∈J be corresponding partitions of unity, and define   φij (y) := φi π(y) φj ϕi (y) . For each i ∈ I, the family Φi := {φij }j∈J is a partition of the nonnegative function φi ◦ π : π −1 (Ui ) → R≥0 , and the family Φ := {φij }(i,j)∈I×J is a partition of unity in E, subordinate to the Euclidean cover W := {Wij := ϕi−1 (Vj )}(i,j)∈I×J .

(3.39)

A key point to notice is that if β ∈ c (E), then for each i ∈ I, the sum j φij β is a finite sum of nonzero differential forms in cv (π −1 (Ui )), and this, because the differential form (φi ◦ π ) β is of compact support in π −1 (Ui ) and that the family Φi := {φij }j∈J is a locally finite partition of φi ◦ π . As a consequence, if for each (i, j) ∈ I × J, we denote by  Vj

: cv (Wij ) → (Ui )

the integration along the fibers of the trivial Euclidean fiber bundle (Wij , Ui , π, Vj ), then the sum    β := φij β (3.40) M

ij Vj

is a well-defined differential form in (B).

76

3 Poincaré Duality Relative to a Base Space

Lemma 3.2.1.2 The sum (3.40) is independent of the choices of atlases U := {Ui }i∈I , V := {Vj }j∈J and partitions of unity {φi }i∈I and {φj }j∈J . In particular, if β ∈ cv (E) is such that |β| ⊆ cv (Wij ), then, for all α ∈ (B),  M

π ∗ (α) ∧ β = α ∧



β. Vj

Proof The proof of the independence upon the choices of atlases is the same as for the Lemma 2.3.2.1. The displayed equality results from:  M

π ∗ (α) ∧ β =

 

  π ∗ (φi α) ∧ β = π ∗ (φi α) ∧ β i M i Vj   

 φi α ∧ β =α∧ β, = i

Vj

Vj

 

after Proposition 3.2.1.1. We have now enough tools to extend Proposition 3.2.1.1 to the general case.

Proposition 3.2.1.3 (and Definitions) Let (E, B, π, M) be a connected, oriented fiber bundle of manifolds. 1. The map defined by (3.40)  M

: cv (E)[dM ] → (B) ,

(3.41)

is a morphism of left (B)-differential graded modules. In particular, for all α ∈ (B) and β ∈ cv (E), we have  M

π ∗ (α) ∧ β = α ∧



β.

(3.42)

M

Furthermore, for every Cartesian diagram E

f

E

B

f

B

π

π

the fiber bundle E  is orientable, and if f preserves orientations, then we have a commutative diagram of morphisms of complexes cv (E)[dM ] M

f∗

cv (E

)[dM ]

⊕

M

f∗

).

(3.43)

3.2 Integration Along Fibers on Fiber Bundles

77

2. The map  ID B,M : (E)[dM ] −→ Hom•(B) cv (E), (B) ,

(3.44)

defined by 

 α∧β , ID B,M (α) := β →

(3.45)

M

is an injective morphism of left (B)-differential graded modules, called the left Poincaré adjunction map relative to B. Proof All the statements result by restriction to the subspaces of an Euclidean cover W := {Wij }(i,j)∈I×J of E together with a corresponding partition of unity Φ := {φij }(i,j)∈I×J , as described in 3.2.1.1 -(3.39). For example, to show that  M

π ∗ (α) ∧ (β + dβ  ) = α ∧

 M

β + (−1)dM d



β



M

we apply formula (3.40) and Lemma 3.2.1.2 to reduce to the case of trivial Euclidean fiber bundles already settled by Proposition 3.2.1.1. We then get  M

π ∗ (α) ∧ (β + dβ  ) =

 ij

=

 ij

=α∧ =α∧

Vj

π ∗ (α) ∧ (φij β + d φij β  )

α∧



 

Vj

Vj

ij M

φij β +



d φij β 

Vj

φij β + (−1)dM d

β + (−1)dM d



β





 Vj

φij β 



M

We check the commutativity of (3.43) in the same way by choosing an Euclidean cover W  := {Wi j }(i  ,j )∈I  ×J  of E  which is a refinement of the inverse image f −1 W := {f −1 (Wij )}(i,j)∈I×J . Details are left to the reader. (2) Same as for the trivial Euclidean case 3.2.1.1-(2)   Comment 3.2.1.4 It is worth noting that the injectivity of the Poincaré adjunction relative to B in 3.2.1.3-(2) establishes the nondegeneracy of what should be called the Poincaré pairing relative to B, i.e. the (B)-bilinear map: ·, ·B,M : (E) × cv (E) → (B) ,

(α, β) →

 M

α∧β,

(3.46)

thus extending the nondegeneracy property of the (absolute) case B := {•}, established in Proposition 2.4.1.1.

78

3 Poincaré Duality Relative to a Base Space

3.2.2 Sheafification of Integration Along Fibers Given a connected, oriented fiber bundle (E, B, π, M), the compatibility of M with open base changes U ⊆ B established in 3.2.1.3-(1), gives rise to the sheafification of the integration along fibers, i.e. to a morphism of  B -differential graded modules:  M

: π!  E [dM ] →  B ,

(3.47)

Proposition 3.2.2.1 Let (E, B, π, M) be a connected, oriented fiber bundle. 1. The morphism (3.47) induces a morphism in the cohomology sheaves  M

: Hq+dM (π! ) → Hq ( B ) = R B [0]q

(3.48)

which is zero for q = 0. For q = 0 it is always surjective, and is an isomorphism if, in addition, M is connected (cf. Proposition 3.1.10.4 -(2)). Furthermore, for M = Rm the morphism (3.48) is an isomorphism for all q. 2. The morphism (3.47) can be filtered so a to get a morphism of convergent spectral sequences p,q

IE(π! )2

p+q+dM

:= H p B; Hq+dM (π! )

H p (B, p,q IE 2

Hcv

M)

H p (B

(E)

M

RB 0

q)

H p+q (B)

where the left vertical arrow is induced by the sheaf morphism (3.48). Proof

(1) The important fact here is that M is a morphism of sheaves, which, as already mentioned, is consequence of the compatibility of integration along fibers and base change. The statements in (1) are then local in nature, and we have commutative diagrams ⊗ id ⊗

c (M)[dM ]

⊆

cv (π

−1 (U ))[d

⊕

M

M]

M

R inducing in cohomology H (U ) ⊗ Hc (M)[dM ] id ⊗

⊕

M

H (U )

Γ (U ; H(π! )[dM ])

R

M

Γ (U R B 0 )

(3.49)

3.2 Integration Along Fibers on Fiber Bundles

79

(3.50)

Fig. 3.1 Godement’s simultaneous flasque resolution of π!  E

where U ⊆ B is any π -trivializing contractible open subspace and where the horizontal ‘⊆’ arrow in the left-hand diagram is the quasi-isomorphism 3.1.10.5. In degree 0, we have Γ (U ; H(π! ))[dM ]0 = HcdM (M) and Γ (U ; R B [0])0 = R, and the right-hand side vertical arrow is the integration M : HcdM (M) → R, which is always surjective and is bijective if M is connected. When M = Rm , we know already that M Hc (M)[dM ] = R[0] (Poincaré Duality), which implies that the morphisms in (3.49) are all isomorphisms. (2) We give a fairly detailed proof of this statement as it will serve as a model for future constructions of spectral sequences. To study the action in cohomology of the morphism of complexes of sheaves  M

: Γ (B; π!  E [dM ]) → Γ (B;  B ) ,

we recall that since the sheaves are  0 (B)-modules, they are Γ (B; −)-acyclic (B.6.3.4-(3)) (cf . fn. (2 ), p. 70) and that we can work in the derived category D+ (B; R) of bounded below complexes of sheaves on B, hence replacing the functor Γ (B; −) by its right derived functor IR Γ (B; −) (cf . Sect. A.2.3) and the complexes of sheaves π!  E and (B) by quasi-isomorphic complexes. For example, the simultaneous Godement resolutions, the bicomplex (3.50), where each column is a flasque resolution of the sheaf at its top, and where, if we apply the cohomology functor H(−) to the rows, then the i’th column 0 → π!  iE → I0i → I1i → · · · becomes the complex of sheaves 0 → Hi (π! ) → Hi (I0∗ ) → Hi (I1∗ ) → · · · which is again a flasque resolution. It is this particularity that explains the terminology simultaneous resolution for the bicomplex (3.50). Likewise, denote by  ∗B → J ∗ be the Godement’s resolution of  ∗B . Godement’s flasque resolution is functorial and exact from the category C∗ (B) of complexes of sheaves on B to the category Cfl,∗ (B) of bicomplexes of flasque

80

3 Poincaré Duality Relative to a Base Space

sheaves on B.4 Applied to the morphism of complexes M : π!  E [dM ] →  B , gives the commutative diagram of morphisms of complexes E [dM ]

B

M

(3.51) Tot

∗

M

∗

Tot

where Tot⊕ K ∗ is the total complex associated with a bicomplex K ∗ (cf . Sect. 5.4.2), and where the arrow in the second row denotes the induced morphism by the Godement’s resolution functor. Furthermore, the vertical arrows are quasi-isomorphisms, which is a standard property of the Tot⊕ functor (cf . Proposition 5.4.3.1). As a consequence, if we apply the functor IR Γ (B; −) to (3.51), we get a commutative diagram E [dM ])

IR Γ (B;

IR Γ (B; Tot⊕

∗

)

IR Γ (B;

M

M

IR Γ (B; Tot⊕

B)

∗

)

where the vertical arrows are still quasi-isomorphisms. The study of the induced morphism in cohomology by the top row can therefore be accomplished in the bottom row. There, the morphism M comes from a morphism of bicomplexes (3.50) and, as such, it respects the row filtration (clearly regular), thus inducing a morphism p of convergent spectral sequences whose IE 1 terms are respectively p,q

IE(π! )1

:= IR Γ (B; Hq (Ip∗ ))

and

p,q

IE 1

:= IR Γ (B; Hq (Jp∗ )) .

But, for q fixed, the -complex Hq (I ∗ ) is a flasque resolution of Hq (π! ), and the q complex Hq (J ∗ ) is a flasque resolution of Hq ( B ), i.e. of R B [0] . Furthermore, by exactness of Godement’s resolution, the morphism M : IE(π! )1 → IE 1 is the one induced by the morphism M : Hq+dM (π! ) → Hq ( B ) of (1). Therefore, in the second page of spectral sequences, we get the morphism p,q

IE(π! )2

H p (B; Hq (π! )) which ends the proof of (2). 4 The

p,q

IE 2

M

H p (B;

M)

H p (B; R B [0]q )  

existence of simultaneous resolutions is easy to establish thanks to the exactness of Godement’s flasque resolution. See Godement [46] §4.3 Resolution canonique d’un faisceau, p. 167, or Bredon [20] §II.2 The canonical resolution and sheaf cohomology, p. 36.

3.2 Integration Along Fibers on Fiber Bundles

81

3.2.3 Thom Class of an Oriented Vector Bundle Let (M, [M]) be a connected, oriented manifold of dimension dM . In 2.4.2 we recalled the definition of the fundamental class of M as the unique cohomological class ζM ∈ HcdM (M) verifying the equality M ζM = 1. The analogue for an oriented fiber bundle of manifolds (E, B, π, M) would be a cohomological class dM ζπ ∈ Hcv (E) verifying M ζM = 1. But, although it is easy to construct differential forms ζπ ∈ dcvM (E) verifying M ζπ = 1 (exercise ! ( , p. 340)), it may not always be possible to have a closed form (see counterexample 3.7.2.1). dM A particular case where such a cohomology class ζπ ∈ Hcv (E) does exist is when the fiber M is a vector space, for example if (E, B, π, M) is an oriented vector bundle.5 Proposition 3.2.3.1 (Thom Isomorphism) Let (E, B, π, M) be an oriented fiber bundle such that Hc (M)[dM ] = R[0]. Then, the morphism  M

: Hcv (E)[dM ] → H (B) ,

(3.52)

is an isomorphism of H (B)-module. There exists therefore a unique cohomology dM (E) such that, class π ∈ Hcv  M

π ∗ ([α]) ∪ π = [α] ,

∀[α] ∈ H (B) .

(3.53)

The class π is called the Thom class of the fiber bundle (E, B, π, M). The inverse map of 3.52 is the isomorphism π ∗ (−) ∪ π : H (B) → Hcv (E)[dM ] ,

(3.54)

which is called the Thom isomorphism. Proof Proposition 3.2.2.1-(2) says us that (3.52) is the abutment of a morphism of spectral sequences which, in the second page, is the morphism H p (B;



m)

p,q

p,q

: IE 2 (π! ) := H p (B; Hq (π! )) → IE 2

:= H p (B; R B [0]q )

induced by the morphism M : H(π! )[dM ] → R B [0], which is an isomorphism since M has the cohomology of a vector space (3.2.2.1-(1)). We can then state that (3.52) is an isomorphism. 5 Recall that a vector bundle is a fiber bundle (E, B, π, V ), where the fiber V is a vector space, and which is defined by a trivializing cover {Φi : π −1 (Ui ) → Ui ×V }i∈I such that the transition maps are linear isomorphisms on the fibers. This enables us to give an intrinsic definition of a structure of vector spaces of the fibers of E, and, in particular, to define the zero section σ : B → E, x → 0V , which is an obvious homotopy equivalence.

82

3 Poincaré Duality Relative to a Base Space

(Note that IE 2 (π! ) and IE 2 are concentrated in q = 0, so that the spectral sequences (IE r (π! ), dr ) and (IE r , dr ) already degenerate for r ≥ 2.)   3.2.3.1

On Tubular Neighborhoods

Giving a closed embedding of manifolds ι : N → M, a tubular neighborhood of N consists of a vector bundle (E, N, π, RdM −dN ) and an open map τ : E → M which is a diffeomorphism onto its image Uτ := im(τ ) and is such that the restriction of τ to the zero section σ (N) is the embedding ι : N → M. τ

E σ

π

N

Uτ

jτ ⊆

M

⊆ ι

N

In that case, we have an induced projection map π : Uτ → N and we can define the compact vertical cohomology Hcv (Uτ ) of Uτ . The map τ is an open tube in M around N , and the manifold M is said to admit tubular neighborhoods if for every closed embedding N → M and every neighborhood U ⊇ N there exists a an open tube τ in U around N . Note that, under our assumptions in 2.2.1, manifolds admit tubular neighborhoods.6 When the conditions of orientability of the fiber bundle (Uτ , N, π, RdM −dN ) are also satisfied, for example M and N are orientable (3.1.7.2), we can apply Proposition 3.2.3.1 and state that the morphisms of graded spaces ι∗ : H (Uτ ) → H (N) and

(−) ∪ τ : H (N ) → Hcv (Uτ )[dM −dN ] ,

where τ is a second notation for the Thom class π , are isomorphisms. Now, since the differential forms in cv (Uτ ) have compact vertical support, they can be extended by zero to the whole M. Denote by jτ : Uτ ⊆ M the inclusion map, and by jτ,! : cv (Uτ ) → (M) the extension by zero map, which is clearly a morphism of complexes. It therefore induces a morphism in cohomology jτ,! : Hcv (Uτ ) → H (M), allowing us to extend Thom isomorphism (3.54) to the graded morphism (−) ∪ τ : H (N) → H (M)[dM −dN ] .

(3.55)

This morphism depends a priori on the choice of tube τ , since the inclusion Uτ12 ⊆ Uτ1 does not necessarily respect fibers. But, we will show as a consequence of

6 For

N compact, see Spivak [83] ch. 9, Tubular Neighborhoods, Theorem. 20, p. 346. For N general, see Lang [72] ch. IV, §5. Existence Of Tubular Neighborhoods, Theorem 5.1, p. 110.

3.3 Poincaré Duality for Fiber Bundles

83

Poincaré duality relative to the base of the tube, the manifold N , that the morphism (3.55) it is indeed intrinsic (3.6.6.1).

3.3 Poincaré Duality for Fiber Bundles 3.3.1 Sheafification of the Poincaré Adjunction In Proposition 3.2.1.3-(2), we introduced, for a given connected, oriented fiber bundle of manifolds (E, B, π, M), the left Poincaré adjunction relative to B, which is the injective morphism of B -differential graded modules  ID B,M : (E)[dM ] −→ Hom•(B) cv (E), (B) ,

(3.56)

 defined by ID B,M (α) := β → M α ∧ β . We will proceed to sheafify this construction as we did for M in 3.2.2, but this time relatively to the fibration → B. π :E→ The compatibility of M with base changes (3.2.1.3-(1)), implies that, given open subspaces W ⊆ V ⊆ U ⊆ forms α ∈ Γ (U ; π∗  E )  B, and differential and β ∈ Γ (V ; π!  E ), we have M α V ∧ β W = M α W ∧ β W . In other terms, the following diagram where the vertical arrows are the restriction maps, is commutative. Γ (V ; π!  E ) (−)

U V

ID V ,M (α

/ Γ (V ;  B )

W)

 / Γ (W ;  B )

⊕ 

Γ (W ; π!  E )

V)

ID W,M (α

(−)

U V

(3.57)

In this way, α ∈ Γ (U, π∗ E ) determines a morphism of sheaves  ID U,M (α) ∈ Hom[α] π!  E

U , B U



 = Γ U ; Hom [α] (π!  E ,  B )

almost tautologically compatible with the action of  B and the restriction maps Γ (U, π∗ E ) → Γ (U  , π∗ E ), for U  ⊆ U . Hence, the morphism of sheaves ID B,M : π∗  E −→ Hom • B (π!  E ,  B )

(3.58)

which is the announced sheafification of the left Poincaré adjunction (3.56). Notice that the morphism ID B,M is injective since the horizontal arrows in the diagram (3.57) are nonzero are as long as the corresponding restriction of α do not vanish, and this since the Poincaré adjunction (3.56) is injective.

84

3 Poincaré Duality Relative to a Base Space

3.3.2 Deriving the Sheafified Poincaré Adjunction Functors Until now, the fact that  B (resp. (B)) is a differential graded algebra (dga) has played only a marginal role, but since the introduction of the Poincaré adjunction relative to B (3.58), it appears in the functor Hom• B (−, −), which obliges us to work in the category of ( B , d)-differential graded modules (dgm). An ( B , d)-dgm is an  B -graded module M := (M, d) equipped with a differential d of degree 1 and verifying the well-known compatibility relation between differentials d(α m) = (dα)m + (−1)[α] α dm , for all α ∈  B homogeneous and all m ∈ M. A morphism of ( B , d)-dgm’s α : (M, d) → (N, d) is a morphism of graded  B -modules compatible with differentials. Denote by DGM( B , d) (resp. DGM((B), d)) the category of ( B , d)-dgm’s. We have the following commutative diagram of categories and functors

C GM( B )

Tot

Γ (B;−)

/

DGM( B , d)

(I)  C GM(R B )

Tot

/

DGM((B), d)

/

(II)  DGM(R B , 0)

Γ (B;−)

o

Tot

C GM((B))

(3.59)

/

(III)  DGM(R, 0)

o

Tot

 C GM(R)

where Tot is the total complex functor, and Γ (B; −) the global section functor. The vertical arrows are the restriction functors induced by the augmentation morphism of dga’s  : R B →  B and  : R → (B). Notice that since R B is concentrated in degree 0, the category DGM(R B , 0) (resp. DGM(R, 0)) is simply the category of complexes of R B -modules C Mod(R X ) (resp. C Mod(R)), which we also denoted Sh(B; R) (resp. C Vec(R)). The bifunctor Hom• (−, −) is well-defined in each of the categories appearing in diagram (3.59). The definitions are compatible through corresponding functors, and, more importantly, they extend to derived categories using techniques adapted to categories DGM(A, d), where (A, d) is a general differential graded algebra, for example as they are developed in Stacks Project [90] §24.26, and that we briefly recall in A.3. The commutativity of subdiagrams (I) and (III) is discussed in sections 5.3–5.4.6, and we establish that of (II) in Appendix B.9. We denote K DGM( B , d) and D DGM( B , d) the corresponding homotopy category and derived category of DGM( B , d) (and likewise for DGM((B), d)). In the category of sheaves, we also have the sheafified bifunctor Hom • B (−, −), internal bifunctor of DGM( B , d) with derived bifunctor IR Hom •( B ,d) (−, −) : D DGM( B , d) × D DGM( B , d)  D DGM( B , d) where the subscript in the notation ‘IR Hom •( B ,d) ’ recalls that we are deriving in D DGM( B , d) and not in D GM( B ).

3.3 Poincaré Duality for Fiber Bundles

85

The following facts are established in Appendix B (Corollary B.9.1.3). Proposition 3.3.2.1 1. For all M, N ∈ DGM( B ), the natural map IR Hom •( B ,d) (M, N) → IR Hom •R B (M, N) . is an isomorphism in the derived category D(B; R). In particular, IR Hom •( B ,d) (π!  E ,  B )  IR Hom •R B (π!  E ,  B )  IR Hom •R B (π!  E , R B ) . 2. The derived functor of the restriction functor DGM( B , d)  DGM(R B , 0) induced by the augmentation morphism of dga’s  : R B →  B , is an equivalence of derived categories D DGM( B , d)  D DGM(R B , 0) =: D(B; R) , with inverse, the functor induced by the functor  B ⊗R B (−) : Sh(B; R)  DGM( B ) .

3.3.3 The Poincaré Duality Theorem for Fiber Bundles The following theorem extends the Poincaré Duality Theorem for oriented manifolds 2.4.1.3, to oriented fiber bundles. Theorem 3.3.3.1 (Poincaré Duality for Fiber Bundles) Let (E, B, π, M) be an oriented fiber bundle with fiber M of dimension dM . 1. The following morphisms induced by the left Poincaré adjunction (3.58), π∗

ID B,M

E [dM ]

q.i.

IR Hom •

B ,d)

π!

E

B

q.i. Grothendieck-Verdier Duality

q.i.

IR Hom •R B π!

(3.60) E , RB

are isomorphisms in D(B; R). 2. Applying IR Γ (B; −) to (3.60), we get quasi-isomorphisms of complexes [dM ]

IR Γ (B;ID B,M ) q.i. q.i.

IR Hom•

cv q.i.

IR Hom• π!

(3.61) E , RB

86

3 Poincaré Duality Relative to a Base Space

3. The sheaf-cohomology functor H(−) applied to (3.60), gives an isomorphism of local systems  H(ID B,M ) : H(π∗ )[dM ] −→ Hom •R B H(π! ), R B . 

4. If M is of finite type, the previous statements remain true if we swap terms π!  E ↔ π∗  E and cv (E) ↔ (E). The right Poincaré adjunction:

π!  E [dM ]

ID B,M q.i.

 / IR Hom • ( B ,d) π∗  E ,  B

(3.62)

is therefore a quasi-isomorphism in D(B; R) too. Proof (1) The vertical arrow is 3.3.2.1-(1). For the morphism ID B,M , since it concerns sheaves, we need only show that it is locally an isomorphism. We can then confine ourselves to only consider trivializing open subspaces U ⊆ B, and take advantage of the fact that in that case, the inclusions  U ⊗ c (M) ⊆ π!  U ×M and  B ⊗ (M) ⊆ π∗  B×M , are quasi-isomorphisms after Corollary 3.1.10.5. The restrictions of the sheafified left Poincaré adjunction to π −1 (U ) then reads, in D(U ; R), as the isomorphism id ⊗ ID M :  B ⊗ (M)[dM ] →  B ⊗ c (M)∨ , where DM : H (M)[dM ] → Hc (M)∨ is the Poincaré duality isomorphism 2.4.1.3. Hence, the commutative diagram U

⊗

[dM ]

ID U,M

id⊗DM

IR Hom •

U ,d)

U

⊗

U

⊗

c (M)

c

U

∨

which ends the proof of (1). (2) Follows from (1), since by Proposition B.9.1.1-(1), we have  IR Hom•((B),d) (−, −) = IR Γ B, IR Hom •(Ω (B),d) (−, −) .  (3) Applying H(−) to π∗  E → IR Hom • π!  E , R B in (3.60), we get  p,q H(π∗ )  H Hom • (π!  E , I ) ⇐1 IE 2 := hp Hom • (Hq (π! ), I ) 2 Hom • (H(π! ), R B ) , where I is an injective resolution R B , (⇐1 ) is the spectral sequence given by the (regular) -filtration, and (2 ) since Hq (π! ) is locally free (3.1.10.4-(2)),

3.3 Poincaré Duality for Fiber Bundles

87

which implies that Hom • (Hq (π! ), −) is an exact functor, in which case the p,q spectral sequence IE r is concentrated in p = 0, hence degenerated, for all r ≥ 2. (4) Should be obvious. The proof is exactly the same as for the previous statements, so details are left to the reader.   Comments 3.3.3.2 1. Although clear enough, it is still worth emphasizing that, for B = {•}, we recover Poincaré Duality for a manifold. The reader may also notice, that changing the point of view from a point {•} to a general base space B, entails changing the category of oriented manifolds Manor by the category of oriented fiber bundles • Fibor B (3.1.8), changing the category of complexes C Vec(R) = DGM(({ }), d) by the category of differential graded modules DGM((B), d), and, therefore, changing the target category GM(R) of the cohomology functor by GM(H (B)) (cf . Sect. 3.1.3). 2. The attentive reader has noticed the reference to the Grothendieck -Verdier duality in diagram (3.60), which is fully justified since we give its original formulation there (cf . Chap. 8). In fact, we can say that, in a sense, this chapter is a detailed example of the Grothendieck-Verdier Duality formalism in the simplified framework of fiber bundles of manifolds.

3.3.4 Poincaré Duality for Fiber Bundles and Base Change The Poincaré Duality Theorem for fiber bundles is well behaved relative to the base change (3.1.5). More precisely, given a map h : B  → B and a fiber bundle E := (E, B, π, M), let h−1 (E) := (E  , B  , π  , M) (cf . Corollary 3.1.7.2-(1)). By base change h−1 (−) : FibB  FibB  (3.1.5), we have the Cartesian diagram E

h

E

h

B

π

π

B

and we know, from 3.1.10.2-(2), that h−1 (E) is oriented if E is so, in which case integration along fibers is compatible with base change in the sense that the following diagram is commutative cv (E)[dM ] M

h∗

cv (E

)[dM ]

⊕

M

h∗

).

(3.63)

88

3 Poincaré Duality Relative to a Base Space

It is therefore natural to expect some kind of compatibility between Poincaré Duality relative to B and to B  . The following theorem addresses this question. Theorem 3.3.4.1 Let E := (E, B, π, M) and E  := (E  , B  , π  , M) be two oriented fiber bundles of manifolds with same fiber M of dimension dM , and let E

h

E

B

h

B

π

π

(3.64)

be a Cartesian diagram of fiber bundles. Then, there exists a canonical commutative diagram of isomorphisms in D(B  ; R) h−1 (π∗

E )[dM ]

h−1 (ID B,M ) q.i.

⊕

q.i.

π∗

E

h−1 IR Hom •

[dM ]

ID B

,M

q.i.

B ,d)

(π!

B)

E

q.i .

IR Hom •

B

,d) (π!

(3.65) E

B

)

and likewise, if M is of finite type, swapping π!  E ↔ π∗  E . Proof Since we are working in derived categories of sheaves, we can replace the complexes  by the constant sheaf R, and even by the constant sheaf k for any field k since the theorem is still verified in this case. The diagram (3.65) is then a straightforward consequence of three technical facts which we first justify.7 Lemma A There exists natural canonical quasi-isomorphisms: ⎧ ⎨ (i) h−1 (π∗ k E ) −→ π∗ (h−1 k E ) , q.i.

⎩ (ii) h−1 (π! k E ) −→ π! (h−1 k E ) .

(3.66)

q.i.

Proof of Lemma A To prove (i), we begin observing the well-known relations   Hom h−1 (−), h−1 (−) =1 Hom (−), f∗ ◦ h−1 (−)  →2 Hom π∗ (−), π∗ ◦ f∗ ◦ h−1 (−)  =3 Hom π∗ (−), h∗ ◦ π∗ ◦ h−1 (−)  =4 Hom h−1 ◦ π∗ (−), π∗ ◦ h−1 (−)

7 See

also the proof of Proposition 8.2.2.1-(2), p. 252.

(3.67)

3.3 Poincaré Duality for Fiber Bundles

89

where (=1,4 ) are the usual adjunctions between the inverse and direct image functors in categories of sheaves, (→2 ) is the map defined by direct image functor π∗ , and (=3 ) results from the obvious equality π∗ ◦ h∗ = h∗ ◦ π∗ . The morphism id(−) ∈ Hom(h−1 (−) → h−1 (−)) therefore defines naturally a morphism for every complex of sheaves F • ∈ D+ (E; k): h−1 ◦ IR π∗ (F • ) → IR π∗ ◦ h−1 (F • ) .

(3.68)

Taking F • := k E , the morphism (3.68) induces the morphism of cohomology sheaves h−1 (Hi (π∗ (k E ))) → Hi (IR π∗ (h−1 (k E ))) = Hi (π∗ (k E  )) , which is clearly an isomorphism since the sheaves Hi (π∗ k E ) and Hi (π∗ (k E  )) are both locally trivial with the same fiber H (M), because the diagram of fiber bundles (3.64) is Cartesian (cf . Proposition 3.1.10.4-(2)). To prove (ii) we proceed as in (i), replacing π∗ with π! . The equality (=3 ) in (3.67) is then a consequence of the equality π! ◦ h∗ = h∗ ◦ π! , which is easy to prove since (3.64) is the diagram Cartesian (cf . Proposition 3.1.9.2-(1,2b)).  Lemma B Let F ∗ and G be bounded below complexes of sheaves in Sh(B; k), and assume that the cohomology sheaves Hq (F ∗ ) are locally trivial.8 Then the natural morphism  (iii) h−1 IR Hom • (F ∗ , G ) → IR Hom • (h−1 (F ∗ ), h−1 (G ))

(3.69)

is a quasi-isomorphism. Proof of Lemma B We have already seen in the proof of 3.3.3.1-(3) that under the current assumptions in F ∗ and G , we have a canonical quasi-isomorphism H(IR Hom • (F ∗ , G ))  Hom • (H(F ∗ ), G ) . As a consequence, and given that Hq (h−1 (F ∗ )) = h−1 (Hq (F ∗ )) is also locally trivial, we can show that (3.69) is a quasi-isomorphism simply by showing that the natural morphism   h−1 Hom • (H(F ∗ ), G ) → Hom • h−1 (H(F ∗ )), h−1 G ) . is an isomorphism, which is clear by looking at stalk level.



Taking F ∗ := π ! kE and G := k B in lemmas A and B, and recalling the compatibility between integration along fibers and base change (3.63), establishes

8 See

Borel [16], Proposition 10.21, p. 171.

90

3 Poincaré Duality Relative to a Base Space

the existence of the commutative diagram of isomorphisms in D+ (B; k) h−1 (ID B,M )

h−1 (π∗ k E )[dM ] q.i.

q.i.

 / h−1 IR Hom • (π! k E , k B )

⊕



q.i.

 / IR Hom • (π! k E  , k B  )

ID B  ,M

π∗ k E  [dM ]

q.i.

 

hence the diagram (3.65).

Comment 3.3.4.2 Statement (ii) in Lemma A, is a well-known result valid for any F ∗ ∈ D+ (E) rather than the constant sheaf k E and for any Cartesian diagram (3.64), not necessarily of fiber bundles.9 By contrast, care must be taken with (i) since it does not generally hold. Here, the hypothesis that E is a fiber bundle is critical.

3.4 Poincaré Duality Relative to a Formal Base Space Although absolute and relative viewpoints emerge from the same formalism, an important difference arises when comparing the statements of the absolute (2.4.1.3) and the relative (3.3.3.1) Poincaré duality theorems, which is that the second states only quasi-isomorphisms of complexes, saying nothing about the induced morphisms in cohomology. More precisely, in the relative case it is stated (for example) that the left Poincaré adjunction induces an isomorphism in cohomology H (E)[dM ]



/ h IR Hom• ((B),d) (cv (E), (B)) ,

and we know, besides, that there exists a natural map (see 5.4.7.2-(1)) ξ

h IR Hom•((B),d) (cv (E), (B)) −−−→ Hom•H (B) (Hcv (E), H (B)) ,

9 See

Kashiwara-Schapira [61], Prop. 2.5.11, p. 106, or Borel [16] Prop. 10.7, p. 159.

3.4 Poincaré Duality Relative to a Formal Base Space

91

leading us to wonder if the composition DB,M := ξ ◦ H (B; ID B,M  ): H (E)

H (B;ID B,M )

h IR Hom•

cv

(3.70)

( ) ξ

( )

DB,M

•

HomH (B) (Hcv (E), H (B)) ,

is also an isomorphism. But it turns out that ξ is generally not an isomorphism, which means we cannot always expect the natural morphism of H (B)-modules DB,M : H (E) −−→ Hom•H (B) (Hcv (E), H (B)) ()

(3.71)

to be an isomorphism, contrary to the absolute case B = {•}. The following section recalls an important property of a base space B, particularly relevant in equivariant cohomology where B will be a classifying space IBG, which allows us a better understanding of (3.71).

3.4.1 Formality of Topological Spaces Following Deligne-Griffiths-Morgan-Sullivan,10 we recall the concept of formality of differential graded algebras and of topological spaces, in a way suited to our needs. Definition 3.4.1.1 1. A dg-algebra (A, d) is said to be formal if there exists a zig-zag diagram (A, d)

(A2 , d) (A1 , d)

(A4 , d) · · · (An , d)

(A3 , d )

h (A , d )

of quasi-isomorphic morphisms of dg-algebras, beginning at (A, d) and ending at its cohomology viewed as a dg-algebra with zero differential. 2. A manifold M is said to be formal if its de Rham complex ((M), d) is a formal dg-algebra. More generally, a topological space X is said to be k-formal if the complex Ω(X; k) of its Alexander-Spanier cochains with coefficients in k is a formal dg-algebra.

10 DGMS

[36], Section 4. Formality of Differential Algebras, p. 260.

92

3 Poincaré Duality Relative to a Base Space

Examples 3.4.1.2 Compact Kähler manifolds (e.g. smooth projective varieties), Lie groups, classifying spaces IBG of Lie groups G are examples of R-formal spaces (cf . Sect. 4.1.1.3 and especially Theorem 4.9.2.1-(2)).11 The following Proposition is proved in Appendix A as Corollary A.3.3.2. Proposition 3.4.1.3 Let (A, d) be a formal dg-algebra such that the cohomology algebra H (A) := h(A, d) is of finite homological dimension. Let (M, d) be an (A, d)-dg-module and denote by H (M) the H (A)-graded module h(M, d). 1. There exists a convergent spectral sequence ⎧ p,q p • q ⎪ ⎨ IE 2 := h IR HomH (A) (H (M), H (A)) ⇓ ⎪ ⎩ p+q • h IR Hom(A,d) ((M, d), (A, d)) 2. If dimproj (H (M)) ≤ 1, then there exists an isomorphism in D DGM(H (A)): IR Hom•(A,d) ((M, d), (A, d))  IR Hom•H (A) (H (M), H (A)) . and, the spectral sequence (IE r , dr ) in (1) degenerates for r ≥ 2. Returning to our preliminary discussion in 3.4, we can now state the following enhancement of the Poincaré duality of fiber bundles when the base manifold B is a connected and simply connected formal space. Theorem 3.4.1.4 (Poincaré Duality for Fiber Bundles with Formal Base Space) Let (E, B, π, M) be an oriented fiber bundle of manifolds where M is of dimension dM and where B is a formal manifold such that H (B) is of finite homological dimension. Then, the following holds: 1. The Poincaré adjunction (3.3.3.1-(1)) induces a convergent spectral sequence: p,q

IE 2

q

:= hp IR Hom•H (B) (Hcv (E), H (B)) ⇒ H p+q+dM (E) .

2. If dimproj (Hcv (E)) ≤ 1 as H (B)-gm, then the Poincaré adjunction induces an isomorphism in D DGM(H (B)): DB,M : (E)[dM ]  IR Hom•H (B) (Hcv (E), H (B)) ,

(3.72)

and the spectral sequence (IE r , dr ) in (1) degenerates, i.e. dr = 0 for r ≥ 2.

11 From

Sullivan [85] §12. Formal computation and Kaehler manifolds, p. 317.

3.4 Poincaré Duality Relative to a Formal Base Space

93

Furthermore, it Hcv (E) is a projective H (B)-gm, then isomorphism (3.72) is canonical and induces an isomorphism of H (B)-gm’s  H (E)[dM ]  Hom•H (B) Hcv (E), H (B) .

(3.73)

3. If M is of finite type, the statements (1) and (2) remain true if we swap the terms ΩG (M) ↔ G,c (M) and H (E) ↔ Hcv (E). Proof Apply Proposition 3.4.1.3 to Poincaré Duality of Fiber Bundles 3.3.3.1.

 

3.4.2 Poincaré Duality Relative to Classifying Spaces We explain how Equivariant Poincaré duality for G-manifolds, where G is a compact Lie group, can be seen as a particular case of Poincaré Duality relative to the classifying space IBG. When writing Sect. 3.3 on Poincaré Duality for fiber bundles, we were aware that the same results can be stated in categories larger than that of fiber bundles of manifolds. The reason being that we essentially only need separateness, local contractibility, and paracompactness of topological spaces. This is why we included Appendix B which extends the considerations of Sects. 3.1 to 3.3 to fiber bundles of mild spaces. The only significant change to be done is to substitute the complexes (−) of sheaves of differential forms by the complexes Ω (−; k) of sheaves of Alexander-Spanier cochains with coefficients in the field k. It is therefore possible to state Poincaré Duality theorem for fiber bundles, as stated in 3.3.3.1 and 3.4.1.4, in the category Fibor IBG of fiber bundles (E, IBG, π, M), where M is an oriented manifold of dimension dM (or more generally a mild Poincaré Duality space), and where the base space IBG is a classifying space for a compact connected Lie group G. We will later see that IBG (thoroughly discussed in 4.6), is not a manifold, but, rather, a CW-complex inductive limit of compact manifolds. To fully understand the relevance of Poincaré duality for fiber bundles in equivariant cohomology, we need recall the Borel construction (cf . Sect. 4.7). This is a covariant functor (−)G : G-Man  FibIBG from the category G-Man of Gmanifolds and G-equivariant maps, which associates with a G-manifold M, the fiber bundle (MG , IBG, π, M), where MG denotes the homotopy quotient MG := IEG ×G M and where π : MG → IBG is the locally trivial fibration [x, m] → [x] of fiber space M (cf . Definition 4.7.1.1). The equivariant cohomology of M with coefficients in k, denoted by HG (M; k), is then defined as the composition of the two functors HG (−;k)

G-Man M

(−)G

FibIBG

H (−;k)

(MG ,IBG,π,M)

GM(H (IBG; k)) H (MG ; k)

94

3 Poincaré Duality Relative to a Base Space

When the G-manifold M is oriented, the fiber bundle MG := (MG , IBG, π, M) is oriented (cf . Proposition 4.7.3.1) and Poincaré Duality relative to IBG applies to MG (3.4.1.4). But in addition, and this is fairly important, the classifying space IBG is an R-formal space (4.9.2.1-(2)) and since H (IBG; R)  S(g ∨ )G is a polynomial algebra, it has finite homological dimension, and fill the conditions for applying Theorem 3.4.1.4. The following immediate corollary of theorems 3.3.3.1 and 3.4.1.4 can then be stated. Corollary 3.4.2.1 (Equivariant Poincaré Duality) Let G be a compact Lie group, and let M be an orientable G-manifold of dimension dM . Denote by • (MG , IBG, π, M), the associated Borel fiber bundle (4.7). • G (M) := Ω(MG ; R) and G,c (M) := Ωcv (MG , R), the complexes of Alexander-Spanier cochains on MG of respectively closed and proper supports and with coefficients in R; and then G := G ({•}) = Ω(IBG; R). • HG (M) := h(G (M)), HG,c (M) := h(G,c (M)) and HG := HG ({•}). 1. The left Poincaré adjunction isomorphism 3.3.3.1-(2) in D DGM(ΩG ): DB,M : G (M)[dM ]  IR Hom•(G ,d) (G,c (M), G ) ,

(3.74)

induces a convergent spectral sequence: p,q

IE 2

q

p+q+dM

:= hp IR Hom•HG (HG,c (M), HG ) ⇒ HG

(M) .

2. If dimproj (HG,c (M)) ≤ 1 as HG -graded module, then the Poincaré adjunction induces an isomorphism in D DGM(HG ): DB,M : G (M)[dM ]  IR Hom•HG (HG,c (M), HG ) ,

(3.75)

and the spectral sequence (IE r , dr ) in (1) degenerates, i.e. dr = 0 for r ≥ 2. Furthermore, if HG,c (M) is a projective HG -graded module, then (3.75) induces an isomorphism of HG -graded modules  HG (M)[dM ]  Hom•HG HG,c (M), HG . 3. If M is of finite type, the statements (1) and (2) remain true if we swap the terms G (M) ↔ G,c (M) and HG (M) ↔ HG,c (M). Sketch of Proof The only difference with respect to theorems 3.3.3.1 and 3.4.1.4 is that here we are working in the category of fiber bundles above the classifying space IBG, which is not a manifold but a CW-complex, and which obliges us to replace de Rham differential forms by, for example, Alexander-Spanier cochains. Theorem 3.3.3.1 is then valid over any field, and Theorem 3.4.1.4 over a field k, such that IBG is k-formal, i.e. of characteristic not dividing the cardinality of the Weyl group W (G) (cf . Sect. 8.4). (The reader will find detailed proof of this corollary by a

3.5 Gysin Morphisms for Fiber Bundles

95

somewhat different route in Chap. 8 devoted to Equivariant Poincaré Duality over arbitrary fields (cf . Theorem 8.4.1.3).)  Comment 3.4.2.2 When G is the circle group S1 , the ring H (IBG; R) is isomorphic to the polynomial algebra R[X] which is of homological dimension 1. In that case, the hypothesis dimproj (HG,c (M)) ≤ 1 in 3.4.2.1-(2) is automatically verified so can be omitted.

3.5 Gysin Morphisms for Fiber Bundles 3.5.1 Gysin Morphisms Relative to a Base Space Although we will not develop this section as thoroughly as in the absolute case, where B := {•}, we want to make clear that the definition of Gysin morphisms, which we base on the quasi-isomorphisms given by Poincaré adjunctions, can be repeated word for word in the relative case. Given a proper morphism f : (E  , B, π  , M  ) → (E, B, π, M) in Fibor B E

f

E π

π

B we know after 3.1.10.2-(2) that the pullback morphism f ∗ : (E) → (E  ) induces a morphism f ∗ : cv (E) → cv (E  ) in DGM((B), d). We can then apply the relative Poincaré Duality theorem 3.3.3.1-(2) and consider the diagram )[dM ]

Γ (B;ID B,M )

[dM ]

Γ (B;ID B,M )

q.i.

IR Hom•

cv (E f∗

f∗ q.i.

IR Hom•

(3.76)

cv

inducing a canonical morphism in D DGM((B), d) f∗ : (E  )[dM  ] → (E)[dM ]

(3.77)

which we call the Gysin morphism relative to B associated with f . In the same way, for every morphism f : (E  , B, π  , M  ) → (E, B, π, M) in Fibor B (proper or not), but where M is of finite type, the same formalism, exchanging cv (E) ↔ (E) and applying the relative Poincaré Duality theorem 3.3.3.1-(4),

96

3 Poincaré Duality Relative to a Base Space

induces a morphism of ((B), d)-dgm’s f! : cv (E  )[dM  ] → cv (E)[dM ]

(3.78)

which is canonical in the derived category D DGM((B), d) and which we call the Gysin morphism relative to B for proper supports associated with f . Proposition 3.5.1.1 Denote by • Fibor B,pr the category of oriented fiber bundles over B and proper maps; • Fibor f.t.B the category of oriented fiber bundles over B with finite type fiber spaces and arbitrary maps; • (E, B, π, M) a fiber bundle of manifolds. 1. The correspondence:  (−)∗ : Fibor B,pr  D(DGM((B), d))

with

E  E∗ := ((E), d)[dM ] f  f∗ ,

is a covariant functor. It is the Gysin functor relative to B for proper maps. 2. The correspondence:  (−)! : Fibor f.t.B  D(DGM((B), d))

with

E  E! := (cv (E), d)[dM ] f  f! ,

is a covariant functor. It is the Gysin functor relative to B for arbitrary maps. Proof Straightforward, as in the absolute case (cf . Sect. 2.6.2), since the definitions of Gysin morphisms are based on Poincaré adjunctions which are functorial.   Comment 3.5.1.2 Unlike the absolute case, it is not clear if the need of finiteness hypothesis can be waived in the definition of the Gysin morphism f! . Indeed, although the restrictions fU ! : cv (EU )[dM  ] → cv (EU )[dM ] exist above trivializing open subspaces U ⊆ B, whether M is of finite type or not, they are only defined in the derived categories D(U ) and the question of glueing them together in a morphism of D(B) is uncertain.

3.5.2 Gysin Morphisms for Fiber Bundles and Base Change In Theorem 3.3.4.1 we established that Poincaré Duality for fiber bundles commutes with base change, and this immediately gives the same property for Gysin morphisms. Indeed, for every map h : B  → B, the base change functor or h−1 (−) : Fibor B,pr  FibB  ,pr

3.6 Examples of Gysin Morphisms

97

is well defined (cf . Corollary 3.1.7.2-(1)) and when applied to a proper morphism of fiber bundles f : (E  , B, π  , M  ) → (E, B, π, M), where M and M  are equidimensional, gives rise to the inverse image of the diagram of complexes of sheaves in D+ (B; R), corresponding to (3.76), which we used in defining the Gysin morphism f∗ associated with the pullback f ∗ : cv (E) → cv (E  ), i.e. we have a commutative diagram in D+ (B  ; R) h−1 π∗ (E ) [dM ]

h−1 (ID B,M ) q.i.

h−1 IR Hom•

(B),d)

π! (E

(B)

f∗

h−1 (f∗ )

h−1 π∗ (E) [dM ]

h−1 (ID B,M ) q.i.

h−1 IR Hom•

(B),d)

π!

(B)

which coincides, thanks to Theorem 3.3.4.1, with the diagram defining the Gysin morphism h−1 (f )∗ associated with h−1 (f )∗ : cv (h−1 (E)) → cv (h−1 (E  )). We have thus justified the following Proposition. Proposition 3.5.2.1 Gysin morphisms are compatible with base change. More precisely, given a map h : B  → B and a morphism of oriented fiber bundles f : (E  , B, π  , M  ) → (E, B, π, M), where M and M  are equidimensional, we have the base change diagram h−1 (E) h−1 (f )

h−1 (E

E

h f

h−1 (π)

)

π

E

h

h−1 (π )

π h

B

B

where all the parallelogram subdiagrams are Cartesian, and the following diagrams combining pullback and Gysin morphisms are commutative f∗

)

(if f is proper)

h∗

h−1 (E))

⊕

h−1 (f )∗

cv (E) h∗

h

h−1 (E

))

cv

(h−1 (E))

f!

cv (E

⊕ h−1 (f )!

)

h cv

(h−1 (E

))

98

3 Poincaré Duality Relative to a Base Space

3.6 Examples of Gysin Morphisms In this section, fiber bundles are implicitly assumed to be connected fiber bundles of manifolds with finite type fibers.

3.6.1 Adjointness of Gysin Morphism The discussion on Poincaré adjoint pairs 2.5 is also meaningful in the relative case since it only depends on the nondegeneracy of the Poincaré adjunction at the level of cochain complexes (cf . Comment 3.2.1.4). This property gives a useful way to discover explicit definitions of Gysin morphisms. Proposition 3.6.1.1 Let f : (E  , B, π  , M  ) → (E, B, π, M) be a morphism of oriented fiber bundles above B. 1. A set-theoretic map f" : cv (E  )[dM  ] → cv (E)[dM ] verifying,  M

f ∗ (α) ∧ β = 

 M

α ∧ f" (β) ,

(3.79)

for all α ∈ (E) and β ∈ cv (E  ), is automatically a morphism of (B)-dgmodules inducing the Gysin morphism f! in D(DGM((B), d)). Furthermore,  f" f ∗ (α) ∧ β = α ∧ f " (β) ,

(3.80)

for all α ∈ (E) and all β ∈ cv (E  ). 2. If f is proper, a set-theoretic map f" : (E  )[dM  ] → (E)[dM ] verifying,  M

f ∗ (β) ∧ α = 

 M

β ∧ f" (α) ,

(3.81)

for all α ∈ (E  ) and β ∈ cv (E), is automatically a morphism of (B)-dgmodules inducing the Gysin morphism f∗ in D(DGM((B), d)). Furthermore,  f" f ∗ (β) ∧ α = β ∧ f " (α) ,

(3.82)

for all α ∈ (E  ) and all β ∈ cv (E). Proof Consequence of the injectivity of the relative Poincaré adjunctions at cochain level 3.2.1.3-(2), which implies the nondegeneracy of the relative Poincaré pairing (3.2.1.4). For example, the fact that f" commutes with differential results from the

3.6 Examples of Gysin Morphisms

99

obvious equalities (−1)[α]

 M

α ∧ f" (dβ) =

 M

=d

f ∗ (α) ∧ dβ  M

=d

f ∗ (α) ∧ β − 



M



α ∧ f" (β) −

= (−1)[α]

 M

M

d f ∗ (α) ∧ β



M

dα ∧ f" (β)

α ∧ d f" (β) ,

which imply that f" ◦ d = d ◦ f" .

 

Comment 3.6.1.2 Note that while adjointness completely determine the Gysin morphisms at cochain level, the same is not always true in cohomology, in other words, the formulas (3.80) and (3.82) do not define Gysin morphisms by themselves, for which we would need to have Poincaré duality isomorphisms   H (E)[dM ]  Hom•H (B) Hcv (E), H (B)  Hcv (E)[dM ]  Hom•H (B) H (E), H (B) . And this actually happens when B is a formal space, that H (B) is of finite homological dimension, and that Hcv (E) is a projective H (B)-module (3.4.1.4-(2)). For example, when B = {•} (2.6.2.1-(2)). The pairs (f ∗ , f! ) and (f ∗ , f∗ ) are then Poincaré adjoint pairs in cohomology (cf . Proposition 2.5.1.1).

3.6.2 Constant Map and Locally Trivial Fibrations Let (E, B, π, M) be and oriented fiber bundle. The relative version of the constant map cM : M → {•} is the morphism of fiber bundles E

π

B

π

idB

B

and formula (3.79), with α ∈ (B) and β ∈ cv (E), gives the equality: α ∧ π" (β) =

 {•}

α ∧ π" (β) =

 M

π ∗ (α) ∧ β = α ∧



β. M

by 3.2.1.3-(1). Hence, the identification of Gysin morphism and integration π! = M : cv (E)[dM ] → (B) . The pair (π ∗ , M ) is therefore a Poincaré adjoint pair in D DGM((B), d).

100

3 Poincaré Duality Relative to a Base Space

3.6.3 Open Embedding Let (E, B, π, M) be and oriented fiber bundle, and let j : U ⊆ E be an open embedding of fiber bundles above B. The formula (3.80), with α ∈ (E) and β ∈ cv (U ), then gives, in (B), the equality:  M

j ∗α ∧ β =

 M

α ∧ j! β ,

(3.83)

where j! β ∈ c (M) is the extension by zero of β. The relative Gysin morphism in cohomology j! : cv (U )[dU ] → cv (M)[dM ] naturally extends the pushforward morphism j! : c (U ) → c (E) (2.6.3.1-(1)). The pair (j ∗ , j! ) is a Poincaré adjoint pair in D DGM((B), d).

3.6.4 Proper Base Change Let (E, B, π, M) and (E  , B  , π, M) be oriented fiber bundles, where M is equidimensional, let g : B → B  be a proper map and let E

g

E

π

B

π g

B

be a Cartesian diagram (with the obvious abuse of notation). The statements in following proposition are part of the so-called (proper) base change theorems. Proposition 3.6.4.1 If g : B  → B is proper, then g : E  → E is proper, and ⎧ ⎨ (i) g ∗ ◦ π! = π! ◦ g ∗ : cv (E)[dM ] → (B  ) , (ii) g ∗ ◦ π! = π! ◦ g ∗ : c (E)[dM ] → (B  ) , ⎩ (iii) π ∗ ◦ g∗ = g∗ ◦ π ∗ : (B  ) → (E)[dM ] . Proof The map g : E  → E is proper, since a compact subspace K ⊆ E can always be expressed a finite union of compact subspaces Ki , where Ki ⊆ π −1 (Ui ) for some open trivialization (π(−), ϕi (−)) : π −1 (Ui ) → Ui × M (cf . Sect. 3.1(F-1)). It then suffices to prove that g −1 (Ki ) is compact. For this, we replace Ki by the product of compact spaces X × Y with X = π(Ki ) and Y := ϕi (Ki ). But

3.6 Examples of Gysin Morphisms

101

then g −1 (X × Y ) = g −1 (X) × Y is compact since g : B  → B is proper (cf . Proposition 3.1.6.1). Notice that formulas (i,ii,iii) now make sense since the two g’s are proper maps. Identity (i) is the particular case of Proposition 3.5.2.1 where f := (π : E → B), and we know by 3.6.2 that π! is given by integration along fibers M . Notice also that the equality of differential forms g∗

 M

  ω = g ∗ (ω) ,

∀ω ∈ (E) ,

M

 12 which results by a local check in B , then fully justifies (i). Statement (ii) results from the obvious equality M (c (E)) ⊆ c (B), and (ii) and (iii) are equivalent by adjointness.  

3.6.5 Zero Section of a Vector Bundle Let V := (V , B, π, Rn ) be an oriented vector bundle. A notable difference of V compared to the case of a general fiber bundle E := (E, B, π, M) is the existence of a distinguished section of π , the zero section map σ : B → V , which is a homotopy equivalence. id σ

B

π

V

B

π id

B

id

We therefore have the Gysin morphisms σ! and π! relative to B, which are inverse of each other (since their composition is the identity), and that both are adjoint to homotopy equivalences, namely the pullbacks σ ∗ and π ∗ , π∗ q.i.

σ∗ q.i. id σ! q.i.

π!

cv (V )[n] q.i. id

12 Bott-Tu

[18] I §6 pp. 61–63.

(3.84)

102

3 Poincaré Duality Relative to a Base Space

where, an adjoint σ" of σ ∗ at the level of cochains, has to verify 

α ∧ σ" (β) = σ ∗ (α) ∧ β = σ ∗ (α ∧ π ∗ β) ∧ 1 = n



R

Rn

α ∧ π ∗ (β) ∧ σ" (1) .

Therefore, by uniqueness of the adjoint, we have σ" (β) = π ∗ (β) ∧ σ" (1), where σ" (1) ∈ ncv (V ) is a cocycle such that π(! σ" (1)) =

 Rn

σ" (1) = 1 .

We then set, after 3.2.3.1, σ" (1) := Φπ , where Φπ ∈ ncv (V ) is a cocycle representing the Thom class π of (V , B, π, Rn ). We have thus proved the following proposition. Proposition 3.6.5.1 The Gysin morphism σ! : (B) → cv (V )[n] relative to B is represented by the morphism of (B)-dgm’s σ! (−) := π ∗ (−) ∧ Φπ , which induces the Thom isomorphism in cohomology (3.2.3.1-(3.54)). When B and V are also oriented manifolds, we can consider the absolute point of view and seek for an adjoint σ" : (B) → (E)[n] of the restriction morphism σ ∗ : c (V ) → c (B) , which is generally not a quasi-isomorphism.13 We then write  V

σ" (α) ∧ β =

 B

α ∧ σ ∗ (β) =

 B

1 ∧ σ ∗ (π ∗ (α) ∧ β) =

 V

σ" (1) ∧ π ∗ (α) ∧ β ,

and conclude, as in the relative case, that σ∗ (α) := Φπ ∧ π ∗ (α) , represents the Gysin morphism σ∗ : (B) → (V )[n].

13 For

example, if B is compact and dB < dV , then Hc0 (B) = 0 and Hc0 (V ) = 0.

3.6 Examples of Gysin Morphisms

103

Proposition 3.6.5.2 Let (π, V , B) and (π, V  , B  ) be oriented vector bundles with zero section maps σ : B → V and σ : B  → V  . Let g

V  −−→ V ⏐ ⏐  π π g

B  −−→ B be a Cartesian diagram with g : B  → B proper. Then g : V  → V is proper and the following equalities hold ⎧ ∗ ∗  ⎨ (i) g ◦ σ! = σ! ◦ g : c (B) → c (V ) , (ii) g ∗ ◦ σ! = σ! ◦ g ∗ : (B) → cv (V  ) , ⎩ (iii) σ ∗ ◦ g∗ = g∗ ◦ σ ∗ : (V  ) → (B) .  

Proof Corollary of 3.6.4.1 since σ! is the inverse of π! .

3.6.6 Closed Embedding Given a closed embedding ι : N ⊆ M of connected oriented manifolds, we seek for a map ι" : (N )[dN ] → (M)[dM ] adjoint to the restriction morphism ι∗ : c (M) → c (N) . For that, fix a tubular neighborhood Uτ ⊇ N of N in M (3.2.3.1), and decompose ι : N ⊆ M as the composition of the zero section σ : N → Uτ and the open inclusion jτ : Uτ → M. For β ∈ c (M) and α ∈ (N ), we then have:  N

α ∧ ι∗ (β) =

 N

α ∧ ι∗ (φ β)

where φ ∈ 0cv (Uτ ) is any function verifying φ = 1 in a neighborhood of the support of a Thom form Φτ of the vector bundle π : Uτ → N (cf . Sect. 3.2.3.1). In that case, φ β ∈ cv (Uτ ) and ι∗ (φ β) = σ ∗ (φ β). We can then write  N

α ∧ ι∗ (φ β) =

 Uτ

Φτ ∧ π ∗ (α) ∧ (φ β) =

 Uτ

 Φτ ∧ π ∗ (α) ∧ β ,

which proves the following proposition. Proposition 3.6.6.1 The map  jτ ! ◦ Φτ ∧ π ∗ (−) : (N)[dN ] → (M)[dM ] ,

104

3 Poincaré Duality Relative to a Base Space

is Poincaré adjoint to the restriction morphism ι∗ : c (M) → c (N ) and represents the Gysin morphism ι∗ : (N)[dN ] → (M)[dM ], which is independent of the choice of the tubular neighborhood Uτ , in the derived category D Vec(R).

3.7 Applications We now give two important applications of the existence of Gysin morphisms in the form of exercises whose solutions are to be founded in Helpful Hints. We encourage the reader to solve them without looking at the solutions.

3.7.1 Gysin Long Exact Sequence Let i : F ⊆ M be a closed embedding of oriented manifolds. Assume F compact.14 Put U := M  F and j : U ⊆ M the open injection map. 1. (a) Let F denote the set of open neighborhood of F . Restriction morphisms RVW : (W ) → (V ) for all W ⊇ V ⊇ F , give rise to a filtrant inductive M : system {RVW | W ⊇ V in F } and a canonical morphism of complexes RF (M) → lim V ∈F (V ). Show that the short sequence −→ M RF

j!

0 → c (U ) −−→ c (M) −−→ lim F (V ) → 0 −→ where j! is the extension by zero, (3.6.3), is exact. ( , p. 341) (b) The restrictions RFV : (V ) → (F ) for V ⊇ F , define a morphism of the inductive system {RVW | W ⊇ V in F } to (F ). Denote by RFF := lim R V . Show that the morphism −→ F F RFF : lim F (V ) → (F ) , −→ is a quasi-isomorphism. Hint. Use the Tubular Neighborhood Theorem ( cf. question 1d). ( , p. 342) (c) Derive from the previous questions the the long exact sequence of compactly supported cohomology ( , p. 342) j!

i∗

ck

· · · → Hck (U ) −−→ Hck (M) −−→ Hck (F ) −−→ Hck+1 (U ) → · · · (3.85)

14 Although

not necessary, compactness of N simplifies the formulation of the exercise.

3.7 Applications

105

Fig. 3.2 Tubular neighborhood

π

M

S

π

B2 π

π

F (d) Endow M with a Riemannian metric d : M × M → R. For  ∈ R, denote B (F ) := {m ∈ M | d(m, F ) < } . Since F is compact, the Tubular neighborhood theorem states that for  small enough, B2 (F ) is a fiber bundle with fiber RdM −dF over F via the geodesic projection π : B2 (F ) → F . By restriction, denote π  : S (F ) → F

(3.86)

the fiber bundle with fiber the sphere SdM −dF −1 ⊆ RdM −dF (see Fig. 3.2). Let σ : S → B2  F and j : B2  F → U denote the inclusion maps, and consider the diagram S

σ

B2

F

π

j ⊂

U j ⊂

π i ⊂

F

(3.87)

M

Show that in the long exact sequence (3.85), the connecting morphism c : H (F ) → Hc (U )[1], is the following composition of morphisms H (F )

π∗

Hc (S )

σ! [−dS ]

Hc (B2

F )[1]

j ! [1]

Hc (U )[1]

c

where σ! and j! denote the Gysin morphism respectively associated with σ and j . ( , p. 342) 2.(a) Dualizing and shifting the long exact sequence of compactly supported cohomology (3.85), justify the exactness of the Gysin long exact sequence δ[−1]

i∗ [−dM ]

j∗

δ

−−→ H (F )[dF − dM ] −−−−−−→ H (M) −−→ H (U ) −−→

(3.88)

where i : F → N and j : U → N are the canonical injections and δ is adjoint to the shift of the connecting morphism c in (3.85). ( , p. 343)

106

3 Poincaré Duality Relative to a Base Space

(b) Show that the connecting morphism δ : H (U ) → H (F )[−(dM − dF − 1)] is the restriction to S followed by integration along fibers of π  ( , p. 344) H (U ) # α → δ(α) =

 S

α

S

∈ H (F ) .

3.7.2 Lefschetz Fixed Point Theorem Let M be a compact connected oriented manifold. Denote by δ : M → M × M the diagonal embedding x → (x, x) and let $M := Im(δ). Given f : M → M, denote Gr(f ) : M → M × M the graph map x → (f (x), x). The Lefschetz class of f is by definition L(f ) := Gr(f )∗ (δ∗ (1)) ∈ H dM (M) , and the Lefschetz number of f is the number f :=

M

L(f ) .

1. Explain the following equalities ( , p. 344) f :=

 M

Gr(f )∗ (δ∗ (1))

=

 M×M

(3.89)

δ∗ (1) ∪ Gr(f )∗ (1) = (−1)dM

 M

δ ∗ (Gr(f )∗ (1)) .

2. Assuming that f has no fixed points, show that the Gysin morphism Gr(f )∗ : H (M)[dM ] → H (M × M)[2dM ] factors through Hc (M × M  $M ) and f = 0. ( , p. 344) 3. Let B := {ei }i∈I be a graded basis of H (M) and let B  := {ei }i∈I denote the Poincaré dual basis of B, i.e. such that ei ∪ ej = δi,j ζM , where ζM denotes the fundamental class of M (2.4.2). Using the projection formula for diagonal map δ : M → M × M show that  δ∗ (1) = (−1)deg(ei ) ei ⊗ ei , i∈I



Deduce the equality: M

δ∗ (1)

$M

=

 k∈N

 (−1)k dim H k (M) . ( , p. 345)

4. Combining (3.89) with the last result, show the Lefschetz fixed point formula f =

 k∈N

 (−1)k Tr f ∗ : H k (M) → H k (M) .

In particular, if this sum does not vanish, then f has fixed points ! ( , p. 345) Notice that if f := idM , then f is the Euler characteristic χM of M.

3.8 Conclusion

107

Exercise 3.7.2.1 Let M be a compact connected oriented manifold. The notations are the same as in 3.7.2. Denote by (M , $M , π, RdM ) a tubular neighborhood of the diagonal $M in M × M, and define the Euler class of the pair ($M , M × M) by Eu($M ) := δ! (1)

$M

dM ∈ Hcv (M ) .

1. Show that if the Euler characteristic of M is nonzero, then Eu($M ) = 0. 2. Let M := M  $M , and let π  : M → M be the restriction of π to M . Then, consider the diagram of open/closed embeddings j ⊂

M π

M π

M

δ ⊃

M (3.90) id

(a) Show that π  : M → M is an oriented fiber bundle. (b) Describe the fiber F of π  . (c) Consider the long exact sequence of compact vertical cohomology associated with (3.90), and show that there exists a natural surjective map dM H dM −1 (M) → → Hcv (M ) . dM (d) Give an example of manifold M such that Hcv (M ) = 0. Conclude that although there are differential forms ζπ  ∈ dcvM (M ) verifying F ζπ  = 1, none of them can be a cocycle, thus providing the counterexample announced in 3.2.3. ( , p. 345)

3.8 Conclusion We have reached the end of the preliminaries on Poincaré duality and Gysin morphisms in the nonequivariant setting and using the de Rham model for cohomology, both in the absolute and in the relative case. As shown, the key ingredient in the approach is to define the Poincaré adjunction morphism in some derived category of differential graded modules. In the next chapter, we will concentrate on G-manifolds and de Rham G-equivariant cohomology, and so recall the Cartan Model for equivariant cohomology. We will then introduce the Poincaré pairing and the corresponding adjunctions, in such a way that we can continue to apply the same approach. Chap. 5 is entirely devoted to this, while in Chap. 6, the G-equivariant Gysin functors will be defined following the procedures described in 2.8. But, before going into those subjects, the next chapter begins with quick historical review of the origins of equivariant cohomology theory.

Chapter 4

Equivariant Background

4.1 Significant Dates in Equivariant Cohomology Theory In this section, we recall some of the more important dates in the genesis of equivariant cohomology. We do not aim for completeness and do not address developments after the 1990s.

4.1.1 Cartan’s ENS Seminar (1950)1 In lectures n◦ 19/20 of the Séminaire Cartan at the École Normale Supérieure de Paris,2 delivered in May and June of 1950, Henri Cartan is interested in principal G-bundles p : E → B, where G is a compact and connected (see p. 7) Lie group of Lie algebra g, and where B is a manifold. His focus was to settle an algebraic framework for studying the relationship between the cohomologies of E , B and G, extending the approach of the Chern–Weil homomorphism ch : S(g)G → H (B), and hence the construction of characteristic classes, well beyond the category of manifolds. To construct Chern–Weil homomorphisms, we need only the de Rham complex ((E ), d) equipped with interior products ι(X) and Lie derivatives θ (X) for all X ∈ g, and with an infinitesimal connection f : g ∨ → 1 (E ). Since the operators ι(X) and θ (X) admit purely algebraic descriptions, the underlying manifold E can be disregarded, and with this in mind, Cartan introduces,

1 See

Chevalley’s review [29], and also Chapter III of Tu [91], The Cartan Model, p. 141.

2 Lecture 19 on May 15 [25], and lecture 20 in two sessions: May 23 and June 19 [26]. The contents

of these lectures were published with some additions in [32] Colloque de topologie (espaces fibrés), held in Brussels on June 5–8 1950, (1951). © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Arabia, Equivariant Poincaré Duality on G-Manifolds, Lecture Notes in Mathematics 2288, https://doi.org/10.1007/978-3-030-70440-7_4

109

110

4 Equivariant Background

in his first lecture, the category DGA(g) of g-differential graded algebras (g-dga), which are dga’s equipped with operators ι(X) and θ (X) satisfying axioms which reflect the way they interact in the context of manifolds (cf . Sect. 4.2.3). In the same way, the concept of infinitesimal connection for principal G-bundles is extended to every E ∈ DGA(g) in what Cartan calls algebraic connection. This is a θ equivariant linear map f : g ∨ → E 1 verifying ι(X)f (λ) = λ(X), for all X ∈ g. The correspondence Connalg : DGA(g)  Set associating with E ∈ DGA(g) the set Connalg (E) of its algebraic connections is a representable functor. It therefore characterizes a g-dga (W (g), d): the Weil algebra of g, and an isomorphism of functors Connalg (−) ∼ = MorDGA(g) (W (g), −) .

(4.1)

As a graded algebra, W (g) := (g ∨ ) ⊗ S(g ∨ ) where (g ∨ ) and S(g ∨ ) are respectively the exterior and the symmetric algebras on g. The grading of W (g) is defined by setting g ∨ in degree 1 in (g ∨ ), and in degree 2 in S(g ∨ ). As a differential algebra, Cartan shows that (W (g), d) has the cohomology of a contractible G-space endowed with a canonical connection f0 (corresponding to id : W (g) → W (g) by (4.1)). The full subcategory of g-dga’s admitting algebraic connection is therefore a category of W (g)-modules and the canonical connection f0 appears as a universal connection.3 Taken together, this all already justifies Cartan saying (see top lines in Fig. 4.1, p. 113): the algebra W (g) plays the role of a cochain algebra of a fiber bundle p : IEG → IBG which would be classifying for all principal G-bundles. (See 4.9.) Cartan’s second lecture focuses on different ways of relating the cohomologies of E , B and G. Of these, the one interesting us is the construction of a g-dga canonically quasi-isomorphic to ((B), d) which takes as its main ingredients the de Rham complex ((E ), d) and the Weil algebra (W (g), d). This is the purpose of §5 in [28], where Cartan observes that for a principal G-bundle p : E → B, the pullback morphism p∗ : (B) → (E ) identifies (B) by with the subcomplex (E )bas of basic elements of (E ), which is the name Cartan gives to g-invariant and horizontal differential forms.4 The point here is that, for G connected, the notion of basic element is again purely algebraic and can be extended to the whole category DGA(g). The question then arises of how to algebraically determine E bas for any given E ∈ DGA(g). 4.1.1.1

The Cartan-Weil Morphisms

For E := (E ), Cartan proposes the g-dga W (g) ⊗ (E ) as algebraic ‘de Rham like’ model for the cohomology of the space IEG × E , product of the CW-complex IEG and the manifold E . Then, if f is a connection for p : E → B, the 3 See

Kumar [70], A remark on universal connections. differential form in the total space E of a principal G-bundle is horizontal if it is killed by the interior products with G-fundamental fields (Sect. 4.4.2).

4A

4.1 Significant Dates in Equivariant Cohomology Theory

111

representability equivalence (4.1) associates with it the Weil morphism of g-dga’s denoted (by abuse) f : W (g) → (E ), which leads Cartan to consider the diagram of morphisms of g-dga’s

(4.2)

where i(ω) := 1 ⊗ ω and f(α ⊗ ω) := f (α) ∧ ω. By restriction to basic elements, the bottom row gives the sequence of morphisms of complexes (4.3) where f ◦ i is the identity. We call i and f the Cartan-Weil morphisms. The morphism i can clearly be defined, in the same way, for any E ∈ DGA(g), and likewise for f as long as E admits an algebraic connection. The main result concerning this sequence (4.3) is Cartan’s Theorem 3, in §5 of [28]:5,6, 7 Theorem 4.1.1.1 (Cartan) If E ∈ DGA(g) admits algebraic connections, then the Cartan-Weil morphism i : E bas → (W (g) ⊗ E)bas is a quasi-isomorphism. In particular, the Cartan-Weil morphism  H (f) : H (W (g) ⊗ E)bas → H (E bas ) is an isomorphism too, inverse of H (i), hence independent of the connection. Following immediately the theorem, Cartan more closely examines (in §6 [28]) the complex (W (g) ⊗ E)bas for any given E := (E, d) ∈ DGA(g). He shows that there exists a natural isomorphism,8 which we call Cartan isomorphism, ∨ G (W (g) ⊗ E)bas −→  ((S(g ) ⊗ E) , dg )

(4.4)

5 Cartan gave summary indications of the proof. Later Michel André in his Ph.D. thesis [6] directed

by Claude Chevalley gave complete proofs of all the statements in Cartan’s lectures. The theorem is also proved in Tu [91] Appendix A, Theorem A.2, p. 264. 6 An important, but rarely mentioned fact about sequence (4.3), is that it remains meaningful for any g-submodule K ⊆ E which is also a graded ideal of E, and that, in that case, the theorem remains true for K instead of E (cf . Appendix C). In particular, for any G-equivariant fiber bundle π : E → B, the morphism cv (E)bas → (W (g) ⊗ cv (E))bas is a quasi-isomorphism although there are generally no algebra homomorphisms W (g) → cv (E). 7 The theorem corroborates the idea that algebraic connections on g-dg-algebras are the right analogue to free actions on G-manifolds. 8 The isomorphism is completely justified in Cartan [28] §6 Transformation de l’algèbre différentielle B, p. 63. We recall this proof in Sect. 4.3.2, p. 124. See also Tu [91], The Cartan Model in General §21-1,2, pp. 167–172.

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4 Equivariant Background

where dg (P ⊗ ω) = P ⊗ dω +

 i

P ei ⊗ ι(ei ) ω

(4.5)

where d is the differential in E, {ei } is a basis of g of dual basis {ei }, and ι(ei ) is the interior product corresponding to ei ∈ g (cf . Sect. 4.3.2). Another important by-product of diagram (4.2) results from the restriction of the subdiagram (I) to basic elements. We have W (g)bas = S(g ∨ )G and the restriction of f to this subalgebra is, by construction, the well-known Chern–Weil homomorphism. Hence, ch = f bas . We therefore have commutative diagrams

which, besides proving that the usual Chern–Weil homomorphism in cohomology is independent of the infinitesimal connection, show that the abstract Cartan-Weil formalism is the right framework for dealing with connections and characteristic classes. The following commutative diagram summarizes the Cartan-Weil approach.

(4.6)

Here, the third row is the representability theorem of the set of equivalence classes of principal G-bundles above a manifold B.9 The second row represents the construction of characteristic classes through the Chern–Weil homomorphism, where the multiple arrow (†) recalls that for a given principal G-bundle there are many infinitesimal connections. The first row, the representability of the set

9 See

Steenrod [84], §19.3 Classification Theorem, p. 101 and the Historical Note in p. 105, where the Theorem for Lie groups is independently granted to Steenrod, Whitehead and jointly by Chern and Sun. See also Husemöller [56] §Preface to the First Edition p. xi.

4.1 Significant Dates in Equivariant Cohomology Theory

113

Fig. 4.1 Excerpt from the last paragraph in Cartan’s lecture [32]

of algebraic connections, and the corresponding upper subdiagram represent the contribution of Cartan’s lectures to characteristic classes.

4.1.1.2

The Cartan Complex

It is worth emphasizing that for any G-manifold M, whether G acts freely or not, the de Rham complex (M) is well equipped with a structure of g-dga. We can hence still consider the complex (W (g) ⊗ (G))bas (4.4), which we denote G (M) := ((S(g ∨ ) ⊗ (M))G , dg )

(4.7)

Although this was clear in Cartan’s lectures, the subject was out of focus at the time, as research was mainly concentrated on principal G-bundles rather than on general G-manifolds, and still less on general topological G-spaces. The complex G (M) (4.7) is commonly referred to as the Cartan complex of M, and also as the Cartan Model for the cohomology of the homotopy quotient MG := IEG×G M, since it is indeed quasi-isomorphic to the complex of AlexanderSpanier cochains Ω(MG ; R) (cf . 4.8.3.1, 4.10.1.2).

4.1.1.3

Homotopy Quotients and Formality of the Classifying Space

We will see in Theorem 4.9.2.1 that the dg-algebras (G (M), d) and (Ω(MG ; R), dg ) can be joined by a zig-zag () of quasi-isomorphisms of dga’s and, in the particular case where M := {•}, we will then have (Ω({•}G ; R), d)  (G ({•}), dg ) = (S(g ∨ )G , 0) , where {•}G = IBG. But here, the complex (S(g ∨ )G , 0) is canonically isomorphic to its cohomology, whence the R-formality of IBG (cf . Sect. 3.4.1). This is a remarkable fact which has not been sufficiently highlighted in Cartan’s lectures, although Cartan was clearly aware of it as he says in the very last lines of his first lecture (see bottom lines in Fig. 4.1): The algebra S(g ∨ )G plays the role of the cochain algebra of the base space of a universal fiber bundle, with the particularity that the elements of S(g ∨ )G are all cocycles.

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Fig. 4.2 Excerpt from A. Borel [15], IV-§3 , p. 55 (1960), where Borel refers to the sources of the homotopy quotient

4.1.2 Borel’s IAS Seminar (1960) Some years later, in 1958–1959, Armand Borel, who had, since his arrival in Paris in 1949, been an active participant in the Cartan Seminar and in the Leray’s courses at the Collège de France, held his Seminar on transformation groups at the Institute for Advanced Study in Princeton (Borel [15]). There, Borel drew attention to the advantages of considering for any locally compact G-space X, the orbit space of IEG × X under the diagonal action of G: XG := IEG ×G X , as the homotopically best-suited substitute for the orbit space X/G, whether G acts freely or not on X . The space XG is the total space of the following two surjective maps [X]

XG := IEG ×G X

π p [IBGx ]

IBG

X/G ,

where the fibers are shown in brackets. More precisely • π : XG → → IBG, (w, x) → w, a locally trivial fibration of fiber space X, and • p : XG → → X/G, (w, x) → x, where the fibers are the classifying spaces IBGx of the different stability groups Gx for x ∈ X. As Borel says in its introduction: XG allows us to tie together the cohomology groups of X, X/G, and the fixed point set F := XG , with those of the classifying spaces of the stability groups and of G.

4.1.2.1

The Borel Construction

The space XG , which Borel called twisted product, is known today as the homotopy quotient, the homotopy orbit space and also the Borel construction (see Fig. 4.2).

4.1 Significant Dates in Equivariant Cohomology Theory

115

Beyond its immediate aim, which was the homological study of the set of fixed points F := XG , the seminar laid most of the foundations of what would later be known as the equivariant cohomology of locally compact G-spaces. Orbit types, slice theorems, spectral sequences and fixed points theorems, were already present, if not yet in their final form, at least at a level that would appeal to other mathematicians for further development. The restriction map H (XG ) → H (FG )

(4.8)

appears in almost every section of applications of the Borel Seminar, often with restrictive conditions to ensure it is an isomorphism. And in the case of the circle group action, G := T 1 , while it is clear that Borel was aware that (4.8) is an isomorphism modulo H (IBG)-torsion, he never stated it in those terms.

4.1.3 Atiyah-Segal: Equivariant K-Theory (1968) In 1968, the bases of equivariant K-theory are set out in the works of Atiyah-Segal and Segal [8, 82], where the following enhanced analogue to (4.8) appears for the first time. Localization theorem ([8, 82]) 10 Let G be a compact Lie group and let X be a locally compact G-space. The localized restriction map KG (X)p → KG (G.XS )p ,

(4.9)

where p is a prime ideal of KG (•), S is minimal among the subgroups of G such that p is the inverse image of a prime ideal of KS (•) under the restriction map KG (•) → KS (•) , and XS is the set of S-fixed points in X, is an isomorphism.

4.1.4 Quillen: Equivariant Cohomology (1971) As K-theory has a cohomological behavior, equivariant cohomology soon came to light. We owe it to Daniel Quillen [80] who, merging the ideas of Atiyah-Segal and Borel, defines11 the equivariant cohomology HG (X) of a G-space X with coefficients in a ring A, as the ordinary cohomology of Borel’s construction XG , i.e. HG (X) := H (XG ; A) . 10 Proved 11 In

in Segal [82], Section §4. Localization, Proposition 4.1, p. 144. the first page of Part I, page 549 of the journal.

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4 Equivariant Background

Quillen proves the analogue to the localization theorem (4.9) for the case where G is an elementary p-group, and for the case where G is a torus T . Theorem ([80]) Assume either X is compact or paracompact with dimh (X) < +∞ and that the set of identity components of the isotropy groups of points of X is finite. Then the inclusion of XT in X induces an isomorphism   HT (X) HT (•) − 0 −1 → HT (XT ) HT (•) − 0 −1 .

(4.10)

4.1.5 Hsiang’s Book (1975) 1975 saw the appearance of Wu-yi Hsiang’s book ‘Cohomology theory of topological transformation groups’ [55] in which the third chapter promptly introduces the reader to the foundations of equivariant cohomology for locally compact G-spaces. It includes a version of the localization theorem more in the vein of AtiyahSegal that Hsiang calls the ‘Borel-Atiyah-Segal localization theorem’. It is stated as follows: Theorem 12 Assume either X is compact or paracompact with dimh (X) 0 (d = 2). As a consequence, the homomorphism dg is a differential on (S(g ∨ ) ⊗ C)g . It is commonly referred to as the Cartan differential. Definition 4.3.2.2 Given  a g-dgm C := (C, d, θ, ι), the Cartan complex associated with C is the complex Cg , dg , where Cg := (S(g ∨ ) ⊗ C)g

and

dg (ω))(Y ) := d(ω(Y )) + ι(Y )(ω(Y ) , ∀Y ∈ g .

Its cohomology will be called the g-equivariant cohomology of C, and will be denoted by  Hg (C) := h Cg , dg . Notice that: • Hg (C) is a graded Hg (R) = S(g ∨ )g -module. • After Cartan’s theorem 4.1.1.1, if C admits an algebraic connection, then Hg (C) = H (C bas ). In particular, Hg (W (g)) = S(g ∨ )g . • If A := (A, d, θ, ι) is a g-dg-algebra, then Ag is an S(g ∨ )g -dg-algebra and Hg (A) is an S(g ∨ )g -algebra.

4.3.3 Induced Morphisms on Cartan Complexes A morphism of g-complexes α : (C, d, θ, ι) → (D, d, θ, ι) induces a canonical S(g ∨ )g -linear morphism of complexes αg : Cg → Dg , by αg = id ⊗ α. Furthermore, if α is a morphism of g-dg-algebras, then αg is a morphism of dgalgebras.

4.3 Equivariant Cohomology of g-Complexes

127

Theorem 4.3.3.1 With the above notations, we have the following. 1. The correspondence (C, d, θ, ι)  (Cg , d) and α  αg is a covariant functor from DGM(g) into DGV(R). 2. For every g-complex (C, d, θ, ι) ∈ DGM+ (g), there exists a spectral sequence converging to Hg (C) with  p,q  p ∨ p+q IE 0 = S (g ) ⊗ C q g , d0 = 1 ⊗ d ⇒ Hg (C) . 3. Let G be a compact Lie group, let g := Lie(G), and let C and D be g-split g-complexes (4.2.5.1) in DGM+ (g). Then, the following statements hold. a. The (IE 2 , d2 ) spectral sequence term in (2) is given by  p,q

p+q IE 2 = S p (g ∨ )g ⊗ H q (C) , d2 = i ei ⊗ ι(ei ) ⇒ Hg (C) . b. If H m (C) = 0 for all odd (or for all even) m, then Hg (C) = S(g ∨ )g ⊗ h(C) . c. If α : C → D is a quasi-isomorphism, then αg is a quasi-isomorphism. 4. Let T be a commutative compact Lie group and g := Lie(G). a. For every g-complex (C, d, θ, ι), the subcomplex (C g , d) is stable under θ and ι, i.e. (C g , d, θ, ι) is a well-defined g-complex. In the next statements C and D are g-complexes in DGM+ (g). b. The (IE 2 , d2 ) spectral sequence term in (2) is given by  p,q

p+q IE 2 = S p (g ∨ ) ⊗ H q (C g ) , d2 = i ei ⊗ ι(ei ) ⇒ Hg (C) . c. If H m (C g ) = 0 for all odd (or for all even) m, then Hg (C) = S(g ∨ ) ⊗ h(C g ) . d. α : C → D is a quasi-isomorphism, then αg is a quasi-isomorphism. Proof

 (1) Clear. (2) For m ∈ Z, let Km = S ≥m (g ∨ )⊗C g . Each Km is clearly a sub complex of (Cg , dg ) and Cg = K0 ⊇ K1 ⊇ · · · is a regular decreasing filtration of (Cg , dg ) since C is bounded below (see [46] §4 pp. 76-) giving rise to the stated spectral sequence. (3a) The assumption that G is compact ensures that each (finite dimensional) p ∨ g-module Proposition 4.2.5.5-(2) may be used, and  p ∨ S (g g) is semisimple. (S (g ) ⊗ C) , 1 ⊗ d is quasi-isomorphic to (S p(g ∨ )g ⊗ C, 1 ⊗ d . Consequently (IE 0 , d0 ) in (2) is quasi-isomorphic to S(g ∨ )g ⊗ C, 1 ⊗ d

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4 Equivariant Background

and IE 1 = S p (g ∨ )g ⊗ H q (C). But the differential d1 : IE 1 → IE 1 is equal to zero since the S(g ∨ ) vanishes in odd degrees, therefore IE 1 = IE 2 , which completes the proof of the claim. p,q Since the differential dr is of total degree 1 and that IE r = 0 if p or q is odd for all r ≥ 2, then we have dr = 0 for r ≥ 2, and IE 2 = IE ∞ . Follows immediately from (3-i) and 4.2.5.5-(1). We must check that θ (Y )ι(X)C g = 0 for all X, Y ∈ g, but, on C g we have θ (Y )ι(X) = θ (Y )ι(X) + ι(X)θ (Y ) = ι([Y, X]) = ι(0) since g is abelian and from property (iii) of g-complexes (see Sects. 4.2.3–(4.13)). Left to the reader.   p,q

(3b) (3c) (4a)

(4b,c,d)

p,q

p+1,q

4.3.4 Split G-Complexes Let G be a Lie group (not necessarily compact) with Lie algebra g. It’s worth noting that the proof of 4.3.3.1-(3) uses the split condition 4.2.5.1 only on the finite dimensional vector spaces S p (g ∨ ) ⊆ S(g ∨ ) endowed with the structure of g-module obtained by differentiating their natural G-module structure. We are thus led to extend the split definition 4.2.5.1 to G-modules. Definition 4.3.4.1 For any inclusion of G-modules N ⊆ M we write ‘N |M’ whenever the natural map HomG (V , M) → HomG (V , M/N )

(4.22)

is surjective for all finite dimensional G-modules V . A complex of G-modules (C, d) is said to be G-split is B i |Z i |C i , for all i ∈ Z. Exercise 4.3.4.2 Let G be a compact Lie group. Denote by G0 the connected component of the identity element e ∈ G. Show that a complex of G-modules (C, d) is G-split if and only if it is G0 -split. ( , p. 348) The proof of the following proposition is then the same as for Proposition 4.2.5.5. Proposition 4.3.4.3 Let G be a compact Lie group, and let (C, d) be a G-split complex of G-modules such that the action of G in h(C, d) is trivial. Then, 1. The inclusion C G ⊆ C is quasi-isomorphism. 2. If V is a semisimple finite dimensional G-module, the inclusions VG ⊗C

⊇

V G ⊗ CG

⊆

(V ⊗ C)G

Hom•R (V G , C) ⊇ Hom•R (V G , C G ) ⊆ Hom•G (V , C) are quasi-isomorphisms.

4.4 Equivariant Differential Forms

129

4.4 Equivariant Differential Forms 4.4.1 Fields in Use Manifolds, Lie groups and Lie algebras, vector spaces, linear maps, tensor products and related material will be defined over the field of real numbers R.

4.4.2 G-Fundamental Vector Fields Let G be a Lie group with Lie algebra g. On every G-manifold M, an element Y ∈ g defines a vector field, called the G-fundamental vector field on M associated with Y , by the formula d t → exp(tY ) · m t=0 . Y' (m) := dt

(4.23)

The vector field Y' is G-invariant and the map (− ' ) : g → X(M)G , where X(M) denotes the Lie algebra of vector fields on M, is a Lie algebra homomorphism. When M := G, the map (− ' ) : g → X(G)G is an isomorphism onto the set of left invariant vector fields. Moreover g' (x) = Tx (G) for every x ∈ G, and, as a consequence, a differential form ω ∈ i (G) is completely determined by the functions ω((− ' ), . . . , (− ' )) : G → (g ∨ ). Hence, a canonical identification (G) = 0 (G) ⊗ (g ∨ ) .

(4.24)

4.4.3 Interior Products and Lie Derivatives – The interior product with a vector field ξ ∈ X(M) is the map ι(ξ ) : (M) → (M)[−1] , defined by ι(ξ )(ω)(ξ1 , . . . , ξd−1 ) := ω(ξ, ξ1 , . . . , ξd−1 ) ,

∀ω ∈ d (M) , ξi ∈ X(M) ,

it is a graded derivation (aka antiderivation) of degree −1 , i.e. ι(ξ )(ω1 ∧ ω2 ) = ι(ξ )(ω1 ) ∧ ω2 ) + (−1)d1 ω1 ∧ ι(Y )(ω2 ) ,

130

4 Equivariant Background

for all ω1 ∈ d1 (M) and ω2 ∈ (M). The map ι : g → MorGV(k) ((M), (M)[−1]) ,

Y → ι(Y' ) ,

(4.25)

verifies conditions (i)-(4.13) and (iv)-(4.14) for g-complexes. – The Lie derivative with respect to ξ ∈ X(M), is the map L(ξ ) : (M) → (M) , defined by L(ξ ) := d ◦ ι(ξ ) + ι(ξ ) ◦ d , it is a graded derivation of degree 0 , i.e. L(ξ )(ω1 ∧ ω2 ) = L(ξ )(ω1 ) ∧ ω2 ) + ω1 ∧ L(Y )(ω2 ) , for all ω1 ∈ d1 (M) and ω2 ∈ (M). The map θ : g → EndgrGV(k) ((M)) ,

Y → L(Y' ) ,

(4.26)

is a Lie algebra representation by R-derivations and verifies conditions (ii,iii)(4.13) (p. 121) and (v)-(4.14) (p. 121) for g-complexes. The operators θ (Y ) and ι(Y ) stabilize c (M), and the quadruples ((M), d, θ, ι)

and

(c (M), d, θ, ι) ,

are g-complexes since all the conditions in Sect. 4.2.3 are satisfied.

4.4.4 Complexes of Equivariant Differential Forms Let G be a compact Lie group. The complex of G-equivariant differential forms, resp. compactly supported, of M, modeled on the Cartan complex (4.3.2.2), is the complex:   G (M), dG := S(g ∨ ) ⊗ (M) G , dg ,   resp. G,c (M), dG := S(g ∨ ) ⊗ c (M) G , dg ,

(4.27)

dG (ω))(Y ) := d(ω(Y )) + ι(Y )(ω(Y ) ,

(4.28)

with ∀Y ∈ g .

Its cohomology, denoted by HG (M), resp. HG,c (M), is the G-equivariant cohomology, resp. compactly supported, of M.

4.4 Equivariant Differential Forms

131

Note that the complexes G (M), G,c (M) and the cohomology vector spaces HG (M) and HG,c (M) are S(g ∨ )G -graded modules (cf . 5.1.2). Convention 4.4.4.1 When M := {•}, we shrink notations to G := G ({•}) and HG := HG ({•}) Notice that in that case dG = 0, so that G = S(g ∨ )G = HG . Comment 4.4.4.2 The reader will have noticed that in (4.27), we do not assume G connected, and that we have replaced the g-invariants involved in the definition of Cartan complexes 4.3.2.2, by G-invariants. This is possible since the actions of G on g and on M are differentiable, and that it is this differentiability that induces the action of the Lie algebra g on the g-complexes (M)g and c (M)g . As a consequence, when G is connected there is no difference between G-invariance and g-invariance and the results in Cartan’s lectures apply. On the contrary, when G is not connected, we deviate from Cartan’s lectures and we need to justify the relevance of our definition and its relationship with the connected case. This is the aim of the next section.

4.4.5 On the Connectedness of G Let G be a Lie group and denote by G0 the connected component of the identity element e ∈ G. We recall that G0 is an open normal subgroup of G and that W := G/G0 is a discrete group (finite if G is compact). Let M be a G-manifold, and denote by ? (M) either (M) or c (M). The group G acts on g := Lie(G0 ) by the adjoint representation, and on M by diffeomorphisms, whence its action on S(g ∨ ) ⊗ ? (M), and the equality:  (S(g ∨ ) ⊗ ? (M))G = (S(g ∨ ) ⊗ ? (M))g W .

(4.29)

Lemma 4.4.5.1 1. The map dG : S(g ∨ ) ⊗ ? (M) → S(g ∨ ) ⊗ ? (M), defined in 4.4.4-(4.28) by the formula (P ⊗ ω) → P ⊗ dω +

 i

P ei ⊗ ι(ei ) ω ,

is an S(g ∨ )-linear G-equivariant map. 2. The differential in the Weil algebra (W (g), d) is G-equivariant. Proof We denote by a dot ‘· ’ the (left) actions of G.

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4 Equivariant Background

For all g ∈ G, we have: • For all λ ∈ g ∨ and all X ∈ g: λ, X = g · λ, g · X .

(4.30)

Hence, if {ei } and {ei } are dual basis of g, then so are {g · ei } and {g · ei }. • For all ω ∈ ? (M), X ∈ g and x ∈ M, we have:  ' ' (g · ω)(x)(X(x), . . . ) = ω(g −1 · x) Tx g −1 (X(x)), ...) −−−−−−→ = ω(g −1 · x) g −1 · X(g −1 · x), . . . ) , whence  g · ι(X)(ω) = ι(g · X)(g · ω) ,

(4.31)

 g · θ (X)(ω) = θ (g · X)(g · ω) ,

(4.32)

and also

since θ (X) = d ◦ ι(X) + ι(X) ◦ d. (1) For all P ∈ S(g ∨ ) and ω ∈ ? (M), we have:   dG g · (P ⊗ ω) = dG (g · P ) ⊗ (g · ω) = (g · P ) ⊗ d(g · ω) +



=1 (g · P ) ⊗ · (dω) +  =2 g · dG (P ⊗ ω) ,

i



(g · P ) ei ⊗ ι(ei )(g · ω)  g · P (g −1 · ei ) ⊗ ι(g −1 · ei )(ω)

i

where, for (=1 ) we used (4.31), and for (=2 ) we used (4.30). (2) Since the differential d of the Weil algebra (W (g), d) is an R-derivation, and since G acts by algebra isomorphisms, the G-equivariance of d results by simply checking this property on the elements λ ⊗ 1 ∈ 1 (g ∨ ) ⊗ 1 and 1 ⊗ λ ∈ 1 ⊗ S 2 (g ∨ ). After Cartan’s description19 of the differential d, we have, for all g ∈ G,  g · d(λ ⊗ 1) = g · dK (λ) ⊗ 1 + 1 ⊗ λ = dK (g · λ) ⊗ 1 + 1 ⊗ · λ

 = g · dK (λ) ⊗ 1 + 1 ⊗ · λ = d g · (λ ⊗ 1) ,

19 See

Cartan [27], §6 L’algèbre de Weil d’une algèbre de Lie, formulas (9–10), p. 23–24.

4.4 Equivariant Differential Forms

133

where dK is the Koszul differential on λ(g ∨ ), well-known to be G-equivariant. We also have: 

 ei ⊗ θ (ei )(λ) g · d(1 ⊗ λ) = g · =

i

 i

 (g · ei ) ⊗ θ (g · ei )(g · λ) = d 1 ⊗ (g · λ) ,  

by (4.30) and (4.32).

Thanks to the lemma, we see that, whether the group G is connected or not, the complexes   S(g ∨ ) ⊗ ? (M), dG G = (S(g ∨ ) ⊗ ? (M), dG )g W

(4.33)

are well defined. This all immediately leads us to ask if this extension of the definition of Cartan complexes is compatible with other parts of Cartan’s lectures (Sect. 4.1.1), among which, if, given a principal G-bundle (E, B, π, G), the Cartan-Weil morphisms i, f (in (4.3), p. 111, and Appendix C) and the Cartan morphism Ξ  (in (4.18), p. 125):

(4.34)

are W -equivariant quasi-isomorphisms? The answer is yes. Indeed, the morphisms i and Ξ  are W -equivariant, since i : ? (E) → W (g)⊗? (E), ω → 1⊗ω, and Ξ : W (g)⊗? (E) → S(g ∨ )⊗? (E) (cf . (4.18), p. 125) are G-equivariant. The same is true for f provided that we can choose the infinitesimal connection f : g ∨ → 1 (E) to be G-invariant, and we know this is possible when G is compact (which is always the case in this book), since we can then G-average any infinitesimal connection. The following proposition summarizes all these observations. Proposition 4.4.5.2 Let G be a compact Lie Group. Denote by G0 the connected component of the identity element e ∈ G, and set W := G/G0 . Let  (−)Bas := (−)bas W = (−)hor ∩ (−)G . Then, for every principal G-bundle (E, B, π, G), the morphisms in (4.34) are W equivariant and induce the following quasi-isomorphic morphisms of dg-algebras, (4.35)

134

4 Equivariant Background

(resp. of dg-modules if we replace (M) by c (M)), and where i  and Ξ  denote the restrictions of i and Ξ  to the subspaces (−)Bas ⊆ (−)bas . In particular, HG0 (? (M)) is a W -module, and we have canonical isomorphisms 

H (B)  HG (E)  HG0 (E)W Hc (B)  HG,c (E)  HG0 ,c (E)W .

(4.36)

Proof The fact that π ∗ : (B) → (E)Bas is an isomorphism is standard in differential geometry, and is true whether G is compact or not. The morphism Ξ  is an isomorphism by elementary algebraic reasons. At the end, compactness of G is needed to justify the existence of a connection of principal G-bundle f : g ∨ → 1 (E), hence to apply Cartan’s theorem 4.1.1.1. We therefore get W -equivariant quasi-isomorphisms (4.37) Compactness of G implies also that the group W is finite, hence that the restrictions i  and Ξ  of the quasi-isomorphisms i and Ξ  in (4.37), to W -invariants remain quasi-isomorphic, proving (4.35) for (M). The same arguments apply, mutatis mutandis, to c (M) in lieu of (M), replacing Cartan’s theorem by its enhancement in Appendix C. The isomorphisms (4.36) then easily follow.  

4.4.6 Splitness of Complexes of Equivariant Differential Forms Theorem 4.4.6.1 (20 ) 1. Let G be a compact Lie group. For every G-manifold M, the complexes of Gmodules ((M), d) and (c (M), d) are G-split (4.3.4). 2. For all m ∈ N, the inclusions S ≥m (g ∨ )G ⊗ C ⊇ S ≥m (g ∨ )G ⊗ C G ⊆ (S ≥m (g ∨ ) ⊗ C)G , where C denotes ((M), d) or (c (M), d), are quasi-isomorphisms if and only if G acts trivialy on h(C), for example if G is connected.21

20 A

different approach of this theorem can be found in Tu [91], Appendix C, p. 283. the connectedness hypothesis see (4.4.6.3)–(1).

21 On

4.4 Equivariant Differential Forms

135

Proof (1) After Exercise 4.3.4.2 we can assume G connected. The pushforward action of G on i (M) is defined, for g ∈ G and ω∈i , by g∗ (ω) := (g −1 )∗ (ω) , where (g −1 )∗ denotes the usual pullback of differential forms. The pushforward action is such that we have (g1 g2 )∗ = g1∗ ◦ g2∗ for all gi ∈ G. If V is a (smooth) finite dimensional representation of G over C, we make the group G act on Hom(V , i (M)) by the formula  (g · λ)(v) = g∗ λ(g −1 v) ,

∀λ ∈ Hom(V , i (M)) ,

so that λ is a G-module morphism if and only if g · λ = λ. We claim that there exists a ‘symmetrization’ operator * : Hom(V , i (M)) → Hom(V , i (M))G , such that * 2 = * and *(λ) = λ if and only if λ is a G-module morphism. Indeed, let λ be a linear map from V to i (M). For every i-tuple of vector fields {χ1 , . . . , χi } over M and each v ∈ V , the real function M # x →

 G

   g∗ λ(g −1 v) (x) χ1 (x), . . . , χi (x) dg ∈ R ,

where dg is a G-invariant form of top degree on G, such that 1 = G dg, is a smooth function, since V is finite dimensional, depending linearly on v ∈ V , and multilinearly and antisymmetrically on the χ∗ ’s. We therefore have an i-differential form which we denote by *(λ)(v) :=

 G

 g∗ λ(g −1 v) dg ,

(4.38)

and whose fundamental properties are • *(d ◦ λ) = d ◦ *(λ); • *(λ) : V → i (M) is a G-module morphism; • *(λ) = λ, if λ is already a G-module morphism. We can now resume the proof that Z i (M)|i (M). Given a G-module morphism μ ∈ HomG (V , B i+1 (M)), there always exists a linear map λ : V → i (M) lifting μ, i.e. such that μ = d ◦ λ, but then we apply the symmetrization operator * and we get μ = *(μ) = *(d ◦ λ) = d ◦ *(λ) , which shows that the G-module morphism *(λ) lifts μ. For Zci (M)|ic (M), note that, since V is finite dimensional, the supports of the elements in λ(V ) are contained in one and the same compact subset C ⊆ M,

136

4 Equivariant Background

but then the supports of the g∗ (λ(g −1 v)) in (4.38) are contained in G · C which is obviously compact. Therefore, given λ : V → c (M), we get a linear map *(V ) : V → c (M) which is a G-module morphism, and the preceding arguments apply to the compactly supported case. To prove that B i (M)|Z i (M), we need only, from 4.2.5.4-(2), show that every cocycle is cohomologous to a G-invariant cocycle. But before doing so, let us recall a general homotopy argument. Given a smooth map ϕ : R × M → N , if ω ∈ i (N) the pullback ϕ ∗ ω belongs to i (R × M), i.e. is a section of the exterior algebra bundle of the cotangent bundle T ∗ (R × M) of R × M. Now, the canonical decomposition T ∗ (R × M) as the direct sum of cotangent bundles T ∗ (R) ⊕ T ∗ (M), leads to a canonical decomposition of the i-th exterior power of the cotangent bundle  # # #i ∗ T (R × M) = R ⊗ i (T ∗ M) ⊕ T ∗ (R) ⊗ i−1 (T ∗ M) . Consequently, the pullback ϕ ∗ (ω) canonically decomposes as ϕ ∗ (ω)(t, x) = α(t, x) + dt ∧ β(t, x) , # # where α (resp. β) is a section of the vector bundle i T ∗ (M) (resp. i−1 T ∗ (M)) over the base space R × M. When ω is in addition a cocycle, so is ϕ ∗ (ω) and, in view of the previous decomposition, this amounts to the following two conditions dα(t, x) = 0 ,

∂ α(t, x) = dβ(t, x) , ∂t

where d is the coboundary in (M) (t is then assumed constant). In particular, denoting by ϕt : M → N the map x → ϕ(t, x), we get ϕt∗ (ω) − ϕ0∗ (ω) = α(t) − α(0)  t  t

 t  ∂ α(t) dt = dβ(t) dt = d β(t) dt , = 0 ∂t 0 0

(4.39)

and the cocycles ϕt∗ (ω) are all cohomologous to ϕ0∗ (ω). It is worth noting that this process gives a canonical element  (x) = 1 i−1 (M), depending on ω, such that ϕ ∗ (ω) − ϕ ∗ (ω) = d . 1 0 0 β(t, x) dt ∈  Under the hypothesis of the theorem, a first consequence of the previous observations, is that if ω ∈ Z i (M), then g ∗ ω is cohomologous to ω for all g ∈ G. Indeed, since G we assumed connected, there is a smooth path γ : R → G such that γ (0) = e and γ (1) = g, and then taking ϕ : R × M → M, (t, x) → γ (t) · x, we conclude that g ∗ ω = γ1∗ (ω) ∼ γ0∗ (ω) = ω. More generally, given a diffeomorphism φ : RdG → G onto an open subset U ⊆ G, we define a smooth multiplicative action of R over U by setting t g := φ(t · φ −1 (g)) for all t ∈ R and g ∈ U , and consider, for each g ∈ U , the map ϕg : R × M → M, ϕg (t, x) = (t g)x.

4.4 Equivariant Differential Forms

137

Following that, if ω is a cocycle of i (M), we then have g ∗ ω − g0∗ ω = d



1

 β(t, g) dt ,

(4.40)

0

with g0 := φ(0) and where β(t, g) denotes a family of elements of i−1 (M) depending smoothly on (t, g) ∈ R × U , i.e. for any (i − 1)-tuple (χ1 , . . . , χi−1 ) of vector fields over M, the following map is smooth: R × U × M # (t, g, x) → β(t, g, x)(χ1 (x), . . . , χi−1 (x)) ∈ R . We now come to a key point. If, in addition, we have a compactly supported function ρ : U → R, then, for any top degree form dg on G, we have  G

ρ(g) g ∗ ω dg =

   ρ(g) g ∗ ω − g0∗ ω dg + ρ(g) dg g0∗ ω G G

  1    =d ρ(g) β(t, g) dg + ρ(g) dg g0∗ ω



G 0

G

1 where G 0 ρ(g) β(t, g) dg is a smooth differential form over M. But, as we already show that g0∗ ω ∼ ω, since G is connected, we may conclude that  G

ρ(g) g ∗ ω dg ∼



 ρ(g) dg ω , G

which is satisfied by any compactly supported function ρ : G → R whose support is contained in any open subset of M diffeomorphic to RdG . If we now make use of the fact that G is compact (which we haven’t done so far), we can choose the measure dg to be G-invariant such that G dg = 1, and we can fix a smooth partition of unity {ρi } subordinate to a finite good cover of G (fn. (15 ), p. 28). Then, for any cocycle ω ∈ Z i (M), we have *(ω) :=

 G

g ∗ ω dg = ∼



  i G

G

∗ i ρi (g) g ω dg =



ρi (g) dg ω =

  i G



G

i

ρi (g) g ∗ ω dg 

(4.41)

ρi (g) dg ω = ω ,

and *(ω) is then obviously a G-invariant cocycle, thus completing the proof that B i (M)|Z i (M) as G-modules. If we denote by |−| the support of a differential form, we see in the preceding lines that for t ∈ [0, 1] and g ∈ G, we have 

|β(t)| ⊆ γ ([0, 1]) · |ω|

in (4.39),

|ρ(g)β(t, g)| ⊆ ([0, 1] |ρ|)|ω| in (4.40).

138

4 Equivariant Background

Therefore, if |ω| is compact, the arguments show that *(ω) − ω is in fact the differential of a compactly supported differential form, i.e. we have also proved that Bci (M)|Zci (M), which ends the proof of the fact that the complexes of G-modules ((M), d) and (c (M), d) are G-split (2) After (1), the complex C is G-split and since G acts trivially on h(C) and its action on S(g ∨ ) stabilizes each S n (g ∨ ), which is finite dimensional, we can apply Proposition 4.2.5.5. The inclusions S n (g ∨ )G ⊗ C ⊇ S n (g ∨ )G ⊗ C G n ∨ G ⊆ the same for S ≥m (g ∨ ) = (S (g n ) ⊗∨ C) are then quasi-isomorphisms, and 0 ∨ n≥m S (g ). The converse is clear since C = S (g ) ⊗ C is the cokernel of the inclusion S ≥1 (g ∨ ) ⊗ C ⊆ S ≥0 (g ∨ ) ⊗ C. Details are left to the reader.   Corollary 4.4.6.2 Let G be a compact connected Lie group.22 1. The correspondence M  ((M), d, θ, ι), f  f ∗ is a contravariant functor from the category of G-manifolds into the category of G-split g-complexes. 2. The correspondence M  (c (M), d, θ, ι), f  f ∗ is a contravariant functor from the category of G-manifolds and proper maps to the category of G-split g-complexes. 3. The correspondence that assigns to a G-manifold M, the spectral sequence IE G (M) associated with the Cartan complex (G (−), dG ) (4.3.3.1), is contravariant and functorial on the category G- Man. With a G-equivariant map f : M → N , the correspondence associates a morphism of spectral sequences IE G (f ) : IE G (N ) → IE G (M) , which converges to the pullback f ∗ : HG (N ) → HG (M) and which reads at the IE 2 page level as the homomorphism idS(g ∨ ) ⊗ f ∗ : ⎧ p,q p+q ⎪ IE G (N )2 = S p (g ∨ )G ⊗ H q (N) ⇒ HG (N ) ⎪ ⎪ ⎨ ⏐ ⏐ ⏐ ⏐ IE G (f ) id f ∗ f ∗ ⎪ ⎪ ⎪ ⎩ p,q p+q IE G (M)2 = S p (g ∨ )G ⊗ H q (M) ⇒ HG (M) 4. The correspondence that assigns to a G-manifold M, the spectral sequence IE G ,c (M) associated with the Cartan complex (G,c (−), dG ) (4.3.3.1), is contravariant and functorial on the category G- Manpr . With a proper G-equivariant map f : M → N , the correspondence associates a morphism of spectral sequences IE G,c (f ) : IE G,c (N ) → IE G,c (M) ,

22 On

the connectedness hypothesis see 4.4.6.3-(1).

4.5 Cohomological Properties of Homotopy Quotients

139

which converges to the pullback f ∗ : HG,c (N ) → HG,c (M) and which reads at the IE 2 page level as the homomorphism idS(g ∨ ) ⊗ f ∗ : ⎧ p,q q p+q IE (N )2 = S p (g ∨ )G ⊗ Hc (N ) ⇒ HG,c (N ) ⎪ ⎪ ⎪ G,c ⎨ ⏐ ⏐ ⏐ ⏐ IE G,c (f ) id f ∗ f ∗ ⎪ ⎪ ⎪ ⎩ p,q q p+q IE G,c (M)2 = S p (g ∨ )G ⊗ Hc (M) ⇒ HG,c (M) Proof The statements are simple consequences of Theorems 4.4.6.1 and 4.3.3.1 interchanging g and G, and Propositions 4.3.4.3 and 4.3.4.3. For the statements (3,4), the only point to check is that the induced morphisms f ∗ : (ΩG,? (N)g , dG ) → (ΩG,? (M)g , dG ) respect the filtration by S(g ∨ )-degrees, which is obvious.   Comments 4.4.6.3 1. In Theorem 4.4.6.1 and its Corollary 4.4.6.2 connectedness hypothesis on G can be weaken to require simply the action on C of every g ∈ G to be homotopic to the identity map. This occurs in particular when, G being connected, we are interested in the K-equivariant cohomology HK (M) of a G-manifold, where K is a closed subgroup of G, connected or not. 2. In view of Exercise 4.3.4.2, we can even completely delete the hypothesis of connectedness of G in Corollary 4.4.6.2, except that in the statements concerning the spectral sequences, will have to write: p,q

IE G,? (M)2

 q p+q = S p (g ∨ )g ⊗ H? (M) W ⇒ HG,? (M)

where W := G/G0 .

4.5 Cohomological Properties of Homotopy Quotients 4.5.1 Local Triviality of G-Spaces23 We recall the well-known concept of G-space in order to fix notations and terminology. Given a topological group G, a left G-space is a topological space X together with a homomorphism of groups ρ : G → Homeom(X) such that the map G × X → X, (g, x) → g · x := ρ(g)(x) is continuous. The G-orbit of x ∈ X is the set G · x := {g · x | x ∈ X}. Two orbits are then either equal or disjoint, so that the

23 A

reference for this section is Hsiang [55] §1.2 Generalities of Fibre Bundles and Free G-Spaces and §1.3 The Existence of Slice and its Consequences on General G-Spaces, p. 6–12.

140

4 Equivariant Background

Fig. 4.3 A local slice

 orbit decomposition X = x∈X G · x can be rewritten as a partition parametrized by the set X/G of G-orbits. Endow X/G with the quotient topology and denote by ν : X → X/G ,

(4.42)

the projection map which associates with x ∈ X, its G-orbit ν(x) := G · x.

4.5.2 Slices When the group G acts freely on X we say that X is a free G-space. A classical question is then to find conditions under which (4.42) is a left principal G-bundle, which means, by definition, that there exists an open cover U := {Ui }i∈I of X/G, and, for each i ∈ I, a trivializing homeomorphism

which is G-equivariant if we endow G × Ui with the obvious left G-action defined by g · (h, z) := (gh, z), for all g, h ∈ G and z ∈ Ui . A section of ν above Ui , for example σi (z) := Φi−1 (1G , z), is called a slice (see Fig. 4.3). To give a structure of topological left principal G-bundle is therefore equivalent to giving a family of slices {σi : Ui → X}i∈I such that the G-equivariant maps Ψi : G × Ui → X ,

mi (g, z) := g · σi (z) ,

are open embeddings, for all i ∈ I. All these concepts can be given, mutatis mutandis, for right actions in which case we obtain the notion of right principal G-bundle. We can also work with G-manifolds simply by adding the usual differentiability conditions in the constructions.

4.5 Cohomological Properties of Homotopy Quotients

141

4.5.3 Existence of Slices This question has a positive answer when G is a compact Lie group acting on a manifold M.24 In that case differentiable local slices always exist, and, beyond being mere locally closed subspaces, they are locally closed submanifolds, which allows us to endow the quotient topological space M/G of a canonical structure of manifold (see 4.5.3.1-(1)). The idea of the construction as it appears in the paper by Koszul [69],25 consists of endowing M with a G-invariant Riemannian metric (. , .)M (possible since G is a compact Lie group and the action is differentiable),26 and then defining a slice at a point x ∈ M as the image by the exponential expx : Tx (M) → M of a small open ball U centered at the origin of the normal space Nx (G · x) ⊆ Tx (M). Compactness of G · x and the fact that G acts by isometries then allow to show that U can be chosen (small enough) so that the map G×U →M,

(u, z) → g · exp(z) ,

is a diffeomorphism, G-equivariant by construction. The map expx : U → M is then a slice centered at x defined by the metric (. , .)M . In this construction slices are completely determined by the Riemannian metric. The observation is particularly useful when considering an equivariant closed embedding of G-manifolds N → M. In that case, a G-invariant Riemannian metric (. , .)N in N can always be extended (with the help of partitions of unity) to a Riemannian metric (. , .)M in M, which can then be chosen to be G-invariant by Gaveraging, hence without modifying (. , .)N . As a consequence, if N and M are free G-manifolds, the differentiable slices of (N, (. , .)N ) can be extended as Riemannian submanifolds of differentiable slices of (M, (. , .)M ). This shows, in particular, that the induced map N/G → M/G is a closed embedding of manifolds. We gather all these observations in a single proposition. Proposition 4.5.3.1 (Existence of Differentiable Slices) Let G be a compact Lie group. (G acts either on the left or on the right.) 1. For every free G-manifold M, the topological quotient M/G admits a canonical structure of manifold such that the projection map ν : M → M/G is a principal G-bundle. Local sections of ν are then slices. 24 Koszul

[69] and Gleason [45] for M a completely regular topological space.

25 Besides the work of Gleason and Koszul, a number of papers on the same subject were published

at about the same time. Among the most cited are those of Montgomery and Yang [77], Mostow [78] and Palais [79], all in 1957. 26 See Tu [91] §25.4 Equivariant Tubular Neighborhoods, Theorem 25.11, p. 210.

142

4 Equivariant Background

2. Given a G-equivariant closed embedding N → M of free G-manifolds, the induced diagram N νN

N/G

M νM

(4.43)

M/G

is a morphism of principal G-bundles and a Cartesian diagram where horizontal arrows represent closed embeddings of manifolds. Furthermore, every principal G-bundle connection for νN extends to a principal G-bundle connection for νM . Proof (1) We must first check that the manifold structure is independent of the choice of the invariant Riemannian metric. Given two slices S1 and S2 meeting the same orbit G · x ⊆ M and small enough such that G · S1 = C · S2 , we must show that the map Φ : S1 → S2 which associates with z ∈ S1 the only Φ(z) ∈ S2 such that Φ(z) ∈ G · z, is a differentiable map, in which case the map id : ν(S1 ) → ν(S2 ) is a diffeomorphism and the manifold structures thus defined on the set ν(Si ) ⊆ M/G coincide. To see this, we return to the definition of slices which we can assume contained in the same tangent space Tx (M). We have two G-invariant Riemannian metrics (. , .)i on M, the corresponding exponential maps expi : Tx (M) → M, and, by definition, the slices Si := expi (Ui ), where Ui is a small ball centered at the origin of the orthogonal complement Ni of the tangent space the orbit Tx (G · x) at x, for each of the two induced Euclidean metric on Tx (M). We thus have decompositions Tx (M) = Tx (G · x) ⊕ N1 = Tx (G · x) ⊕ N2 , and the map id : ν(S1 ) → ν(S2 ) reads, through the exponential maps, as the map θ : N1 → N2 such that, for z ∈ N1 , we have θ (z) − λ(z) z ∈ Tx (G · x) for some λ(z) ∈ R>0 , and such that exp1 (z) and exp2 (θ (z)) are in the same G-orbit (see Fig. 4.4). Elementary reasons then show that the function λ : N1 → R is differentiable, which implies that θ is also differentiable . We have proved that given a slice S := expx (U ) in a free G-manifold M, the map ϕS : ν(S) → N  RdM −dG , defined by the inverse of ν ◦ expx : U → ν(S), is a chart (ν(S), ϕS ) of M/G, such that, for every x  ∈ G · S, the charts (ν(S  ), ϕS  ) defined by slices S  centered at x  , all define the same structure of manifold on the neighborhood of ν(y). In other terms, the collection of charts A := {(ν(S), ϕS )}, indexed by the slices S ⊆ M, is a canonical atlas of differentiable manifold for the topological quotient space M/G. (2) The preliminary discussion already explained how to extend slices from N to M. The canonicity of the associated manifold structures on N/G and M/N, established in (1), then immediately justifies the existence of the commutative diagram of closed embeddings (4.43).

4.5 Cohomological Properties of Homotopy Quotients

143

Fig. 4.4 Exponential map

To show that (4.43) is Cartesian, it suffices to note that, after (1) we can endow N with a Riemannian metric induced by M, in which case if Sx is a slice of M centered at x ∈ N, then Sx := Sx ∩ N is automatically a slice of N . The restriction of (4.43) to the open subspace νM (Sx ) ⊆ M/G is then easily seen to be isomorphic to the diagram of trivial principal G-bundles G × Sx

G × Sx

p2

p2

(4.44)

Sx

Sx

which is clearly Cartesian. We can now address the question of the existence of extensions of principal Gbundles connections. A principal G-bundle connection for νN is a G-invariant 1form fN ∈ (1 (N) ⊗ g)G such that ι(Y )(fN ) = Y ,

∀Y ∈ g .

When N is the trivial principal G-bundle G × Rd , denote by p1 : N → G the projection p1 (g, x) := g. We have the decomposition   1 (N) ⊗ g := 0 (N ) · p1∗ (1 (G)) ⊗ g ⊕ 0 (N ) ⊗ 1 (Rd ) ⊗ g and since 1 (G) ∼ = 0 (G) ⊗ g ∨ (cf . 4.4.2–(4.24)), we can conclude that fN =

 i

p1∗ (ei ) ⊗ ei + ω ,

(4.45)

for some ω ∈ Ω 0 (N ) ⊗ Λ1 (Rd ) ⊗ g, where {ei } is a basis of g of dual basis {ei }.

144

4 Equivariant Background

Call an embedding N ⊆ M Euclidean if it is isomorphic to a canonical  embedding G×Rd ×{0} ⊆ G×Rd ×Rd . In that case to extend the connection fN to M, we can begin extending the term ω in (4.45) to M, which amounts to extending Ω 0 (N) to Ω 0 (M). This can be done in several ways, for example if q : M → →N q:M→ → N is the projection q(g, x, y) = (g, x, 0), we can set fM :=

 i

 ∗ ∗ p1∗ (ei ) ⊗ ei + g (q (ω)) dg , G

and the equality fN = fM N is clear. When N ⊆ M is a general closed embedding of free G-manifolds, we know, after (4.44), that the local view corresponds to an Euclidean embedding. We can therefore state that a principal G-bundle connection fN on N admits local extensions. The idea then is to consider a covering U := {Ui }i∈I of M/G by trivializing open subspaces for νM of two kinds, either Ui ∩ N = ∅, in which case −1 we set fM,i to be any connection on νM (Ui ), or Vi := Ui ∩ N = ∅ and the embedding Vi ⊆ Ui is Euclidean, in which case we set fM,i to be any extension of fN . Then, if {φi }i∈I is a partition of unity subordinate to U , the G-invariant differential form 

  g∗ φi fM,i ) dg ∈ 1 (M) ⊗ g G , fM := i

G

 

is a well-defined extension of fN . Corollary 4.5.3.2 Let G be a compact Lie group.

1. Given a sequence of G-equivariant closed embeddings of free G-manifolds M0 → M1 → · · · → Mn → · · · , the induced diagram M0 ν0

M0 /G

M1

···

ν1

M1 /G

Mn

···

lim-ind νn

νn

···

Mn /G

lim n Mn

···

( lim n Mn )/G

is a sequence of closed embeddings of principal G-bundles. Furthermore, there exists a family {fn }n∈N , where fn ∈ (1 (Mn ) ⊗ g)G is a principal G-bundle connection, such that fn+1 Mn = fn , for all n ∈ N. 2. Let IE be a right free G-manifold with quotient manifold IB := IE/G and projection map νIE : IE → IB. For every left G-space X, the diagonal action of G on IE × X, g · (z, x) := (z · g −1 , g · x), is a free action. Denote by IE ×G X the quotient space and by ν(IE×X) : IE × X → IE ×G X be the corresponding projection map.

4.5 Cohomological Properties of Homotopy Quotients

145

Then, the diagram IE × X

ν(IE×X)

IE ×G X

p1

p1

IE

νIE

p1 (z, x) := z p1 ([z, x]) := [z] ,

with

(4.46)

IB

where we denote [z, x] := ν(IE×X) (z, x) and [z] := νIE (z), is a Cartesian diagram of fiber bundles of vertical fiber X and horizontal fiber G. 3. The correspondence IE ×G (−) : G-Top  TopIB which associates with a left G-space X the fiber bundle (IE ×G X, IB, p1 , X), and with a G-equivariant map α : N → M, the map IE ×G (α) : IE ×G N → IE ×G M ,

[z, x] → [z, f (x)] ,

is a covariant functor whose restriction to G-Man takes its values in the subcategory FibIB (cf. 3.1.8). Proof (1) Corollary of 4.5.3.1-(2). The only new point to check is that the natural map lim-indn (Mn /G) → (lim-indn Mn )/G is a homeomorphism (see (28 ) p. 146). Since the map is clearly continuous and bijective, we need only justify that it is closed. This means that if Z ⊆ (lim-indn Mn )/G is such that its restrictions ˜ Zn := Z ∩ (Mn /G) are all closed, then Z is of the form Z = lim-indn νn (Z), ˜ where Z ⊆ lim-indn Mn is closed and G-stable. The only possibility is then to take Z˜ = lim-ind Z˜ n with Z˜ n := νn−1 (Zn ), which clearly fulfills the requirements. The existence of principal bundle connections is a well-known fact, so that a connection f0 exists and we can apply 4.5.3.1-(2) to justify, by induction on n ∈ N, the existence of the family of extensions {fn }n∈N . (2) The diagonal action of G on IE × X is free since it is so on IE, we can then apply 4.5.3.1-(1) to both projection maps νIE and p1 . If U ⊆ IB is a common trivializing open subspace, the diagram (4.46) becomes (U × G) × X

ν(IE×X)

p1

U ×G

(U × G) ×G X p1

νIE

U

U ×X p1

U

146

4 Equivariant Background

where the outer diagram is Cartesian with the announced fibers. The diagram (4.46) is thus locally Cartesian, hence Cartesian by 3.1.6.1–(2). Statement (3) is clear.

 

4.6 Constructing Classifying Spaces27 Let G be a compact Lie group. Among the different constructions of a universal fiber bundle IEG for G the two most frequently cited are the following.

4.6.1 The Milnor Construction Here IEG is the inductive limit28 of a system of closed embeddings {IEG(n) → IEG(n+1)}n∈N , where IEG(n) is the join G ∗ · · · ∗ G on (n+1) copies of G, equipped with the (free) diagonal right action of G. As a topological space, IEG(n) is compact, oriented and (n−1)-connected.29,30 Moreover, as soon as we fix a triangulation of G, the spaces IEG(n) inherit a canonical structure of CW-complex. The embeddings IEG(n) → IEG(n+1) are then cellular maps, and the inductive limit IEG := lim-indn IEG(n) is a CW-complex, which is then contractible since it is n-connected for all n ∈ N.31 The classifying space of G is then the topological quotient IBG := IEG/G, limit of the inductive system defined by the closed embeddings IBG(n) → IBG(n+1), where IBG(n) := IEG(n)/G.32 Although the Milnor construction has undeniable advantages,33 the fact that it uses CW-complexes which are not manifolds makes it unsuitable for the purposes

27 See

also Tu [91] Appendix §A.10 Approximation of IEG, p. 271. an inductive system of topological spaces {fj,i : Xi → Xj }i(j , the topology of X∞:= lim-indi Xi is the smallest topology such that the maps fi : Xi → X∞ are continuous, i.e. Z ⊆ X∞ is closed (resp. open) if and only if fi−1 (Z) ⊆ Xi is closed (resp. open), for all i. 29 This is Lemma 2.3 in Milnor [75, p. 432]. 30 A space X is said to be n-connected if its first n homotopy groups are trivial, i.e. if we have πi (X) = 0 for i ≤ n. By Hurewicz Theorem, an n-connected space X is n-acyclic over any field k, which means that the n first k-Betti numbers of X are those of a singleton {•}. See Hatcher [53] ch. 4.1 Homotopy Theory, Theorem 4.32, p. 366. 31 A topological space which is n-connected for all n ∈ N is called weakly contractible. The celebrated Whitehead’s Theorem establishes that a CW-complex which is weakly contractible is contractible. See Hatcher [53] ch. 4.1 Homotopy Theory, Theorem 4.5, p. 346. 32 See Husemöller [56] ch. 4, §11–13, pp. 54–60, for a thorough discussion of Milnor’s construction of the universal and of the classifying bundles. Other references are Husemöller et al. [57] ch. 7, pp. 75–81, as well as the original article of Milnor [75]. 33 It is functorial on the category of groups and greatly simplifies the proof of the classification theorem of principal G-bundles, see Hussemöller [56] Theorem 12.2, p. 57. 28 Given

4.6 Constructing Classifying Spaces

147

of the remainder of this chapter, which is to compare the equivariant de Rham cohomology of G-manifolds with the ordinary cohomology of their homotopy quotients. For this reason we prefer the following alternative classical construction of universal bundles for compact Lie groups.

4.6.2 Stiefel Manifolds Based on the fact that a compact group G can be embedded in the group of orthogonal matrices O(r) for r big enough,34 we consider, for all n ∈ N, the Stiefel manifold IEG(n) := Vr (Rr+n ), which consist of all the orthonormal rtuples (v1 , . . . , vr ) of vectors in the Euclidean space Rr+n . By identifying Rr+n := Rr+n × {0} ⊆ Rr+n+1 , Stiefel manifolds become naturally nested, and we can write IEG(n) ⊆ IEG(n+1). The sequence IEG(0) ⊆ IEG(1) ⊆ IEG(2) ⊆ · · · ⊆ IEG(∞) := IEG ,

(4.47)

is then an equivariant sequence of compact, connected, oriented G-manifolds. The classical way to understand the Stiefel manifold Vr (Rr+n ) is as homogeneous space. Indeed, the group O(r+n) acts on the left of Vr (Rr+n ) by its standard representation on Rr+n . The action is easily seen to be transitive, and the stabilizer of the r-tuple (e1 , . . . , er ), canonical basis of Rr ⊕ {0} ⊆ Rr × Rn , is the subgroup {1r } × O(n) ⊆ O(r+n). We can thus write: $ Vr (Rr+n )  O(r+n) , (4.48) {idr }×O(n) which shows that IEG(n) := Vr (Rr+n ) has a natural structure of left O(r+n)space, and right free O(r)-space. Moreover, the inclusions IEG(n) ⊆ IEG(n+1) are compatible with these structures in the obvious way. The description (4.48) is usually used to prove by induction on n ∈ N that the space IEG(n) is (n−1)-connected.35 Furthermore, the manifolds IEG(n) can be equipped with canonical structures of CW-complexes making the embeddings IEG(n) ⊆ IEG(n+1) cellular maps.36 All the remarks on Milnor construction

34 A

corollary of the Completeness Theorem of Peter-Weyl states that every compact topological group has a faithful finite dimensional representation. See Hsiang [55] ch. 1, §1-(D), p. 5. 35 See Husemöller [56] ch. 8 §1–7, Theorem 6.1, p. 95. 36 The quotient G (Rr+n ) := V (Rr+n )/O(r) is the well-known Grassmann manifold, whose r r points parametrize the vector subspaces of dimension r in Rr+n . In Hatcher [54], §1.2 Cell Structures on Grassmannians, p. 31, there is a thorough description of their Schubert decomposition which endows these spaces of a canonical structure of CW-complex. The inclusions Gr (Rr+n ) ⊆ Gr (Rr+n+1 ) are then cellular maps, and Gr (R∞ ) := lim-indn Gr (Rr+n ), which is a realization of the classifying space IBO(r), has a natural structure of CW-complex. By fixing a triangulation of the Lie group O(r), this structure lifts to IEO(r).

148

4 Equivariant Background

then apply and the inductive limit IEG := lim-indn IEG(n) is a contractible CWcomplex endowed with a right free action of O(r), hence of G. The classifying space of G is then the topological quotient IBG := IEG/G, and is also the inductive limit of the family of closed embeddings (4.5.3.2-(1)) IBG(0)

βn

···

IBG(n)

IBG(n+1)

···

IBG(∞) := IBG (4.49)

where the IBG(n)’s are now nice compact manifolds, oriented and furthermore simply connected if G is connected.

4.6.3 Convention From now on, the notations IEG(.) and IBG(.) will refer to the realization of these spaces based on Stiefel manifolds unless otherwise stated.

4.7 The Borel Construction 4.7.1 The Homotopy Quotient Functor Let G be a compact Lie group. As already explained in Sect. 4.1.2, given a Gmanifold M, instead of considering the topological quotient M/G, which generally lacks of good properties when the action of G is not free, one replaces M by the G-space IEG×M on which G acts by the diagonal action, i.e. g · (z, x) := (z · g −1 , g · x). One thus replaces the G-space M by the homotopy equivalent Gspace IEG × M, on which the action of G is free. The topological quotient space, called the homotopy quotient of M, and sometimes the Borel Construction of M, is denoted by (cf . Sect. 4.1.2.1): MG := IEG ×G M := (IEG × M)/G . The natural map: πM

MG := IEG ×G M −→ → IEG/G =: IBG [z, x]

 −−→

[z]

defines a fiber bundle (MG , IBG, πM , M), after 4.5.3.2-(2).

4.7 The Borel Construction

149

Furthermore, if f : M → N is a G-equivariant map, then the induced map fG : MG → NG ,

[z, m] → [z, f (m)] ,

is a well-defined morphism in the category FibIBG , after 4.5.3.2-(3). The correspondence thus defined, M  MG , f  fG , from the category of G-manifolds to the category FibIBG , is clearly covariant and functorial. Definition 4.7.1.1 The functor (−)G : G-Man  FibIBG ,

M  MG , f  fG ,

is the homotopy quotient functor, or Borel construction functor. Exercise 4.7.1.2 Recall that the connected component G0 of the identity element e ∈ G is a normal subgroup of G and that the quotient space W := G/G0 is a discrete group. 1. Show that for every G-manifold M, the group W acts naturally on MG0 and the canonical surjection (M) : MG0 → → MG then induces a homeomorphism (M) : MG0 /W  MG which is functorial for M ∈ G-Man. 2. Show that if W is finite (e.g. G is compact) and if k is a field of characteristic prime to |W |, then there exists a canonical isomorphism of functors H ((−)G ; k)  H ((−)G0 ; k)W .

4.7.2 On the Cohomology of the Homotopy Quotient By Corollary 4.5.3.2, the space MG is the inductive of the manifolds MG (n) := IEG(n) ×G M, for which we denote by ν(n,M) : IEG(n)×M → → MG (n) the corresponding projection map. We have therefore, for each n ∈ N, a Cartesian diagram (after Exercise 3.1.4.3) of fiber bundles of manifolds IEG(n) × M

ν(n,M) [G]

[M] id×cM

IEG(n) × {•}

MG (n) [M] πn

ν(n,{•}) [G]

(4.50)

IBG(n)

where fibers are shown in brackets. These constructions, especially diagram (4.50), are functorial in both entries, IEG(n) in the category of right free G-spaces, and in M in the category of left G-manifolds. In particular, when applied to the G-equivariant sequence IEG(0) ⊆ IEG(1) ⊆ IEG(2) ⊆ · · · ⊆ IEG(n) ⊆ · · · ⊆ IEG(∞) := IEG ,

150

4 Equivariant Background

we obtain the inductive system of fiber bundles with fiber M ···

μn

MG (n) [M]

···

MG (n+1) [M]

πn

IBG(n+1)

MG (∞) [M]

πn+1

βn

IBG(n)

····· ·····

π∞

IBG(∞)

MG [M]

π

(4.51)

IBG

where the horizontal arrows are closed embeddings. We are thus lead to consider the projective systems of complexes of AlexanderSpanier cochains with coefficients in a field k ⎧ ⎪ ⎪ (i) Ω(IBG; k) → ⎪ ⎪ ⎨ (ii) Ω(MG ; k) → ⎪ ⎪ ⎪ ⎪ ⎩ (iii) Ωcv (MG ; k) →

βn∗   lim n∈N Ω ∗ (IBG(n+1); k) − → Ω ∗ (IBG(n); k) ←− μ∗n   (4.52) lim n∈N Ω ∗ (MG (n+1); k) − → Ω ∗ (MG (n); k) ←− ∗ μ  ∗  n ∗ Ωcv (MG (n+1); k) − lim → Ωcv (MG (n); k) ←−n∈N

where the horizontal arrows are the restriction morphisms (see Exercise 4.7.2.1). (Notice that the first line is a particular case of the second since IBG = {•}G . ∗ (M (.); k) by Ω ∗ (M (.); k), Also notice that in the third line we can replace Ωcv G G c since IBG(.) is compact.) Exercise 4.7.2.1 Show that in (4.52), the morphisms (i,ii) are always isomorphisms, while (iii) is injective and generally not surjective. ( , p. 349) Theorem 4.7.2.2 Let G be a compact, connected, Lie group, and let M be a Gmanifold. (Cohomology is Alexander-Spanier’s with coefficients in a field k.) In the following, n ∈ N and m ∈ N ∪ {+∞}, are such that n ≤ m. We denote by μm,n : MG (n) → MG (m) and βm,n : IBG(n) → IBG(m) the maps defined by the projective system (4.51). 1. Let IE(πm∗ ) ⇒ H (MG (m)) denote the Leray spectral sequence associated with πm : MG (m) → IBG(m). The Cartesian diagram MG (n)

μm,n

[M] πn

IBG(n)

MG (m) [M] πm

βm,n

(4.53)

IBG(m)

induces a morphism of associated Leray spectral sequences which reads at the ∗ ⊗ id∗ . IE 2 pages level as the homomorphism βm,n M

4.7 The Borel Construction p,q

IE(πm∗ )2

151

= H p (IBG(m)) ⊗ H q (M) ∗ βm,n

p,q

IE(πn∗ )2

H p+q (MG (m)) μ∗m,n

id∗M

= H p (IBG(n)) ⊗ H q (M)

H p+q (MG (n))

2. Let IE(πm! ) ⇒ Hcv (MG (n)) denote the Leray spectral sequence associated with πm : MG (m) → IBG(m). The Cartesian diagram (4.53) induces a morphism of associated Leray spectral sequences which reads at the IE 2 pages level as the ∗ ⊗ id∗ . homomorphism βm,n M p,q

IE(πm! )2

q

p+q

= H p (IBG(m) ⊗ Hc (M) ∗ βm,n

p,q

IE(πn! )2

Hcv (MG (m)) μ∗m,n

id∗M q

= H p (IBG(n)) ⊗ Hc (M)

p+q

Hcv (MG (n))

3. Given i ∈ N, the projective systems (4.52) induce isomorphisms 

μ∗m,n

H i (MG (m)) −−−→H i (MG (n)) ,

 μ∗m,n i i Hcv (MG (m))−−−→Hcv (MG (n)) , 

∀m ≥ n > i .

(4.54)

4. The morphisms (4.52) are quasi-isomorphism and induce isomorphisms  ⎧ (i) H (IBG) −→ h lim n Ω(IBG(n)) −→ lim n H (IBG(n)) , ⎪ ⎪   ←− ←− ⎪ ⎨  (ii) H (MG ) −→ h lim n Ω(MG (n)) −→ lim n H (MG (n)) , (4.55)   ←− ←− ⎪ ⎪  ⎪ ⎩ (iii) H (M ) −→ h lim Ω (M (n)) −→ lim H (M (n)) . cv G G   ←−n cv ←−n cv G 5. Statements (3,4) are true when G is not connected and the characteristic of the underlying field k is prime to the cardinality of G/G0 (cf. Exercise 4.7.1.2-(2)). Proof (1, 2) Both statements are related to the functoriality of Leray spectral sequences, which we first justify. Lemma A Let (E  , B  , π, M) and (E, B, π, M) be fiber bundles and let E

μ

E π

π

B

β

(4.56)

B

be a Cartesian diagram, with μ : E  → E and β : B  → B closed embeddings.

152

4 Equivariant Background

A.1. For any sheaf F ∈ Sh(B  ; k), Godement’s resolution F → C (F) commutes with the direct image functor β∗ : Sh(B  ; k)  Sh(B; k), i.e. we have a natural identification   β∗ F → C (β∗ F) = β∗ F → C (F) .

A.2. The natural restriction morphism of complexes of sheaves of AlexanderSpanier cochains Ω ∗E → β∗ (Ω ∗E  ) induces morphisms of bicomplexes π∗ Ω ∗E

β∗ π∗ Ω ∗E

π! Ω ∗E

β∗ π! Ω ∗E

C (π∗ Ω ∗E )

β∗ C (π∗ Ω ∗E )

C (π! Ω ∗E )

β∗ C (π! Ω ∗E )

(4.57)

where the vertical arrows are flasque resolutions is Sh(B; k). Proof of Lemma (A.1) The main point is that, since β is a closed embedding, the direct image β∗ preserves stalks (hence is exact). The identification C0 β∗ F = β∗ C0 F is then almost tautological after the definition of Godement’s flasque resolution.37 The exactness of the functors β∗ (−) and C0 (−), then identifies the cokernels 0

β∗ F

0

β∗ F

β∗

β∗ C0 F

β∗ coker

0

C0 β∗ F

coker(β∗

0

on which the Godement Resolution procedure can be iterated. (A.2) By definition of Alexander-Spanier cochains,38 the set Γ (U ; π∗ Ω dE ) consists of the set-theoretic maps f : π −1 (U )d+1 → k, and Γ (U ; β∗ π∗ Ω dE  ) of f  : π −1 (β −1 (U ))d+1 → k. A map f ∈ Γ (U ; π∗ Ω dE ) composed with the embedding β d : (U ∩B  )d → U d then defines a map f  ∈ Γ (U ; β∗ π∗ Ω dE  ). This correspondence f → f  is easily seen to be a morphism of complexes, it is the announced morphism π∗ Ω dE → β∗ π∗ Ω dE  . The morphism for proper supports π! Ω dE → β∗ π! Ω dE  is defined as the restriction of π∗ Ω dE → β∗ π∗ Ω dE  . The fact that is is well-defined results from the properness of μ (we do not even need the diagram (4.56) to be Cartesian), the details are the same as in Proposition 3.1.10.2-(2). 37 See

Godement [46] §4.3 Resolution canonique d’un faisceau, p. 167, or [20] §II.2 The canonical resolution and sheaf cohomology, p. 36. 38 See Godement [46] Ex. 2.5.2, p. 134, or Bredon [20] Alexander-Spanier cohomology p. 24.

4.7 The Borel Construction

153

The diagrams (4.57) then follow applying (A.1) and recalling that the direct image functor β∗ is exact (since β is a closed embedding) which transforms flasque sheaves in flasque sheaves (as do any direct image functor).  We can now prove the statement (1) of the Proposition. Identify the notations of corresponding terms in the two Cartesian diagrams: μm,n

MG (n) [M] πn

MG (m) [M] πm

βm,n

IBG(n)

=

IBG(m)

E

μ

E π

π

B

β

B

By filtering the morphism of bicomplexes    Γ B; C (π∗ Ω ∗E ) → Γ B; β∗ (C (π∗ Ω ∗E  )) = Γ B  ; C (π∗ Ω ∗E  ) by the (regular) decreasing -filtration, we get a morphism of spectral sequences which reads in the IE 1 page as   Γ B; Cp (Hq (π∗ )) → Γ B  ; Cp (Hq (π∗ )) ,

(4.58)

where Hq (π∗ ) denotes the q’th cohomology sheaf of the complex π∗ Ω E , and mutatis mutandis for Hq (π∗ ). These sheaves are a priori locally constant with fiber H (M; k) (3.1.10.4-(2)), but they are also globally constant since the base spaces IBG(.), being the quotients of the simply connected spaces IEG(.) by the free action of a connected group G, are themselves simply connected. As a consequence, the morphism of the IE 2 pages of the associated spectral sequences induced by (4.58) reads like the usual restriction H p (B; H q (M; k)) → H p (B  ; H q (M; k)) .

(4.59)

To finish, we recall that if V is any k-vector space, the exactness of the functor Homk (−; V ) induces an isomorphism SH

p

(B; V ) = Homk (S Hp (B; k), V ) ,

where S H refers to singular homology or cohomology. In the forthcoming proof of statement (3), we will show that the k-Betti numbers of B := IBG are finite, so that we will then be able to write SH

p

(B; H q (M; k)) = Homk (S Hp (B; k), H q (M; k)) = Homk (S Hp (B; k)) ⊗ H q (M; k) = S H p (B; k) ⊗ H q (M; k)

154

4 Equivariant Background

and the statement (1) will thus follow by the well-known equivalence between Singular and Alexander-Spanier cohomologies on CW-complexes.39 The proof of (2) is the same, replacing π∗ by π! and H (M; k) by Hc (M; k). (3) We start with M := {•}, in which case MG (m) = IBG(m). We apply Corollary 4.5.3.2-(2) to IEG × IEG(m), where we must be careful since IEG(m) is, both, a left G-space and a right free G-space (cf . 4.6.2). The action for the identification IBG(m) = IEG(m)/G is the right free action. We thus endow (temporally) IEG(m) with a left action of G by setting g ∗ z := z · g −1 , and we define IEG ×G IEG(m) as in 4.5.3.2-(2), as the space of equivalence classes for the relation (z, x) ∼ (z · g −1 , g ∗x). Since both IEG and IEG(m) are free G-spaces, we can apply the Corollary in two symmetric ways obtaining two locally trivial fibrations IEG ×G IEG(m) [IEG(m)]

[IEG] p1

p2

IBG

IBG(m)

where the fibers are shown in brackets. Notice that the contractibility of fibers of p2 immediately implies that the pullback p∗2 : H (IBG(m)) → H (IEG ×G IEG(m)) is an isomorphism. The study of the restriction H (IBG) → H (IBG(m)) is then equivalent to the study of the pullback morphism p∗1 : H i (IBG) → H i (IEG ×G IEG(m)) ,

(4.60)

associated with the morphism of fiber bundles above IBG p1

IEG×G IEG(m) p1

IBG id [{•}]

[IEG(m)]

IBG for which we know after Lemma A, that p∗1 is the abutment of a morphism of Leray spectral sequences which reads at the IE 2 pages level as the following homomorphism ξ(m) induced by the pullback c∗ : H ({•}) → H (IEG(m)) associated with the constant map c : IEG(m) → {•}: H p (IBG; H q ({•})) ξ(m)p,q

H p (IBG; H q (IEG(m)))

H p+q (IBG) p ∗1

H p+q (IEG ×G IEG(m))

ˇ CW-complexes, Cech, Sheaf, Singular and Alexander-Spanier cohomologies coincide. See Bredon [20], ch. III, Comparison with other cohomology theories, Theorem 2.1, p. 187.

39 On

4.7 The Borel Construction

155

When i = p + q, we conclude that ξ(m)p,q is an isomorphism for all m > i, since IEG(m) is (m−1)-connected. The equivalent to statement (3) for the family (4.60), hence for M = {•}, is thus proved, and, at the same time, this ends the proof of (1) and (2). The statement (3) for general M is now immediate by the spectral sequences ∗ : H i (IBG(m)) → H i (IBG(n)) in (1) and (2) since we now know that the βm,n become isomorphisms as n → ∞. (4) (i) is a particular case of (ii). For (ii) consider the natural diagram of morphisms of graded vector spaces H (MG ) (3)

(1)

h lim n Ω(MG (n)) (2)

lim nH (MG (n)) Thanks to (3), we can immediately claim that (iii) is an isomorphism, and also that, for each i ∈ N, the projective system {H i (MG (n+1)) → H i (MG (n))}n∈N verifies Mittag-Leffler condition.40 Besides, this same condition is also verified at the cochain level, since, the the maps μm,n : M(n) → M(m) being closed embeddings, the restrictions Ω(MG (m)) → Ω(MG (n)) are all surjective maps. When these Mittag-Leffler conditions are satisfied, it is well-known that the cohomology of a projective limit is the projective limit of the cohomologies,41 which immediately implies that (2) is an isomorphism. The fact that (1) is an isomorphism too then results from the equality (3) = (2) ◦ (1). The isomorphisms in (iii) are proved by exactly that same arguments. (5) Statements (3,4) are true for the connected component G0 of the identity element e ∈ G. But then, following Exercise 4.7.1.2 (see also proof of 4.7.3.1), the group W := G/G0 acts naturally on IEG(m)×G0 (N ), for every m ∈ N ∪ {+∞} and every G-manifold N . The complexes Ω(MG0 (m)), Ωcv (MG0 (m)) are W -dg-modules, and all the morphisms are compatible with the action of W . On the other hand, as (|W |, char(k)) = 1, the functor of W -invariants (−)W is exact. This implies that it commutes with cohomology, giving a canonical isomorphism of functors: H ((−)G ; k)  H ((−)G0 ; k)W (cf . 4.7.1.2-(2)), but also that it preserves Mittag-Leffler conditions at complexes and cohomology levels, and that it commutes with projective limits. The statements (3,4) for G then follow.  

projective system of vector spaces {ρn,m : Vm → Vn }m≥n∈N is said to satisfy MittagLeffler condition if for any n ∈ N, the decreasing sequence {ρn,m (Vm )}m≥n of subspaces of Vn is stationary. The condition is trivially verified when the vector spaces Vn are all finite dimensional, and also when Vm → Vn is an isomorphism for n  0. See Kashiwara-Schapira [61] ch. I §1.12 The Mittag-Leffler condition, p. 64, or Weibel [95] §3.5 Derived Functors of the Inverse Limit, Definition 3.5.6, p. 82. 41 See Kashiwara-Schapira [61] Proposition 1.12.4, p. 67, or Weibel [95] Theorem 3.5.8, p. 83. 40 A

156

4 Equivariant Background

4.7.3 Orientability of the Homotopy Quotient Proposition 4.7.3.1 Let G be a compact Lie group and let M be a G-manifold. If M is orientable and the action of G preserves the orientation, then the fiber bundle MG (n) := (MG (n), IBG(n), πM , M) is orientable. This arrives, for example, when G is connected.42 Proof It suffices to consider the case where M is equidimensional. Denote by G0 the connected component of G containing 1G . The connectedness of G0 implies that IBG0 (n) is simply connected, in which case the fiber bundle MG0 (n) := (MG0 (n), IBG0 (n), πM , M) is automatically orientable after Corollary 3.1.7.2-(3c). The quotient W := G/G0 is a finite group which acts naturally on MG0 (n) := IEG(n) ×G0 M by w · [z, x] = [z · w −1 , w · x]. We then have the following diagram of natural morphisms of fiber bundles νM : MG0 (n) [M]

πM

νIB : IBG0 (n)

[W ]

(I) [W ]

MG0 (n)/W [M]

πM

IBG0 (n)/W

MG (n) [M]

πM

IBG(n)

where the subdiagram (I) is Cartesian after Exercise 3.1.4.3. An orientation of the fiber bundle (MG (n), IBG(n), πM , M) over a field k is given by a nowhere vanishing global section of the cohomology sheaf  HdM (π! ) := HdM πM,! Ω (MG0 (n); k) . When M is orientable over k, the fibers of HdM (π! ) are isomorphic to k. The space IBG0 (n), being simply connected, the sheaf HdM (π! ) is the constant sheaf k · ζM , where ζM is the fundamental class of M. We can therefore define a global section o ∈ Γ (IBG0 ; HdM (π! )) by o : x → ζM , which is W invariant since we assume ζM invariant under the action of G. On the other hand, the projection νIB : IBG0 (n) → IBG0 /W , being a group covering, induces an isomorphism νI∗B : Γ (IBG; H(π! )) → Γ (IBG0 ; H(π! ))W , so we can transfer the orientation of the fibers of MG0 to MG (n). When k := R we can use differential forms to show orientability of MG (n). Indeed, by 3.1.7.1 and 3.1.7.2-(3c), there exists ωπ ∈ dM (MG0 (n)) whose restrictions to fibers of πM : MG0 (n) → IBG0 (n) are nowhere vanishing. For all g ∈ G, the differential form g ∗ (ωπ ) defines the same orientation on the fibers of

42 The

same is true for k-orientations over any field k.

4.8 Equivariant de Rham Comparison Theorems

157

πM since we assume G to preserve the orientation of M. As a consequence, the G-average43

  1  ω˜ π := w∗ g ∗ (ωπ ) dg , w∈W |W | G0 still defines the same orientation on the fibers of πM , and is also W invariant. We can then transfer ω˜ π to H dM (MG ) through the pullback homomorphism induced by ∗ : the group covering νM : MG0 (n) → MG (n) which induces an isomorphism νM d d W M M H (MG ) → H (MG0 ) whose restrictions to fibers are nowhere vanishing. Hence the orientation of MG (n), IBG(n), π, M .  

4.8 Equivariant de Rham Comparison Theorems Using Cartan’s method (cf . Sect. 4.1.1), we prove in this section and in Sect. 4.10, three extensions to the classical de Rham Theorem, which compare equivariant cohomologies of a manifold M with corresponding ordinary cohomologies of the homotopy quotient MG . For G compact, not necessarily connected, we give canonical isomorphisms44 (i) HG (M)  H (MG ; R)

(ii) HG,c (M)  Hcv (MG ; R)

(4.61)

(iii) HG,N (M)  HNG (MG ; R) , where (iii) concerns local cohomology, to be discussed in Sect. 4.10. The strategy to prove equivalences (i,ii)-(4.61) follows the approach we used in Sect. 4.7.2, which is based on the fact that IEG = lim-indn∈N IEG(n). In these preliminaries, we assume the compact Lie group G to be connected. Given a G-manifold M, we have, in the category of topological G-spaces, lim n∈N IEG(n)×M [G] ν(n,M)

lim n∈N IEG(n)×G M

= IEG×M

p2

M

[G] νM

(4.62)

= IEG×G M =: MG

where the projection map p2 (z, x) := x is a homotopy equivalence. An important point here is that, in the top row, the spaces IEG(n)×M are free G-manifolds, whether the action on M is free or not.

43 See

Tu [91] §13.2 Integrating over a Compact connected Lie Group, p. 105. (i), see Tu [91], §A.9 Proof of the Equivariant de Rham Theorem in General , p. 269, and Guillemin-Sternberg [50], §2.5 The Equivariant de Rham Theorem, p. 28.

44 For

158

4 Equivariant Background

From (4.62), we deduce the following diagrams of homomorphisms HG (M) (1)

lim n HG (IEG(n)×M)

(4)

(2)

H (MG ; R) (3)

(4.63)

lim n H (IEG(n)×G M; R)

and HG,c (M) (1)

lim n HG,c (IEG(n)×M)

(4)

(2)

Hcv (MG ; R)

(3)

(4.64)

lim n Hc (IEG(n)×G M; R)

where the arrows (3) are the isomorphisms in Alexander-Spanier cohomology of Theorem 4.7.2.2-(4), and the arrows (2) recall the existence, for each n ∈ N, of a canonical isomorphism which we will later make specific (cf . 4.8.1.2). To show that we have canonical isomorphisms (4), the idea is then to show that: Iso 1: Iso 2:

the isomorphisms (2) define an isomorphism of projective systems; for each i ∈ N, the pullback morphism p2∗ : HGi (M) → HGi (IEG(n)×M)

induced by the projection p2 : IEG(n)×M → M, p2 (z, x) := x, is an isomorphism for all n > i. We will address these questions in the following two sections.

4.8.1 Question Iso 1 The notations are those introduced in 4.7.2, where, for every n ∈ N, we considered the Cartesian diagram of locally trivial fibrations of manifolds 4.7.2-(4.50) IEG(n)×M

ν(n,M) [G]

[M] id×cM

IEG(n)×{•}

IEG(n)×G M [M] πM

ν(n,{•}) [G]

(4.65)

IEG(n)×G {•} := IBG(n)

to which we apply the approach of Cartan’s lectures, as recalled in Sect. 4.1.1.

4.8 Equivariant de Rham Comparison Theorems

159

For the sequel, it will be convenient to shrink notations. We set: IEM n := IEG(n)×M ,

IEM n := IEG(n)×G M

and

νn := νn,M

Since νn : IEM n → IEM n is a projection of principal G-bundles, the corresponding pullback morphism νn∗ identifies (since G is connected) νn∗

(IEM n ) −−→ (IEM n )bas 

and, applying Proposition 3.1.10.2-(2), which we can do since the diagram (4.65) is Cartesian, we also have νn∗ (c (IEM n )) = c (IEM n )bas , as G is compact. Hence, we deduce the following commutative diagram c (IEM n )

νn∗ c

IEM n

bas

IEM n

bas

⊕

IEM n )

(4.66)

νn∗

where, we recall, c (IEM n ) and c (IEM n ) denote the complexes of differential forms respectively with πm and (id×cM )-proper supports in (4.65). (Notice that the horizontal arrows are actual isomorphisms and not just quasi-isomorphisms.) Given a connection fn for the principal G-bundle νn : IEM n → IEM n , let  fn : W (g) →  IEMn ,

(4.67)

be the corresponding Weil morphism of g-dga’s and consider the analogue to the diagram 4.1.1.1-(4.3) in the present context, i.e. the commutative diagram of CartanWeil morphisms c (IEMn ))

bas q.i.

(W ( ) ⊗

c (IEMn ))

bas

⊕ n ))

bas q.i.

c (IEMn ))

bas

q.i.

⊕ (W ( ) ⊗

n ))

bas

(4.68) n ))

bas

q.i.

with i(ω) := 1 ⊗ ω and f(α ⊗ ω) := f (α) ∧ ω, hence f ◦ i = id in each row. There are two important remarks about this diagram: – Since (IEMn ) is a g-dg-algebra, we can apply Cartan’s Theorem 4.1.1.1, and state that, in the bottom row, both morphisms i and f are quasi-isomorphisms. – The same is true in the top row, although some care must be taken since c (IEMn ) is not an algebra. In Appendix C, we show that Cartan’s theorem is

160

4 Equivariant Background

more generally true for any differential graded ideal of ((IEMn ), d) stable by gderivatives and g-interior products, all properties clearly satisfied by c (IEMn ), since these operators reduce supports. To compare the diagram (4.68) for different values n, recall that we are working on the inductive system of closed embeddings of fiber bundles (4.5.3.2-(1)): ····

IEM n [G]

····

· · · · IEG×M

IEM n+1 [G]

νn

[G]

νn+1

μn

IEM n

····

IEM n+1

νM

(4.69)

MG

which leads us to consider the diagram n+1 ) n+1

n+1 ) q.i. n+1

n)

(I)

q.i.

W( ) ⊗ n+1

∗ n

bas

)bas

bas

id⊗

n ∗ n

bas

q.i.

W( ) ⊗

(II)

q.i.

∗ n

n)

bas

(4.70)

n

n)

bas

where • the subdiagram (I) is commutative by construction, but • the subdiagram (II) need not be commutative. Indeed, for that we need the family of connections F := {fn }n∈N to be compatible with the family of restrictions {n }n∈N , i.e. we need the diagrams n+1 ) fn+1

∗ n

⊕

n) fn

W( ) to be commutative for all n ∈ N. And this is indeed possible after Corollary 4.5.3.2-(1) which states the existence of such families F, which we call projective system of connections for {IEM n }n∈N (cf . fn. (3 ), p. 110). To finish, notice that since the subdiagrams in (4.69) are Cartesian, we can exchange  ↔ c in (4.70) and still have a commutative diagram with vertical quasi-isomorphisms. We can now state the main result in this section.

4.8 Equivariant de Rham Comparison Theorems

161

Proposition 4.8.1.1 Let G be a compact connected Lie group. For every Gmanifold M, the families {i n }n∈N of Cartan-Weil quasi-isomorphisms: 

 i n : (IEG(n)×G M) → W (g) ⊗ (IEG(n)×M) bas  i n : c (IEG(n)×G M) → W (g) ⊗ c (IEG(n)×M) bas

(4.71)

are morphisms of projective systems from the projective system 

μ∗n

(IEG(n)×G M) −−→ (IEG(n+1)×G M)

 n∈N

to the projective system id⊗n∗    W (g) ⊗ (IEG(n)×M) bas −−−→ W (g) ⊗ (IEG(n+1)×M) bas n∈N

Furthermore, if F :={fn }n∈N is a projective system of connections for the inductive  system IEG(n)×M n∈N (4.69), then the families {f n }n∈N of Cartan-Weil quasiisomorphisms: 

 f n : W (g) ⊗ (IEG(n)×G M) bas → (IEG(n)×G M)  f n : W (g) ⊗ c (IEG(n)×G M) bas → c (IEG(n)×G M)

(4.72)

are also morphisms of projective systems. Proof Since (IEG(n)×G M) = (IEG(n)×M)bas and c (IEG(n)×G M) = c (IEG(n)×M)bas , the statements are just a rewriting of the commutative diagram (4.70) and of the same diagram replacing  with c .   The following Corollary is now immediate. Corollary 4.8.1.2 Let G be a compact connected Lie group. For every G-manifold M, the families {H (i n )}n∈N of Cartan-Weil isomorphisms  H (i n ) : H (IEG(n)×G M) → HG (IEG(n)×M) H (i n ) : Hc (IEG(n)×G M) → HG,c (IEG(n)×M) induce isomorphisms of projective limits  lim H (i ) : lim H (IEG(n)× M) −→ lim H (IEG(n)×M) n G G  ←−n ←−n ←−n lim H (i n ) : lim n Hc (IEG(n)×G M) −→ lim n HG,c (IEG(n)×M)  ←− ←− ←−n Furthermore, if F  := {fn }n∈N  is a projective system of connections for the inductive system IEG(n)×M n∈N , then the corresponding family of Cartan-Weil isomorphisms {H (f n )}n∈N verifies lim H (f n ) := lim n H (i n )−1 . ←− ←−n

162

4 Equivariant Background

Remark 4.8.1.3 The classical de Rham Theorem allows replacing in 4.8.1.2 the de Rham cohomologies of the manifolds IEG(n)×G M, by the corresponding ordinary cohomologies, hence completing the answer of question Iso 1.

4.8.2 Question Iso 2 We apply Corollary 4.4.6.2-(3,4) to the G-equivariant proper map p2 (n) : IEG(n)×M → M , which gives morphisms of spectral sequences converging to the morphisms  p2 (n)∗ : HG (M) → HG (IEG(n)×M) p2 (n)∗ : HG,c (M) → HG,c (IEG(n)×M) p,q

whose terms IE 2

p,q

IE G (p2 (n))2

and

p,q

IE G,c (p2 (n))2

where



are respectively ⎧ p ∨ g p+q ⎪ S (g ) ⊗ H q (M) ⇒ HG (M) ⎪ ⎪ ⎨ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ := id p2 (n)∗  p2 (n)∗  ⎪ ⎪ ⎪ ⎩ p ∨ g p+q S (g ) ⊗ H q (IEG(n)×M) ⇒ HG (IEG(n)×M) ⎧ p ∨ g q p+q S (g ) ⊗ Hc (M) ⇒ HG,c (M) ⎪ ⎪ ⎪ ⎨ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ := id p2 (n)∗  p2 (n)∗  ⎪ ⎪ ⎪ ⎩ p ∨ g q p+q S (g ) ⊗ Hc (IEG(n)×M) ⇒ HG,c (IEG(n)×M)

H q (IEG(n)×M) = q

Hc (IEG(n)×M) =

 

a+b=q

H a (IEG(n)) ⊗ H b (M)

a+b=q

H a (IEG(n)) ⊗ Hcb (M)

But then, if n > (p + q), we have n > a, in which case H a (IEG(n)) = 0 since IEG(n) is n-connected (cf . fn. (30 ), p. 146), and the morphisms IE G (p2 (n))r and IE G,c (p2 (n))r are isomorphisms, for all r ≥ 2. We have thus proved the following analogue to Theorem 4.7.2.2-(3), which answers question Iso 2. Theorem 4.8.2.1 Let G be a compact connected Lie group. For every G-manifold M, given i ∈ N, the homomorphisms  p2 (n)∗ : HGi (M) → HGi (IEG(n)×M) , (4.73) i i (M) → HG,c (IEG(n)×M) , p2 (n)∗ : HG,c

4.8 Equivariant de Rham Comparison Theorems

163

are isomorphisms for all m ≥ n > i. Furthermore, the induced homomorphisms ⎧  limn p2 (n)∗ ⎨ HG (M) −−−−−→ lim HG (IEG(n)×M) ← h lim G (IEG(n)×M) , ←−n ←−n ⎩  limn p2 (n)∗ HG,c (M) −−−−−→ lim n HG,c (IEG(n)×M) ← h lim n G,c (IEG(n)×M) , ←− ←− are isomorphisms. In particular, the natural morphisms 

lim p (n)∗ : G (M) → ←−n 2 lim p (n)∗ : G,c (M) → ←−n 2 are quasi-isomorphisms.

lim  (IEG(n)×M) , ←−n G lim  (IEG(n)×M) , ←−n G,c

(4.74)

Proof The isomorphisms (4.73) are already justified. For the rest, the proof is the same as 4.7.2.2-(4). Consider the natural diagram of morphisms of graded vector spaces (1)

HG (M) (3)

h lim n

G (IEG(n)×M)

(2)

lim n HG (IEG(n)×M) The isomorphisms (4.73), immediately imply that (3) is an isomorphism, and also that {HG (IEG(n+1)×M) → HG (IEG(n)×M)}n∈N verifies Mittag-Leffler condition. We can therefore conclude as in 4.7.2.2-(4), showing that the projective system {G (IEG(n+1)×M) → G (IEG(n)×M)}n∈N verifies M-L condition too. But this is clear since, n : (IEG(n)×M) → (IEG(n+1)×M) being closed embeddings of manifolds, the morphisms id ⊗ n∗ : S(g ∨ ) ⊗ Ω(IEG(n+1)×M) → S(g ∨ ) ⊗ Ω(IEG(n)×M) are surjective, as are also their restrictions to G-invariants   id ⊗ n∗ : S(g ∨ ) ⊗ (IEG(n+1)×M) G → S(g ∨ ) ⊗ (IEG(n)×M) G , as we dispose of the averaging operator



G,

both on S(g ∨ ) and on (−).

 

4.8.3 Equivariant Cohomology Comparison Theorem With the last proposition, the strategy proposed at the preliminary discussion in Sect. 4.8 is fully confirmed. We can therefore state our first comparison theorem.

164

4 Equivariant Background

Equivariant de Rham Theorem 4.8.3.1 Let G be a compact Lie group. For every G-manifold M, we have canonical isomorphisms HG (M)  H (MG ; R)

and

HG,c (M)  Hcv (MG ; R) ,

(4.75)

coming from the following diagrams of canonical quasi-isomorphisms G (M)

lim n

(1)

G (IEG(n)×M) (2)

Ω(MG ; R)

(3)

n

n

(4.76)

lim n Ω(IEG(n)×G M; R)

and G,c (M) (1)

lim n

G,c (IEG(n)×M) (2)

Ωcv (MG ; R) (3)

n

n

(4.77)

lim n Ωc (IEG(n)×G M; R)

where • (1) is lim-projn (p2 (n)∗ ), from Theorem 4.8.2.1. • (2) are lim-projn (i n ) and lim-projn (f n ), from Corollary 4.8.1.2. • (3) comes from Theorem 4.7.2.2-(4). Furthermore, the diagram (4.76) is functorial in the category G-Man and the diagram (4.77) is functorial in the category G-Manpr . Proof For G connected, the theorem is an easy consequence of Theorem 4.7.2.2 together with the conclusions of questions Iso 1, 2 in Corollary 4.8.1.2 and Theorem 4.8.2.1. Functoriality is clear since all the constructions were so. For G non-connected, let G0 be the connected component of the identity element e ∈ G, set W := G/G0 , and use the functorial quasi-isomorphisms 

G (−)  G0 (−)W G,c (−)  G0 ,c (−)W

 and

Ω(−G ; R)  Ω(−G0 ; R)W Ωcv (−G ; R)  Ωcv (−G0 ; R)W

established in Proposition 4.4.5.2 and Exercise 4.7.1.2.

 

Remark 4.8.3.2 Theorem 4.8.2.1 gives us an isomorphism: HG (M) → lim n HG (IEG(n)×M) ←−

(4.78)

4.9 Cohomology of Classifying Spaces

165

where, on the right-hand side, the spaces IEG(n)×M are all free G-manifolds. We can therefore apply Corollary 4.8.1.2 which gives the projective family of isomorphisms   H (f n ) : HG (IEG(n)×M) → H (IEG(n)×G M) n∈N , and, composing (4.78) with lim-projn H (f n ), we obtain a canonical isomorphism HG (M) → lim n H (MG (n)) ←−

(4.79)

which is functorial on M ∈ G-Man. Expression (4.79) can therefore be retained as an alternative definition of equivariant cohomology theory of G-manifolds as limit theory of de Rham cohomology theory on the approximations of the homotopy quotient.

4.9 Cohomology of Classifying Spaces 4.9.1 Canonicity of the Cohomology of Classifying Spaces In Sect. 4.6 we recalled two well-known constructions of universal fiber bundles IE for a given compact Lie group G. The resulting topological spaces share the properties of being contractible CW-complexes (cf . fn. (31 ), p. 146), free G-spaces and such that the projection IE → → IE/G is a locally trivial fibration. In 4.6.3 we decided to work with some fixed universal fiber bundle IEG which is the inductive limit of Stiefel manifolds in order to prove the Equivariant de Rham comparison theorems. However, in practice, we sometimes need to change the choice of the space IEG, and although the classification theorem of principal fiber bundles45 tells us already that different constructions of IBG are homotopy-equivalent, it is interesting to return to Cartan’s remark in the very last lines of [27], on an alternative a priori justification of the canonicity of the cohomology of classifying spaces based on Cartan’s theorem 4.1.1.1 and the equivariant de Rham Theorem (4.8.3.1). Corollary 4.9.1.1 Let G be a compact Lie group. For every weakly contractible Gprincipal fiber bundle IE (cf. fn. (31 ), p. 146), there exists a canonical isomorphism H (IE/G; R)  HG ({•}) = S(g ∨ )G . Proof Let IEG denote the universal fiber bundle inductive limit of Stiefel manifolds (4.6.3). Let G act on the right of IEG×IE by (x, y) · g := (x · g, y · g). We then have the two locally trivial fibrations p1

p2

[IEG]

[IE]

IBG ←−−− (IEG × IE)/G −−−→ IE/G ,

45 See

Steenrod [84], §19.3 Classification Thm., p. 101, or Hussemöller [56] Thm. 12.2, p. 57.

166

4 Equivariant Background

where the fibers are shown in brackets. Since these fibers are weakly contractible, the Leray spectral sequences associated with the projections pi degenerate at the second page, which implies that the pullbacks p∗i are isomorphisms p∗1

p ∗2





H (IBG; k) −−→ H ((IEG × IE)/G; k) ←−− H (IE/G; k) . We can then conclude, applying the isomorphism H (IBG; R)  HG ({•}) = S(g ∨ )G given by the equivariant de Rham theorem (4.8.3.1).  

4.9.2 Formality of Classifying Spaces In Sect. 3.4.1, we recalled the definition of R-formal topological spaces, which, for classifying space IBG, says that there exists a diagram Ω(IBG; R) → (A1 , d) ← (A2 , d) → · · · ← (An−1 , d) → (An , d) ← H (IBG, R) ,

where the arrows represent quasi-isomorphic morphisms of dg-algebras. Theorems 4.8.2.1 and 4.7.2.2-(4) established that the morphisms of dg-algebras 

G ({•}) −→ lim n G (IEG(n)) q.i. ←− Ω(IBG; R) −→ lim n Ω(IBG(n); R) q.i. ←−

(4.80)

are quasi-isomorphism, and we also know that G ({•}) = h(G ({•}))  H (IBG; R) ,

(4.81)

where (=) is simply because G ({•}) = (S(g ∨ ), 0), and () is the canonical isomorphism given by the equivariant de Rham Theorem 4.8.3.1. Therefore, to prove that IBG is R-formal, we need only show that the projective limits in (4.80) can be joined by a zig-zag () of quasi-isomorphic morphisms of dga’s, and, moreover, since the corresponding projective systems verify Mittag-Leffler conditions at the levels both of complexes and cohomology, we need only show (see (41 ), p. 155) that there exists a family of ziz-zags {G (IEG(n))  Ω(IBG(n); R)}n∈N , such that the diagrams

⊕

G (IEG(n)) 



G (IEG(n+1)) −→

Ω(IBG(n+1); R) −→ Ω(IBG(n); R) , where the horizontal arrows are the usual restrictions, are commutative.

4.9 Cohomology of Classifying Spaces

167

For this, the easiest way is to work with sheaves. Indeed, given a manifold M, let  M and Ω M denote the complexes of sheaves on M respectively of de Rham differential forms and of Alexander-Spanier cochains with coefficients in R. Since M is locally contractible, the cohomology sheaves are concentrated in degree 0 where we have H0 ( M ) = H0 (Ω M ) = R M . The two natural morphisms of sheaves of dg-algebras λ

:  M −→ ( M ⊗ Ω M ) ,

s → s ⊗ 1 ,

ρM : Ω M −→ ( M ⊗ Ω M ) ,

s → 1 ⊗ s ,

M

q.i.

q.i.

are then easily seen to be quasi-isomorphic by Künneth’s theorem at stalks level. Applying these considerations to a closed embedding of manifolds ι : N → M, leads us to consider the following commutative diagram: ι∗

M λM

q.i.

M

ΩM

ρM

q.i.

⊕

q.i.

ι∗

⊕

ΩM

N ι∗ (λN )

ΩN)

N q.i.

(4.82)

ι∗ (ρN )

ι∗ Ω N ,

where the horizontal arrows in the top and bottom rows are induced by the usual restrictions of differential forms and of Alexander-Spanier cochains respectively. The arrow in the middle row is then determined by the commutativity of the diagram, which, at the level of presheaves, associates ω ⊗  with ω N ⊗  N . A key point in diagram (4.82) is that, by the acyclicity theorem B.6.3.4, the j j sheaves  iM , Ω M and  iM ⊗ Ω M are all Γ (M, −)-acyclic, the first since it is an  0M -module and the two others, since they are Ω 0M -modules. The same is obviously true for N . Consequently, applying the functor Γ (M, −) to (4.82) gives the following diagram of morphisms of dg-algebras, where the vertical arrows are still quasi-isomorphisms:46

q.i.

λM

Γ (M; ρM

M

q.i.

ΩM) q.i.

Ω(M; R)

Γ (N;

⊕

q.i.

λN

ΩN)

N

(4.83)

ρN

Ω(N; R) .

46 The reader will have recognized in these lines the proof of the classical de Rham’s theorem using

tools of Sheaf Theory.

168

4 Equivariant Background

Furthermore, the horizontal rows are surjective. Indeed, while the surjectivity of the top and bottom arrows are standard features for a closed embedding N ⊆ M, the surjectivity of the middle arrow is a bit more subtle because it comes from the surjectivity of the morphism of sheaves ( N ⊗ Ω N ) → → ι∗ ( N ⊗ Ω N ) in (4.82) 0 -modules, the Ω 0 (obvious at stalks), and the fact that, being a morphism of ΩM M 0 modules, the global section functor Γ (M; −) = GM(Ω M ) → GM(Ω 0 (M; R)), which is an equivalence of categories after B.4.1-(3), preserves its surjectivity. If we now apply (4.83) to M := IBG(n+1) and N := IBG(n), for all n ∈ N, we obtain the morphisms of projective systems: Ω(IBG(n))

n∈N {λ}n

n∈N

{ρ}n

Γ (IBG(n);

IBG(n)

Ω IBG(n) )

(4.84)

n∈N

which satisfies Mittag-Leffler conditions, as in (4.80), hence inducing quasiisomorphic morphisms of dg-algebras at projective limits. We have thus established the existence of a zig-zag of quasi-isomorphic morphisms of dg-algebras: lim (IBG(n))  lim n Ω(IBG(n); R) . ←−n ←−

(4.85)

One last step can be achieved by considering the following sequence of quasiisomorphic morphisms of dg-algebras: νn∗

bas

q.i. q.i.

)

(4.86)

n

IEG(n))bas

Ξn q.i.

G (IEG(n))

where • νn : IEG(n) → IBG(n) is the canonical projection. • i n is the Cartan-Weil morphism 4.1.1.1. • Ξn is the Cartan isomorphism 4.1.1-4.4 (see also 4.3.2-(4.16)). It is easy to see that the sequences (4.86), being functorial relative to the closed embeddings IEG(n) → IEG(n+1), induce a morphism of projective systems from {(IBG(n))}n∈N to {G (IEG(n))}n∈N , and since these systems satisfy Mittag-Leffler conditions at both complexes and cohomology levels, we obtain a quasi-isomorphic morphism of dg-algebras lim (IBG(n)) −→ lim n G (IEG(n)) . q.i. ←− ←−n We can now prove the main result of this section.

(4.87)

4.10 Local Equivariant Cohomology

169

Theorem 4.9.2.1 Let G be a compact Lie group. 1. For every M be a G-manifold, there exists a zig-zag of quasi-isomorphic morphisms of dg-algebras: G (M)  Ω(MG ; R)

(4.88)

2. The classifying space IBG is an R-formal space. Proof (1) If we put aside equality (4.81), the preliminary discussion in Sect. 4.9.2 is valid for any G-manifold M in lieu of the singleton {•}. Consequently, for G connected, the concatenation of (4.80), (4.85) and (4.87) gives a canonical zigzag G (M)  Ω(MG ; R) of quasi-isomorphic morphisms of dg-algebras which is functorial on M ∈ G-Man. When G is not connected, we use the fact the finite group W := G/G0 , acts on MG0 (cf . Exercise 4.7.1.2) in a way that if we denote by π : MG0 → MG0 /W = MG the canonical projection, then the pullback π ∗ induces algebra isomorphisms W π ∗ : Ω(MG ; R) −→  Ω(MG0 ; R)

W and π ∗ : H (MG ; R) −→  H (MG0 ; R) ,

which are functorial on M ∈ G-Man, by 4.4.5.2 and 4.7.1.2. Analogously, as explained in Sect. cf . 4.4.5, the group W acts on the complex of G0 -equivariant differential forms G0 (M), and the restriction maps W G (M) −→  G0 (M)

and

W HG (M) −→  HG0 (M) ,

are also isomorphisms (cf . Proposition 4.4.5.2). Therefore, to establish ((1)), we need only show that all the morphisms of dg-algebras in the zig-zag G (M)  Ω(MG ; R) are W -equivariant. This is elementary in (4.80) and in (4.84), hence in (4.85). The W -equivariance of (4.87) results from that of (4.86) which was established in 4.4.5.2. (2) Apply (1) to M := {•}, in which case G ({•})  H (IBG; R) (see (4.81)).  

4.10 Local Equivariant Cohomology Let X be a mild topological space (cf . B.1). Given a closed subspace Z ⊆ X, let U := X  Z, and denote by i : Z ⊆ X and j : U ⊆ X the corresponding inclusion maps. One has an exact triangle of functors in the category D(M; k) IR ΓZ (X; −) → IR Γ (X; −) → IR Γ (U ; −) → where ΓZ (X; −) is the functor of sections with support in Z.

(4.89)

170

4 Equivariant Background

The cohomology of the right derived functor IR ΓZ (X; −) is called the local cohomology of X, relative to X.47 In the case of the constant sheaf k X , the triangle (4.89) gives rise to the long exact sequence of local cohomology:48 → HZi (X; k) → H i (X; k) → H i (U ; k) →

(4.90)

where we see that local cohomology is a relative cohomology. Indeed, we have a canonical isomorphism HZi (X; k)  H i (X, X  Z; k) , as well as a canonical isomorphism in derived category  IR ΓZ (X; −)  cˆ IR Γ (X; −) → IR Γ (X; −) [−1] where c( ˆ −) denotes the mapping cone of a morphism of complexes (cf . A.1.3). We will apply these consideration to the case of a closed inclusion i : N ⊆ M of G-manifolds, and to each column of the following induced inclusion of inductive systems (cf . 4.5.3.1-(2)) ···

NG (n)

νn

iG,n

···

MG (n)

NG (n+1)

·····

NG

iG,n+1 μn

MG (n+1)

(4.91)

iG

·····

MG

Let U := M  N and denote by jG : UG → MG and jG,n : UG (n) ⊆ MG (n) the corresponding open inclusions. Since all the spaces in consideration are mild spaces, diagram (4.91) gives rise to the following projective system of (vertical) long exact sequences of local (ordinary) cohomology with coefficients in a field k:

HNi G (MG )

·····

HNi G (n+1) (MG (n+1))

HNi G (n) (MG (n))

·····

H i (MG )

·····

H i (MG (n+1))

H i (MG (n))

·····

H i (UG )

·····

H i (UG (n+1))

H i (UG (n))

·····

47 The 48 See

terminology is due to Grothendieck, see Hartshorne [52] §1, pp. 1–15. Hartshorne [52] §1, Proposition 1.9, p. 9.

(4.92)

4.10 Local Equivariant Cohomology

171

where we apply Theorem 4.7.2.2-(3) and claim that the restrictions H i (MG ; k) → H i (MG (n); k)

and

H i (UG ; k) → H i (UG (n); k)

are isomorphisms for all n > i. But then, by the Five Lemma, the same is true for local (ordinary) cohomology, i.e. we have isomorphisms HNi G (MG ; k) −→ HNi G (n) (MG (n); k) , 

∀n > i ,

(4.93)

whence, an isomorphism HNG (MG ; k) → lim n HNG (n) (MG (n); k) ←−

(4.94)

We can proceed in the same way with equivariant cohomology using the expression 4.8.3.2-(4.79) which gives the projective system of (vertical) long exact sequences of local (de Rham) cohomology:

(?)

·····

HNi G (n+1) (MG (n+1))

HNi G (n) (MG (n))

·····

HGi (M)

·····

H i (MG (n+1))

H i (MG (n))

·····

HGi (U )

·····

H i (UG (n+1))

H i (UG (n))

·····

(4.95)

where we now apply Theorem 4.8.2.1 and claim that the restrictions HGi (M) → H i (MG (n)) and

HGi (U ) → H i (UG (n))

are isomorphisms for all n > i. But then, by the Five Lemma, we have isomorphisms for local (de Rham) cohomology HNi G (m) (MG (m)) −→ HNi G (n) (MG (n)) , 

∀m ≥ n > i .

We are thus lead to define the G-equivariant cohomology of M with supports in N , or the local equivariant cohomology of M relative to N . (cf . 4.8.3.2) HG,N (M) := lim n HNG (n) (MG (n)) ←−

(4.96)

172

4 Equivariant Background

which clearly verifies, for fixed i ∈ N, that the “restriction” homomorphisms i HG,N (M) −→ HNi G (n) (MG (n)) 

(4.97)

are isomorphisms for all n ≥ i (likewise (4.93)).

4.10.1 The Long Exact Sequence of Local Equivariant Cohomology The definition (4.96) fills the missing term (?) in (4.95), giving the long exact sequence of local equivariant de Rham cohomology for G-manifolds: i → HG,N (M) → HGi (M) → HGi (U ) →

(4.98)

All these facts lead us to propose the following definition motivated by the fact that local cohomology is the relative cohomology of the pair (U, M), as recalled in the preamble of 4.10. The definition completes Definition 4.4.4. Definition 4.10.1.1 Let G be a compact Lie group. Let N ⊆ M be a closed inclusion of G-manifolds. The complex of G-equivariant differential forms of M with supports in N , is, by definition, the complex   G,N (M), dG := cˆ G (M) → G (M  N ) [−1] .

(4.99)

where c( ˆ −) denotes the mapping cone of a morphism of complexes (cf . A.1.3). The following theorem ends the chapter showing that this definition is wellfounded and in agreement with the previous observations. Local Equivariant de Rham Theorem 4.10.1.2 Let G be a compact Lie group, and let N ⊆ M be a closed inclusion of G-manifolds. Then, there exist canonical isomorphisms  h G,N (M), dG  HG,N (M)  HNG (MG ; R) . Proof Set U := M  N , and let j : U → M denote the inclusion map. for n ∈ N, denote by p2 : IEG(n)×(−) → (−) the canonical projection (as in Sect. 4.8.2).

4.10 Local Equivariant Cohomology

173

 • H G,N (M), dG  HG,N (M). The commutative diagram (I) of Cartan complexes, canonically extends to a morphism of exact triangles defined by ∗ :  (M) →  (N) and j (n)∗ : (M (n)) → the mapping cones of jG G G G (NG (n)): G,N (M)

∗ jG

G (M)

(I)

p2∗

q∗ NG (n) (MG (n))

G (n))

j (n)∗

G (U ) p2∗ G (n))

which gives rise to a morphism of long exact sequences hi

G,N (M) (1)

HNi G (n) (M(n))

HGi (M)

HGi (U )

(2)

(3)

H i (MG (n))

H i (UG (n))

where the arrow (1) is an isomorphism for all n > i since this is so of arrows (2, 3) after Theorem 4.8.2.1 and Remark 4.8.3.2. Hence, the isomorphism  lim n HNG (n) (M(n)) =: HG,N (M) , h G,N (M) −→  ←− after the definition 4.96. • HG,N (M)  HNG (MG ; R). This isomorphism follows in the same way as Theorem 4.8.3.1. The ingredients of the proof, i.e. formulas (4.93), (4.94), (4.96) and (4.97), have already been justified.   Exercise 4.10.1.3 A spectral sequence for local equivariant cohomology. 1. Let α : (C1 , d, θ, ι) → (C2 , d, θ, ι) be a morphism of positively graded g-dgm’s. Extend Theorem 4.3.3.1-(3) to construct a convergent spectral sequence 

p,q

IE 2

= S p (g ∨ )g ⊗ H q (c(α)) ˆ , d2 =

i

p+q ei ⊗ ι(ei ) ⇒ Hg (c(α)) ˆ .

2. Let G be a compact Lie group, and let N ⊆ M be a closed inclusion of Gmanifolds. Use (1) to construct a spectral sequence IE G,N,r (M)), such that:  q p+q IE G,N,2 (M)p,q := S p (g ∨ )g ⊗ HN (M) W ⇒ HG,N (M) , where W := G/G0 . 3. Show that there exists a long exact sequence of spectral sequences −→ IE G,N,r (M) −→ IE G,r (M) −→ IE G,r (M  N ) −→

174

4 Equivariant Background

where IE G,r (−) denotes the spectral sequence for equivariant cohomology in Corollary 4.4.6.2-(3). Hint. Exercise 5.4.4 in Weibel [95], page 135. ( , p. 349) Comment 4.10.1.4 We will see in Sect. 6.4.6, which introduces the Gysin long exact sequence associated with a closed inclusion of oriented G-manifolds N ⊆ M, the existence of the canonical isomorphism (6.34) HG,N (M)  HG (N )[dN −dM ] .

Chapter 5

Equivariant Poincaré Duality

5.1 Differential Graded Modules over a Graded Algebra We begin extending the definitions and terminology of Sect. 2.1, by replacing the field k by a graded k-algebra A (2.1.5).

5.1.1 Graded Modules and Algebras over Graded Algebras Let A be a graded k-algebra. An A-graded module, A-gm in short, is a graded space V ∈ GV(k) together with a morphism A → Endgr∗k (V ) of graded k-algebras. Given two A-gm’s V and W , a graded homomorphism of A-gm’s of degree d from V to W is a graded homomorphism of graded spaces α : V → W of degree d (2.1.3), compatible with the action of A, i.e. α(a · v) = a · α(v). The space of such homomorphisms is denoted by HomgrdA (V , W ). The graded space of graded homomorphisms of A-gm’s from V to W is then   Homgr∗A (V , W ) = HomgrdA (V , W ) d∈Z ,

(5.1)

which is again an A-graded module. 5.1.1.1 The the A-gm’s are the objects of the category GM(A) of A-graded modules. The morphisms in GM(A) are the graded homomorphisms of degree 0, hence we set MorGM(A) (V , W ) := Homgr0A (V , W ). The category GM(A) is abelian. The full subcategories of bounded graded modules GMb (A), GM+ (A), GM≥ (A), GM− (A) and GM≤ (A) are as in Sect. 2.1.4. 5.1.1.2 An A-graded algebra is an A-graded module B ∈ GM(A) together with a multiplication, i.e. a family of A-bilinear maps { · : B a × B b → B a+b }a,b∈Z , such that the triple (B, 0, +, · ) verifies the axioms of an A-algebra. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Arabia, Equivariant Poincaré Duality on G-Manifolds, Lecture Notes in Mathematics 2288, https://doi.org/10.1007/978-3-030-70440-7_5

175

176

5 Equivariant Poincaré Duality

The algebra is said to be • positively graded: if B m = 0 for all m < 0, • evenly graded: if B m = 0, for all odd m, • anticommutative:1 if b1 · b2 := (−1)d1 d2 b2 · b1 , for all b1 ∈ B d1 , b2 ∈ B d2 . A morphism of A-graded (resp. unital) algebras α : (B, 0, +, · ) → (B  , 0, +, · ) is a morphism of A-graded modules α ∈ Homgr0A (B, B  ) that is compatible with the multiplication operation, i.e. α(x · y) = α(x) · α(y) (resp. α(1A ) = 1B ). Examples 5.1.1.3 (a) Given a G-manifold M, the complexes G (M) and G,c (M) are positively S(g ∨ )G -graded anticommutative algebras. (b) For any V ∈ GM(A), the A-graded module (Endgr∗A (V ), 0, +, id, ◦) of graded endomorphisms is a noncommutative unital graded A-algebra.

5.1.2 The Category of G -Graded Modules We now turn to the graded algebra (cf . 4.4.4.1), G := G ({•}) = S(g ∨ )G where G is a compact Lie group. The category of G -graded modules is denoted by GM(G ), where MorGM(G ) (−, −) := HomG (−, −).  Definition 5.1.2.1 A direct sum i∈I G [mi ], with mi ∈ Z, is called a free G graded module. Proposition 5.1.2.2 1. An object V ∈ GM(G ) is projective (resp. injective) if and only if the functor Homgr∗ (V , −) : GM(G )  GM(G ) (resp. Homgr∗ (−, V )) is exact. 2 2. The category GM(G ) is an abelian category with enough injective and projective objects. A graded module V ∈ GM≥ (G ) admits a free resolution d2

d1

d0



· · · −−→ F2 −−→ F1 −−→ F0 − → →V → 0 with Fi ∈ GM≥ (G ). In particular, GM≥ (G ) and GM+ (G ) are abelian categories with enough projective objects. 3. The homological dimension3 of GM(G ) is finite. 1A

synonym of graded commutative, frequently used in the works of Cartan [25, 26]. Grothendieck [49], chapter I, Thm. 1.10, p. 135. 3 See Weibel [95], Global Dimension Theorem 4.1.2, p. 91, or Jacobson [59], 6.12 Homological Dimension, p. 375. 2 See

5.1 Differential Graded Modules over a Graded Algebra

177

Proof (1) immediate since Homgr∗G (−, −) can be seen as the product Homgr∗G (−, −) =

 m∈Z

Homgr0G (−, −[m]) =

 m∈Z

Homgr0G (−[m], −) ,

where the shift functor [m] is exact. (2) Enough Projectives. Let {vi }i∈I be a family of homogeneous (2.1.3.1) generators for V ∈ GM(G ) and consider, for each i ∈ I, the map γi : G [−di ] → V , x → xvi which is clearly a morphism in GM(G ). The sum  i∈I

γi :

% i∈I

G [−di ] → →V

(5.2)

represents V as the quotient in GM(G ) of a free, and thus projective, G -gm.

Enough Injectives.4 The correspondence & := Homgr∗Z (V , (Q/Z)[0]) V V

(5.3)

is an additive contravariant functor from the category of left (resp. right) G -gm to the category of right (resp. left) G -gm,5 and is exact, by (1), since Homgr0Z (−, (Q/Z)[0]) = HomZ ((−)0 , Q/Z) and since Q/Z is an injective Z-module. & , v → (γ → γ (v)) is an injective morphism. Lemma 1 The map ν(V ) : V → V Proof of Lemma 1 Since ν(V ) is clearly a morphism of graded modules, it is injective if and only if it does not kill any homogeneous nonzero element. If 0 = v ∈ V d , the subgroup Z · v ⊆ V d is isomorphic to some Z/nZ for n = ±1, and there exists a nonzero homomorphism γ  : Z · v → Q/Z (exercise), restriction of some γ  : V d → Q/Z (thanks to the injectivity of Q/Z). Extend this γ  to the whole of V , assigning the value 0 on the homogeneous factors V e when e = d. This last extension, denoted by γ : V → Q/Z, is a graded morphism of degree −d and verifies ν(V )(v)(γ ) = γ (v) = 0 by construction, so that ν(V )(v) = 0, which completes the proof of Lemma 1.  & is injective. Lemma 2 For any free right G -gm F, the left G -gm F

4 We

reproduce the proof in Godement [46] Thm. 1.4.1, p. 6, in the graded framework. N is a right G -gm, the structure of left G -module of Homgr∗Z (N , (Q/Z)[0]) is given by (x · γ )(y) := γ (yx) for all x ∈ G and y ∈ N . If N is a left G -gm, the structure of right G -module of Homgr∗Z (N , (Q/Z)[0]) is given by (γ · x)(y) := γ (xy) for all x ∈ G and y ∈ N . 5 If

178

5 Equivariant Poincaré Duality

Proof of Lemma 2 We recall (cf . [19] Chap. II, §4, Prop. 1) that for any left G dgm N , the maps Homgr∗ G (N , Homgr∗Z

G , (Q/Z)[0]))

γ v

(x

ξ (xv))

Homgr∗Z (N , (Q/Z)[0]) v

γ (v)(1)

ξ

are isomorphisms inverse of each other. It follows that Homgr∗Z (G , (Q/Z)[0])) is an injective left G -gm if and only if Homgr∗Z (−, (Q/Z)[0]) is exact, but this is equivalent, by (1), to the exactness of the functor Homgr0Z (−, (Q/Z)[0]) =  HomZ ((−)0 , Q/Z) , which is clear since Q/Z is an injective Z-module. & Let V be a left G -gm. Fix some epimorphism of right G -gm π : F → →V & →F & is injective and, composed where F is free as in (5.2). The morphism & π :V & with ν(V ) : V → V , which is also injective by Lemma 1, gives an injective & of left G -gm, where F & is an injective left G -gm by Lemma morphism V → F 2. This completes the proof of the existence of enough injective objects in GM(G ). The fact that GM≥ (G ) and GM+ (G ) are abelian categories is obvious. The proof of the existence of enough projective modules also shows that a graded module V ∈ GM≥ (G ) is a quotient  : F0 → → V of a free G -graded module F0 ∈ GM≥ (G ), and since ker() ∈ GM≥ (G ), we can iterate the procedure. (3) When G is connected, G is a polynomial algebra in rk(G) variables6 and we can apply Hilbert’s Syzygy Theorem.7 Otherwise G = (ΩG0 )G/G0 , and dimh (G ) ≤ dimh (G0 ) since ΩG0 = S(g ∨ )g is a polynomial algebra over a field of characteristic 0 and that G/G0 is a finite group. ( , p. 349)   Exercise 5.1.2.3 Let A be a graded R-algebra which is an integral domain. 1. Let S denote the multiplicative system of homogeneous nonzero elements of A. Show that S −1 A is an injective object of GM(A) (see ch. D). Prove that the canonical inclusion A → S −1 A is an injective envelope for A.8 2. Show that when rk(G) > 0, the degrees of a non trivial injective object of GM(G ) cannot be bounded below (2.1.4). The next two sections are straightforward generalizations of Sects. 2.1.6 and 2.1.8 from graded vector spaces to G -graded modules. 6 This

is Theorem 19.1, p. 171, in Armand Borel’s Ph.D. thesis [13]. The result is based on the identification S(g ∨ )g = S(t ∨ )W , where W := NG (T )/T is the Weyl group of the pair (G, T ), and the celebrated Chevalley’s theorem stating that S(t ∨ )W is a polynomial algebra in rk(G) variables, see Chevalley [30]. 7 See Jacobson [59] Hilbert’s Theorem in p. 385, Corollary, p. 386, and Ex. 2, p. 387. 8 The injective envelope of an object O, also called its injective hull, is the ‘smallest’ injective object containing O. More precisely, it is an injective object I together with a monomorphism  : O → I such that any other monomorphism η : O → J where J is injective, factors through  in a monomorphism ι : I → J, i.e. η = ι ◦ .

5.2 The Category of G -Differential Graded Modules

179

5.2 The Category of G -Differential Graded Modules 5.2.1 Definition An G -differential graded module, G -dgm in short, is a pair (V , d) with V ∈ GM(G ) and d ∈ Endgr1G (V ), the differential, such that (V , d) is a complex of vector spaces, i.e. d 2 = 0. A morphism of G -dgm α : (V , d) → (V  , d  ) is a morphism of G -gm’s and a morphism of complexes, i.e. d  ◦ α = α ◦ d. The G -dgm’s and their morphisms constitute the category DGM(G ) of G differential graded modules. The category DGM(G ) is an abelian category. Comment 5.2.1.1 This definition of differential graded modules is specific to evenly graded algebras with zero differential, as is G . In the general definition of differential graded modules over a differential graded algebra (A, d), the anticommutativity of the differentiation operation is required, i.e. we also need to satisfy the Koszul sign rule by which d(a · x) = d(a) · x + (−1)[a] a · d(x), for homogeneous a ∈ A, in which case the action of d does not necessarily commute with the action of A (cf . 2.1.9).

5.2.2 The Hom• (−, −) and (− ⊗G −)• Bifunctors on G DGM(G ) Given two G -dgm’s (V , d) and (V  , d  ), we define the two G -dgm’s: 

Hom•G (V , V  ), D•



and

 (V ⊗G V  )• , • .

As G -graded modules, defined by  m →

 Homm G (V , V )

(V ⊗G V  )m

:= Homgr0G (V , V  [m]) = Homgr0G (V [−m], V  )  := π (V ⊗R V  )m (5.4)

where π : V ⊗R V  → → V ⊗G V  , v ⊗ v  → [v ⊗ v  ], is the canonical (graded) surjection (see remark 5.2.2.1-(2)). The differentials D• and • are: 

Dm (f ) = d  ◦ f − (−1)m f ◦ d m ([v ⊗ v  ]) = [d(v) ⊗ v  ] + (−1)a [v ⊗ d  (v  )]

(5.5)

where v ⊗ v  ∈ V a ⊗ V b and a + b = m. Notice that the two differentials are G -linear.

180

5 Equivariant Poincaré Duality

5.2.2.1 As in Definition 2.1.8.1, these constructions are natural w.r.t. each entry. They define the two bifunctors: 

Hom•G (−, −) : DGM(G ) × DGM(G )  DGM(G ) , (− ⊗G −)• : DGM(G ) × DGM(G )  DGM(G ) .

(5.6)

The ‘Hom•G ’ functor is contravariant left exact on the first entry, and covariant left exact on the second entry. The ‘⊗’ functor is covariant right exact on each entry (compare with Definition 2.1.8.1). Comments 5.2.2.1 1. The formulas (5.5), by which D and  are G -linear, apply only where G is evenly graded with zero differential, otherwise the formulas must include additional signs and terms related to graded commutativity and graded derivability (cf . 5.2.1.1). 2. Some care must be taken with the tensor product, which can conceal some subtleties. A good way to understand this is to note that V ⊗G V  is the quotient of the graded vector space V ⊗R V  by the subspace W spanned by the tensors P v ⊗ v − v ⊗ P v with P ∈ G and (v, v) ∈ V × V  homogeneous. W is a graded subcomplex of (V ⊗R V  , ) and the canonical surjection π : (V ⊗R V  , ) → → (V ⊗R V  , )/W is an epimorphism of graded complexes, therefore inducing over V ⊗G V  a structure of G -dgm. Again, this setting is specific to the fact that G is evenly graded.

5.2.3 The Duality Functor on DGM(G ) In 2.1.11, we introduced the duality functor Hom•k (−, k) : DGV(k)  DGV(k) and noted that it was exact (2.1.11.2). In the framework of G -dgm’s, the corresponding functor is the G -duality functor Hom•G (−, G ) : DGM(G )  DGM(G )

(5.7)

which is contravariant left exact, but not exact.

5.2.4 The Forgetful Functor If we disregard differentials, an G -dgm is simply as an G -gm, and likewise for morphisms. This is the action of the the forgetful functor, which we denote by o : DGM(G )  GM(G ) . It is an exact functor, which will often be implicit in our considerations.

5.2 The Category of G -Differential Graded Modules

181

5.2.5 On the Exactness of Hom• (−, −) and (− ⊗ −)• The most relevant difference between Sect. 2.1.8 and 5.2.2 is the change of the coefficients ring A from a field k to the graded ring G . When A = k, the categories Vec(k) and GV(k) are both split ((3 ), p. 13), which implies that additive functors on these categories always induce exact functors on the categories C(k) and C(GV(k)), where, in particular, they preserve acyclicity and quasi-isomorphism (cf . Exercise 2.1.6.1). By contrast, when A = G , the category GM(G ) is no longer split, and additive functors are no longer necessarily exact on the categories C(GM(G )) and DGM(G ) (cf . 5.2.5.1-(1)). The functor Hom•k (−, k) is central to proving nonequivariant Poincaré duality and its exactness on C(GV(k)) is crucial for the proof to work. When we follow the same approach in the equivariant setting, we are lead to consider Hom•G (−, G ) which is not exact on DGM(G ) (cf . 5.2.5.1-(2)), causing the attempt at establishing equivariant Poincaré duality to fail. This failure of exactness is the main reason for introducing the derived categories D(GM(G )) and D(DGM(G )) and the corresponding derived duality functor IR Hom•G (−, G ), which we will address in Sects. 5.4.5 and 5.4.6 below. We will see that replacing Hom•G (−, G ) by IR Hom•G (−, G ) suffices to overcome the obstructions in the proof of equivariant Poincaré duality, justifying a posteriori the use of derived categories. Exercises 5.2.5.1 1. Let A be a noetherian ring. a. Show that the simple objects9 in the category Mod(A) of A-modules are the fields A/M, where M is a maximal ideal in A. b. Show that Mod(A) is split (fn. (3 ), p. 13) if and only if A is a finite product of fields. Conclude that Mod(G ) is not split if dim g > 0. c. Show that Mod(A) is split if and only if the functor HomA (S, −) : Mod(A)  Mod(A) is exact for every simple A-module S. (cf . Exercise 2.1.6.1). d. Same as (1c) for the functors HomA (−, S) and S ⊗A (−). 2. Show that if Mod(A) is not split, then the A-duality functor does not preserve quasi-isomorphisms of complexes. Apply to the G -duality functor (5.7).

object S in an abelian category Ab is called simple if S = 0 and if 0 and S are its only subobjects.

9 An

182

5 Equivariant Poincaré Duality

5.3 Comparing the Categories C(GM(G )) and DGM(G ) This section is quite technical, so first-time readers might choose to skip it. It examines the differences between the categories of complexes of G -graded modules and of differential graded modules over G . In an abelian category Ab the category C(Ab) of complexes of Ab is defined in the same way as for vector spaces. A complex of objects of Ab is a sequence (X, d) := · · ·

di−2

Xi−1

di−1

Xi

di

Xi+1

di+1

··· .

(5.8)

of objects Xi ∈ Ob(Ab) where di ∈ MorAb (Xi , Xi+1 ) and di+1 ◦ di = 0. A morphism of complexes α : (X, d) → (X  , d  ) is then a family of morphisms  αi ∈ Mor(Ab)(Ci , Ci ) such that di+1 ◦ αi = αi+1 ◦ di . In this presentation, the objects in the complex have no grading, but if we use the subscripts i ∈ Z as degrees, we get a sequence (cf . 2.1.7-(S-1)) 

di−2 di−1 di+1 di · · · −−→ Xi−1 [−i + 1] −−→ Xi [−i] −−→ Xi+1 [−i−1] −−→ · · · ,

(5.9)

where, now, the differentials di are of degree 1. The family {Xi [−i]}i∈Z belongs then to the category GO(Ab) := AbZ of graded objects of Ab, and the category C(Ab) is then (tautologically) equivalent to the category DGO(Ab) of differential graded objects of Ab. This was also the case for vector spaces where we had C(Vec(k)) = DGV(k) (2.1.6).

5.3.1 The Tot Functors When Ab is the category GM(G ), we use the same recipe (5.9) and get objects with two gradings, one coming from the grading as G -gm and the other from the subscript index in the complex. Hence, the two dimensional ladder (5.10). We can then associate with a complex of G -gm’s (X, d) a differential graded module in DGM(G ) in ways that depend on how we collect terms of equal total degree10 in the bigraded ladder (5.10). In the present case where there are only two degrees, we can do this either by taking the direct sum ‘⊕’ or the direct product ‘Π’ of the diagonals of constant total degree.

10 The

total degree of a multi-degree object is the sum of its degrees. Here, it is the sum of the graded module degree plus the index number in the complex.

5.3 Comparing the Categories C(GM(G )) and DGM(G )

183

(5.10)

Fig. 5.1 A complex in C(GM(G ))

For direct sums, we define:11  m−i • Totm , and ⊕ X := i∈Z Xi Tot⊕ (X) := {Totm ⊕ X}m∈Z =

% i∈Z

Xi [−i] ∈ GM(G ) .

(5.11)

• The differential Tot⊕ d : Tot⊕ X → Tot⊕ X[1] is the morphism of G -gm’s : dm (⊕i∈Z xi ) := i(⊕i∈Z di (xi ) .

(5.12)

• If α : (X, d) → (X  , d  ) is a morphism in C(GM(G )), we define Tot⊕ α : Tot⊕ X → Tot⊕ X  ,

 αm (⊕i∈Z xi ) := ⊕i∈Z αi (xi ) .

For direct products, we define Tot mutatis mutandis replacing ⊕ by Π. Proposition 5.3.1.1 The previous settings define faithful exact functors 

Tot⊕ : C(GM(G ))  DGM(G ) , Tot : C(GM(G ))  DGM(G ) .

which coincide on Cb (GM(G )). They preserve cones (Sect. A.1.3), homotopies (Sect. A.2.2) and commute with cohomology: h ◦ Tot⊕ = Tot⊕ ◦h

and

h ◦ Tot = Tot ◦ h .

In particular they preserve acyclicity and quasi-isomorphisms.

11 We

will come back to this in Sect. 5.4.2, where spectral sequences will come on stage.

184

5 Equivariant Poincaré Duality

Proof The additivity and exactness are already visible at the level of complexes of vector spaces where the functors only group together vector spaces with no modification. One has Tot⊕ = Tot on bicomplexes with finite length diagonals, hence the category Cb (GM(G )). The other statements result by a straightforward albeit tedious verification, left to the reader. Notice that the fact that quasiisomorphisms are preserved is a standard consequence of the fact that cones and acyclicity are preserved (Proposition A.1.4.1-(3c)).  

5.3.2 The Hom• (−, −) bifunctor on C(GM(G )) G We start by recalling that in any abelian category Ab with countable products, standard practice is to consider the bifunctor 

Hom•Ab (−, −), D• ) : C(Ab) × C(Ab)  C(Mod(Z)) ,

(5.13)

(as we did for the category Vec(k) (Sect. 2.1.8)) by setting

    HomAb (Xn , Xn+i ) HomiAb (X, d), (X  , d  ) := n

(5.14)

where ‘i’ denotes the index in the resulting complex, and Di (α) := d  ◦ α − (−1)i α ◦ d .

(5.15)

However, in the case of graded categories, for example Ab := GM(G ), this definition would give    HomiGM(G ) (X, d), (X  , d  ) := Homgr0G (Xn , Xn+i ) ∈ Vec(k) , n

which is not what we are interested in for several reasons, among which is that it would give us complexes of vector spaces rather than G -graded modules. This leads us naturally to define instead Hom•G (−, −) : C(GM(G )) × C(GM(G ))  C(GM(G )) ,

(5.16)

   HomiG (X, d), (X  , d  ) := Homgr∗G (Xn , Xn+i ) ∈ GM∗ (G ) , n (5.17) where ‘∗’ denotes the grading of GM(G ), with the same differential (5.15). As in Sect. 5.2.2.1, the ‘Hom•G ’ functor (5.17) is contravariant left exact on the first entry, and covariant left exact on the second entry.

5.3 Comparing the Categories C(GM(G )) and DGM(G )

185

Proposition 5.3.2.1 The compatibility between the Hom•G (−, −) functors on C(GM(G )) and on DGM(G ), is given by the following commutative diagram. C(GM

G ))

× C(GM

Tot⊕

DGM

Hom• (−,−) G

G ))

Tot G)

× DGM

Hom• (−,−) G

G)

C(GM

G ))

(5.18)

Tot

DGM

G)

Furthermore, if we restrict to the full subcategory Cb (GM(G )) × Cb (GM(G )), then the Tot functors can be replaced by Tot⊕ . Proof Since the differentials are not essential to the statement, the proof is elementary and based on the following equalities. Let X := (Xi ), X  := (Xi ) be complexes of G -gm’s. – Tot ◦ Hom•G (−, −). The i-th term in the complex Hom•G (X, X  ), given by the definition (5.17), is HomiG (X, X  )∗ =

 n

  Homgr∗G (Xn , Xn+i ) ∈ GM∗ (G ) ,

(5.19)

and if we apply Tot to Hom•G (X, X  ), we get the ∗G -dgm   i

n

 Homgr0G (Xn , Xn+i [∗ − i]) .

(5.20)

– Hom•G (−, −) ◦ (Tot⊕ ×Tot ). In the ∗G -gm Hom•G (Tot⊕ X, Tot X  ), the term of degree ‘∗’ is : Homgr∗G



n Xn [−n],

=



 m Xm [−m]



n,m



=

  Homgr0G Xn , Xm [n − m + ∗] ,

(5.21)

equal to (5.20), following an obvious change of subscripts. For the last statement, we note that if X ∈ Cb (GM(G )), then, for  each i, there is only a finite number of n such that Xn = 0, in which case the ‘ n ’ symbol in (5.19) can be changed to ‘ n ’. After that, if, in addition, X  ∈ Cb (GM(  G )), then  the set of couples (n, i), for which X =  0, is also finite, and the ‘ i ’ symbol in n+i  (5.20) can be changed to ‘ i ’. Hence, Tot Hom•G (X, X  ) = Tot⊕ Hom•G (X, X  ) .

186

5 Equivariant Poincaré Duality

b    On  G )), we have Tot X = Tot⊕ X and the other hand, for X ∈ C (GM( the ‘ m ’ in (5.21) can be replaced by ‘ m ’. Hence,

Hom•G (Tot⊕ X, Tot X  ) = Hom•G (Tot⊕ X, Tot⊕ X  ) ,  

which ends the proof of the proposition.

5.3.3 The (− ⊗G −)• Bifunctor on C(GM(G )) Given (X, d), (X  , d  ) ∈ C(GM(G )), we define the complex  (V ⊗G V  )• ,  ∈ C(GM(G )) , with i-th term (V ⊗G V  )i :=

 a+b=i

Xa ⊗G Xb ,

where Xa ×G Xb is endowed with the grading induced by the total degree a + b of Xa ⊗R Xb , as in (5.2.2–(5.4)). The differential $i on Xa ⊗ Xb is $i (v ⊗ v) := da (v) ⊗ v + (−1)a v ⊗ db (v) .

(5.22)

In this way we get a bifunctor covariant additive and right exact on each entry (− ⊗G −)• : C(GM(G )) × C(GM(G ))  C(GM(G )) .

(5.23)

Proposition 5.3.3.1 The compatibility of (− ⊗G −)• on C(GM(G )) and on DGM(G ), is described by the following commutative diagram C(GM

G ))

Tot⊕

DGM

× C(GM

G ))

(−

G−

)•

Tot⊕ G)

× DGM

C(GM

G ))

Tot⊕ G)

(−

G −)

(5.24)

•

DGM

G)

Sketch of Proof Results by an elementary check, much simpler than for Proposition 5.3.2.1. Details are left to the reader. 

5.4 Deriving Functors in GM(G )

187

5.4 Derived Functors in the Category GM(G ) Readers with little experience in these subjects may find it helpful to read Appendix A first, as it gives an overview of Derived Categories and Derived Functors. This has resulted in some duplication, for which we apologize. We showed in Proposition 5.1.2.2 that the abelian category GM(G ) has enough projective and injective objects. We now recall the action on GM(G ) of the derived functors of an additive functor F : GM(G ) → Ab

(5.25)

where Ab is an abelian category for which we denote by C(Ab) its category of complexes and by K(Ab) its homotopy category (cf . A.1.1, A.1.5, A.2.2). The right and left derived functors of F are respectively denoted by 

IR F : GM(G )  K(Ab)

(5.26)

ILF : GM(G )  K(Ab) .

Notice that while the target category of F is Ab, the target category of its derived functors is the homotopy category K(Ab). For V , W ∈ GM(G ) the complexes IR F (V ), ILF (W ) ∈ K(Ab) are defined by the following recipe. – Choose a projective (resp. injective) resolution of V (resp. W ) in C(GM(G )). d−2

d−1

d0





d0

d1

d2

· · · −−→ P−2 −−→ P−1 −−→ P0 −−→ V −−→ 0

(5.27)

0 −−→ W −−→ I0 −−→ I1 −−→ I2 −−→ · · · – Set ⎧ ⎫ d−1 d0 ⎨ P V := (· · · −→ P−1 −→ P0 → 0) ⎬ ⎩

d0

d1

I W := (0 → I0 −→ I1 −→ · · · )

⎭

∈ C (GM(G ))

and define 

ILF(V ) := F(P V ) ∈ K(GM(G )) , (5.28) IR F(V ) := F(I V ) ∈ K(GM(G )) .

Notice that, since projective and injective resolutions of objects are always homotopic, these complexes are homotopically independent of the chosen resolutions, so that they are well-defined objects in the homotopy category K(Ab). Their

188

5 Equivariant Poincaré Duality

cohomologies are respectively denoted by (IR i F)(−) := H i (IR F(−))

and

(ILi F)(−) := H −i (ILF(−)) .

5.4.1 Augmentations We easily see from the above definitions that the augmentation morphisms of complexes  : V [0] → I V and  : P V → V [0], give rise to natural morphisms of complexes F() : F(V [0]) → IR F(V ) and F() : ILF(V ) → F(V [0]) , inducing canonical morphisms F(V ) → (IR 0 F)(V )

and

(IL0 F)(V ) → V .

These are isomorphisms if F is respectively left and right exact (see A.2.3.6). Before going further into the subject of derived functors we need to recall some basics of bicomplexes of an abelian category Ab.

5.4.2 Simple Complex Associated with a Bicomplex Since the category C" (Ab) of complexes of an abelian category Ab is additive (in fact abelian), we can consider the category C ," (Ab) := C (C" (Ab)) of complexes of C" (Ab) also called double complexes, or bicomplexes, of Ab. A bicomplex N ," := (N ," , δ ," , d ," ) ∈ C ," (Ab) is generally represented as a two dimensional ladder (5.29) all of whose subdiagrams are commutative. We extend the definition of the Tot⊕ functor of 5.3.1 to C ," (Ab).

(5.29)

Fig. 5.2 A bicomplex in C ," (Ab)

5.4 Deriving Functors in GM(G )

189

The simple (or total) complex associated with N (Tot◦⊕ (N ," ), D◦ ), where, for m ∈ Z, Totm ⊕ (N

) :=

m=i+j

,"

is the complex denoted by

N i,j +1

N i,j

Dm (ni,j ) := di,j (ni,j ) + (−1)j δi,j (ni,j )

di,j

(−1)j δi,j

N i,j

N i+1,j

In this way, we get an exact functor Tot◦⊕ := C ," (Ab)  C◦ (Ab) , where ‘◦ := + "’.

5.4.2.1

Spectral Sequences Associated with Bicomplexes

The bicomplex N ," is said to be of the first quadrant if N i,j = 0 for all (i, j ) ∈ N × N. As explained in Godement [46] (§4.8, p. 86), we assign to such bicomplex, two regular decreasing filtrations of (Tot◦⊕ (N ," ), D◦ ). The first is relative to the row "-filtration Tot◦⊕ (N ," ) := Tot◦⊕ (N ,"≥ ), and the second to the column filtration Tot◦⊕ (N ," )c := Tot◦⊕ (N ≥c," ). The same is true for bicomplexes N ," of the third quadrant or with finite number of nonzero rows or columns. In those cases, each filtration gives rise to a spectral sequence which converges to the cohomology of (Tot◦⊕ N ," , D◦ ), respectively   p,q p q IE 2 := h" h (N 

p,q

IE 2

p q

:= h h" (N

,"

) ⇒ h◦p+q (Tot◦⊕ N

,"

, D◦ ) ,

,"

) ⇒ h◦p+q (Tot◦⊕ N

,"

, D◦ ) ,

where h (resp. h" ) is the cohomology w.r.t. δ (resp. d" ). We now return to the action of derived functors on the category GM(G ). Note that the category Ab in the previous sections, now becomes the category GM∗ (G ) whose objects have a grading ‘∗’, and that the bicomplexes in C ," (GM∗ (G )) will thereby be equipped with three degrees: the column degree ‘ ’, the row degree ‘"’ and the G -graded module degree ‘∗’. Consequently, when using Tot functors we must be careful which pair of degrees we are dealing with.

5.4.3 The IR Hom• (−, −) and (−) ⊗IL (−) Bifunctors G G on GM(G ) Given V , V  , W ∈ GM(G ), the four functors Hom•G (V , −) ,

Hom•G (−, W ) ,

V ⊗  G (− ) ,

(−)⊗G V  ,

where the first two are left exact and the other two are right exact.

(5.30)

190

5 Equivariant Poincaré Duality

To shrink notations, we will now write ‘Hom• ’ for ‘Hom•G ’, and ‘⊗’ for ‘⊗G ’. Given projective resolutions P" (V ) → V , P" (V  ) → V  and an injective resolution W → I (W ) in GM(G ), we have natural morphisms of bicomplexes Hom• (P" (V ), W ) → Hom• (P" (V ), I W ) ← Hom• (V" , I W ) P" (V ) ⊗ V 

→

P" (V ) ⊗ P (V  )

←

V" ⊗ P (V  )

(5.31)

where ‘V" ’ is a shortcut for ‘V [0]" ’, and similarly for ‘V  ’ and ‘W ’. Applying Tot⊕ for (", ), we obtain canonical morphisms in C(GM(G )) Hom• (P" (V ), W ) → Tot⊕ Hom• (P" (V ), I W ) ← Hom• (V , I W ) P" (V ) ⊗ V 

 → Tot⊕ P" (V ) ⊗ P (V  ) ←

V ⊗ P (V  ) (5.32)

The following proposition is a classical result (Godement [46] §4.8). Proposition 5.4.3.1 The morphisms (5.32) are quasi-isomorphisms. Proof Since two projective (resp. injective) resolutions of a given object are homotopic, and since G is of finite dimensional cohomology, we can choose the resolutions to be of finite length. In that case the diagonals of constant (" + ) degree in the corresponding bicomplexes are of finite length, the associated spectral sequences (5.4.2.1) converge, and we can conclude. For example, in the first line of (5.32), we note that the morphisms of complexes are compatible with row and column filtrations of bicomplexes. In the case of Hom• (P" (V ), W ) → Tot◦⊕ Hom• (P" (V ), I W ) ,

(5.33)

for each i ∈ Z, the morphism Hom• (Pi (V ), W ) → Tot◦⊕ Hom• (Pi (V ), I W ) is a quasi-isomorphism since Pi (V ) is projective. Therefore, the induced map on the IE terms of the associated spectral sequences (5.4.2.1) is an isomorphism, which implies that (5.33) is a quasi-isomorphism. In the case of Tot◦⊕ Hom• (P" (V ), I W ) ←− Hom• (V , I W ) , the same idea works, except that now we must consider the row filtration and use the IE spectral sequence. We deal with the second line in (5.32) in the same way.   As a consequence of this proposition, in each line of (5.32), the three complexes represent the same object in the derived category D(GM(G )). These are classi-

5.4 Deriving Functors in GM(G )

191

cally denoted by the following notations: V , V  , W ∈ GM(G ) 

⎧ ⎫ • ⎨ IR HomG (V , W )) ⎬ ⎩

⎭

V ⊗ILG V 

∈ D(GM(G )) .

These constructions are natural w.r.t. each entry and, if we fix one entry, they can be extended naturally to bounded above complexes of G -gm in the free entries to be replaced by projective resolutions, and symmetrically on the other entry.12 We therefore have two well-defined bifunctors IR Hom•G (−, −) : D(GM(G )) × D+ (GM(G ))  D(GM(G )) (− ⊗ILG −)• : D(GM(G )) × D− (GM(G ))  D(GM(G ))

(5.34)

which are biadditive and have the usual variances and exactnesses. Comment 5.4.3.2 In (5.34), there are other possible domains of definition, for example, for IR Hom•G (−, −), we can also choose D− (GM(G ))×D(GM(G )). The compatibility with (5.34) in the intersection D− (GM(G )) × D+ (GM(G )) is then warranted by Proposition 5.4.3.1.

5.4.4 The Ext• and Tor• Bifunctors These are obtained by composing functors (5.34) with the cohomology functors hi : D(GM(G ))  GM(G ) , where we must take care that the cohomology degree is relative to the subscript index in the complex and not to the degree of G -gm’s. The definitions of Ext and Tor, are then given by ⎧  ⎫ i • ⎨ Exti,∗ G (−, −) := h IR HomG (−, −) ⎬ ⎩

Tor∗G ,i (−, −)

 := h − ⊗ILG − −i

⎭

∈ GM∗ (G ) ,

(5.35)

where ‘i’ is the subscript index in the complex and ‘∗’ is the G -gm grading.

12 In

a category with enough projective objects a bounded above complex is always quasiisomorphic to a bounded above complex of projective objects, and symmetrically for injective objects, see Weibel [95] §10.5 Derived Functors (p. 390) and §10.7 Ext and RHom (p. 394).

192

5 Equivariant Poincaré Duality

5.4.5 The Duality Functor on D(GM(G )) By restriction of (5.34) we get the (derived) duality functor IR Hom•G (−, G ) : D(GM(G ))  D(GM(G )) ,

(5.36)

IR Hom•G (−, G ) := Hom•G (−, I G ) ,

(5.37)

defined by

where G I G is an injective resolution. Comment 5.4.5.1 After Proposition 5.4.3.1, the restriction of this functor to D− (G ) is canonically isomorphic to IR Hom•G (−, G ) := Hom•G (P" (−), G ) .

(5.38)

5.4.6 The Duality Functor on D(DGM(G )) Refer to Sect. A.2.5 for definitions of the homotopy category K(DGM(G )) and the derived category D(DGM(G )) associated with DGM(G ). As stated in Theorem A.2.5.3, the derived duality functor IR Hom•G (−, G ) acting on D(DGM(G )) (5.2.3), is defined as in D(GM(G )) (5.4.3), considering an injective resolution G I G . Hence, we set IR Hom•G (−, G ) := Hom•G (−, Tot I G ) ,

(5.39)

which defines the derived duality functor IR Hom•G (−, G ) : D(DGM(G ))  D(DGM(G )) ,

(5.40)

which, composed with the cohomology functor hi : D(DGM(G ))  GM(G ), give the i’th extension functor i • ∗ Exti,∗ G (−, G ) := IR HomG (−, G ) : D(DGM(G ))  GM (G ). (5.41) 13 The family {Exti,∗ G (−, G )}i∈N is a ∂-functor in D(DGM(G )).

means that, applied to exact triangles in D(DGM(G )), they give rise to long exact sequences of cohomology. We will use this fact in the proof of Proposition 5.4.7.2-(2). See Weibel [95] §10.5 Derived Functors (p. 390).

13 Which

5.4 Deriving Functors in GM(G )

193

The relationship between the actions of IR Hom•G (−, −) on C(GM(G )) and on DGM(G ) is given in Proposition 5.3.2.1, which we restate on a restricted category where, modulo Tot⊕ , the two IR Hom• coincide. Proposition 5.4.6.1 On Db (GM(G )) × Db (GM(G )), one has  IR Hom•G Tot⊕ (−), Tot⊕ (−) = Tot⊕ IR Hom•G (−, −) Proof Since G is of finite homological dimension (Proposition 5.1.2.2), bounded complexes admit projective and injective resolutions of finite lengths, in which case the second part of Proposition 5.3.2.1 establishes the equality.  

5.4.7 Spectral Sequences Associated with IR Hom• (−, G ) G Let  : G I G be an injective resolution of finite length in GM(G ). For (V , d) ∈ DGM(G ), we can see the right-hand side in the definition   IR Hom•G (V , d), G := Hom•G (V , d), Tot⊕ (I G , δ ) ,

(5.42)

a bicomplex with rows indexed by the degree ‘"’ of V and columns by the index ‘ ’ of the injective resolution. The differentials d and δ commute, d increases the "-degree and leaves the -degree unchanged, while δ does the opposite. Proposition 5.4.7.1 Let (V , d) ∈ DGM(G ). 1. The filtration by increasing the -degree in I (G ) is regular, giving rise to a convergent spectral sequence 

p,q

IE 2

p,q

:= ExtG (hV , G )) ⇒ IR p+q Hom•G (V , G ) .

(5.43)

2. Assume (V , d) endowed with a regular filtration such that the associated spectral sequence (IE r (V ), dr ) is degenerated, i.e. there exists N ∈ N, such that dr = 0, for all r > N . Then, endowing IR Hom•G ((V , d), G ) with the induced filtration, we get a degenerated, hence convergent, spectral sequence IR Hom•G (IE r (V ), G ) ⇒ IR Hom•G (hV , G ) .

(5.44)

3. If V is projective as G -gm,14 then the following morphism of G -dgm’s induced by the augmentation  : G → I G , is a quasi-isomorphism: ()

Hom•G ((V , d), G ) −→ IR Hom•G ((V , d), G ) . 14 A

projective G -gm is always free, cf . Jacobson [59] corollary of Theorem 6.21, p. 386.

194

5 Equivariant Poincaré Duality

Proof (1) The -filtration is regular simply because (I G , δ ) is of finite length. The first page of the associated spectral sequence, is then (IE 1 , d1 ) := IR Hom•G (hp V , (I G , d )) , p,

and the statement follows (cf . 5.4.4). (2) The proof uses the exactness of the functor IR Hom•G (−, G ) on DGM(G ). It entails that, when the functor is applied to a degenerate spectral sequence, it gives a degenerate spectral sequence, the terms of which are those in (5.44). (3) Since I G is of finite length, we need only, by the mapping cone construction, show that if W := (0 → W0 → · · · → Wr−1 → Wr → 0) is a finite exact sequence of GM(G )-modules, then the G -dgm Hom•G ((V , d), Tot⊕ W ) ,

(5.45)

is acyclic. The following proof is dual to that of the Theorem A.2.5.3. By definition, a p-cocycle in (5.45) is a morphism α : V → Tot⊕ W [p] of G dgm. Now, recalling that Tot⊕ W = rk=0 Wk [ − k] (5.3.1), let us look at the component αr of α in the last G -gm Wr , which is necessarily a sub-G -module of cocycles. We then have the diagram V [−1]

d[−1] hr

Wr−1 [p − 1]

dr−1 [p]

V

d

V [1]

αr

Wr [p]

dr [p]

0

with dr−1 [p] surjective since Tot⊕ W is acyclic. We then factor αr through a morphism of G -gm’s hr : V → Wr−1 [p − 1], which is possible since V is projective, and we extend hr in a morphism of G -gm’s h = V → Tot⊕ W [p − 1], by zero on the coordinates j = r. The morphism α is now homotopic to the morphism of G -dgm’s α − (h[1] ◦ d + d ◦ h) : V → τ 0 ,

and, this conclusion being independent of the support of α, the nondegeneracy of the equivariant Poincaré pairing is thus proved.  

5.6.2 G-Equivariant Poincaré Duality Theorem Theorem 5.6.2.1 Let G be a compact Lie group, and let M an oriented G-manifold of dimension dM . 1. The G -graded morphism of complexes17  ID G,M : G (M)[dM ] −→ Hom•G G,c (M), G is an injective quasi-isomorphism.

16 See

Tu [91] §13.2 Integrating over a Compact connected Lie Group, p. 105. with Allday-Franz-Puppe [3] §4. The Main Result. Lemma 4.10, p. 6579.

17 Compare

(5.67)

204

5 Equivariant Poincaré Duality

2. The morphism ID G,M induces in GM(G ) the Poincaré duality morphism in Gequivariant cohomology (see 5.4.7.2-(1))  D G,M : HG (M)[dM ] −→ Hom•HG HG,c (M), HG

(5.68)

which is an isomorphism if ExtiHG (HG,c (M), HG ) = 0 for all i > 0, for example if HG,c (M) is a free HG -module.18 3. If G is connected, then there exist spectral sequences converging to HG (M)[dM ] ⎧ p,q   ⎨ IE 2 (M) = ExtpH HG,c (M), HG q ⇒ HGdM +p+q (M) G ⎩

p,q

p

−q

d +p+q

IF 2 (M) = HG ⊗R Hom•R (Hc (M), R) ⇒ HGM

(M)

where, in the first, q refers to the graded vector space grading. 4. If, in addition, M is of finite type, the right Poincaré adjunction  ID G,M : G,c (M)[dM ] −→ Hom•G G (M), G is an injective quasi-isomorphism, and mutatis mutandis for (2) and (3). Proof The injectivity of adjunction morphisms ID G,M and ID G,M is consequence of the nondegeneracy of the equivariant Poincaré pairing of Proposition 5.6.1.1. For the rest, since the action of W := G/G0 is exact, we will assume, as usual, that the group G connected. (1) We recall the filtration of the Cartan complex that we used earlier in the proof of 4.3.3.1-(2): Let m ∈ N. An equivariant form in (G (M), dG ) is said to be of index m if it belongs to the subspace  G (M)m := S ≥m (g ∨ ) ⊗ (M) G . We easily check that each G (M)m is stable under the Cartan differential dG , that G (M) = G (M)m for all m 0 and that we have a decreasing filtration G (M) = G (M)0 ⊇ G (M)1 ⊇ G (M)2 ⊇ · · ·

(5.69)

Furthermore, iG (M) ∩ G (M)m = 0 whenever m > i, so that (5.69) is a regular filtration (see [46] §4 pp. 76-). Similarly, λ ∈ Hom•G (G (M), G ) is said to be of index m whenever  , λ S a (g ∨ ) ⊗ c (M) G ⊆ ≥a+m G

18 See

∀a ∈ N ,

Ginzburg [44] Corollary 3.9, Brion [23] Proposition 1, Franz [42] Corollary 1.5.

5.6 Equivariant Poincaré Duality

205

and we denote Hom•G (G,c (M), G )m the vector subspace of such maps. As before, each of these is a subcomplex of (Hom•G (G (M)), D) and the resulting decreasing filtration · · · ⊇ Hom•G (G,c , G )m ⊇ Hom•G (G,c , G )m+1 ⊇ · · ·

(5.70)

verifies, for each λ homogeneous of degree i,  a + dim M + i ≥ deg λ (S a (g ∨ ) ⊗ c (M))G ≥ a + i ,

∀a ∈ N ,

so that the filtration (5.70) is also regular. An immediate verification shows that ID G,M is a morphism of graded filtered modules, i.e.   ID G,M G (M)[dM ]m ⊆ Hom•G G,c (M), G m ,

∀m ∈ Z ,

therefore giving rise to a morphism between the associated spectral sequences19 whose IF 0 terms are   p (i)  IF 0 := (S p (g ∨ ) ⊗ (M))G , 1 ⊗ d [dM ] (5.71)  p p (ii)  IF 0 := Hom•G S(g ∨ ) ⊗ c (M))G , 1 ⊗ d , G , p

≥p

>p

where G := G / G as G -gm. Lemma The complexes (5.71) are respectively quasi-isomorphic to   p (i’) G ⊗ (M), d [dM ] ,  p (ii’) Hom•G G ⊗ (c (M), d), G . Proof of Lemma 1 (i’) is a straightforward consequence of 4.4.6.1 applied to the following inclusions where the differentials come from ((M), d): p

p

G ⊗ (M) ⊇ G ⊗ (M)G ⊆ (S p (g ∨ ) ⊗ (M))G .

19 See

Godement [46], §4 Thm. 4.3.1, p. 80.

206

5 Equivariant Poincaré Duality

(ii’) Consider the following commutative diagram of complexes where the differentials are those coming from (c (M), d) only: Hom• G (S(

∨

)⊗

c (M))

p G

G

q.i.

IR Hom• G (S(

ξ

Hom• G

G

⊗

G

)⊗

c (M)

G

p G

q.i.

IR Hom• G

G

⊗

q.i.

⊗

c

p G

q.i.

IR Hom• G

c (M))

p G

G

ξ

q.i.

ξ

Hom• G

∨

G

c (M)

G

p G

ξ

⊗

c

p G

(5.72) – The vertical arrows ξ are those induced by the inclusions G ⊗ c (M) ⊇ G ⊗ c (M)G ⊆ (S p (g ∨ ) ⊗ c (M))G . which we know to be quasi-isomorphisms after (4.4.6.1). We can then immediately claim that the ξ ’s in the right-hand side of the diagram, concerned by the derived duality functor, are quasi-isomorphisms also. – The horizontal arrows (), induced by the augmentation morphisms (cf . 5.4.1), are also quasi-isomorphisms. Indeed, since G is compact, we have a decomposition S(g ∨ ) = G ⊗R H, where H denotes the (graded) subspace of G-harmonic polynomials of S(g ∨ ).20 Hence, we have   S(g ∨ ) ⊗R c (M) G  G ⊗R H ⊗ c (M) G .

(5.73)

We thus see that in the diagram (5.72), the first entries in the duality functors are all free G -gm’s, in which case, we can apply 5.4.7.1-(3) and conclude that the morphisms () are all quasi-isomorphisms as expected. 

By the lemma, the action of the morphism of spectral sequences IF (ID G,M ) induced by ID G,M at the IF 0 pages, i.e. the morphism IF 0 (ID G,M )  p ∨  p S (g ) ⊗ (M) G [dM ] −−−−−−−−→ Hom•G (S(g ∨ ) ⊗ c (M))G , G ,

20 This

is Kostant Theorem 0.2, in Kostant [67], p. 521, and in Kostant [68], p. 330.

5.6 Equivariant Poincaré Duality

207

for the differential 1 ⊗ d, is equivalent to the action of 1 ⊗ ID M in p

G ⊗ (M)[dM ]

1⊗ID M

  / Hom• G ⊗ c (M), p = Hom• c (M), p G R G G

From this we see the action of IF (ID G,M ) in the IF 1 pages : p,q

IF 1 p G

⊗ H q+dM (M)

IF 1 (ID G,M )

1⊗DM

p,q

IF 1

−q

p G)

Hom•R (Hc

where we recognize in the second row the classical Poincaré duality 2.4.1.3. The morphism IF 1 (ID G,M ) is therefore an isomorphism, which implies that ID G,M is a quasi-isomorphism of complexes as announced in (1). We can go a little further and easily determine the IF 2 pages. Indeed, since the p,q p+1,q is zero because G is evenly graded with zero differential d1 : IF 1 → IF 1 differential, we have an equality of page IF 2 = IF 1 , in which case p,q

−q

p

d +p+q

IF 2 (M) = HG ⊗R Hom•R (Hc (M), R) ⇒ HGM

(M) ,

as stated in (3). (2) Thanks to (1) D G,M factors through the natural morphism h Hom•G (G,c (M), G ) → Hom•HG (h(G,c (M)), HG ) and we can apply Proposition 5.4.7.2, since G (M) is a free G -gm, as was noted in the previous paragraphs. (3) According to (1), ID G,M stablishes a canonical quasi-isomorphism  G (M)[dM ] −−→ Hom•G G,c (M), G , q.i.

(5.74)

and the spectral sequence IE(M) is simply the IE spectral sequence of Proposition 5.4.7.1 converging to the cohomology of the right-hand side of (5.74). The existence of the spectral sequence IF (M) was justified at the end of the proof of (1). Statement (4) follows by the same arguments.  

208

5 Equivariant Poincaré Duality

5.6.3 Torsion-Freeness, Freeness and Reflexivity Proposition 5.6.2.1-(2,4) shows that freeness of equivariant cohomology as HG gm is sufficient for equivariant Poincaré duality to hold. The question then arises whether some weaker condition could be equivalent to duality. The following two properties are related to this question and have been thoroughly studied in Allday et al. [3]. AFP-1. Torsion-freeness. An HG -gm V is said to be torsion-free if Ann(v) := {P ∈ HG | P · v = 0} = 0 ,

for all v ∈ V .

The torsion-freeness of equivariant cohomology, which is clearly a necessary condition for duality, since the modules Hom•HG (−, HG ) are torsion-free, is also a sufficient condition for the injectivity of the Poincaré duality morphism (Prop. 5.9 [3], see Exercise 7.2.2), but not for duality, as examples of Franz-Puppe [43] show. AFP-2. Reflexivity. An HG -gm V is said to be reflexive if the natural map  V → Hom•HG Hom•HG (V , HG ), HG is an isomorphism. For a finite type manifold M, while the reflexivity of HG (M) and HG,c (M) are clearly necessary conditions to HG -duality, the converse, which is also true, is much more subtle. The equivalence between duality and reflexivity has been established in [3] (Prop. 5.10) for G abelian, and in Franz [42] (Cor. 5.1) for general G and real coefficients. The following diagram illustrates the relationship between the different kinds of nontorsions in equivariant cohomology and significant properties of the equivariant Poincaré pairing. {free} ⊆ {reflexive} *

* Perfect Poincaré pairing

⊆ {torsion-free} +

⊆

*

* Nondegenerate Poincaré pairing

+

It is worth noting that in [41], Franz gives the first known examples of compact manifolds having reflexive but nonfree equivariant cohomology.

5.6 Equivariant Poincaré Duality

209

5.6.4 T -Equivariant Poincaré Duality Theorem When G is a compact connected torus T = S1 × · · · × S1 , we have: ⎧ T = S(t ∨ ) ⎪ ⎪ ⎪ ⎨ T (M) = S(t ∨ ) ⊗R (M)T ⎪ ⎪ ⎪ ⎩ T ,c (M) = S(t ∨ ) ⊗R c (M)T so that  Hom•T (T ,c , T ) = HomS(t ∨ ) S(t ∨ ) ⊗ c (M)T , S(t ∨ )  = Hom•R c (M)T , S(t ∨ )  = S(t ∨ ) ⊗R Hom•R c (M)T , R The left adjunction ID T (M) associated with the T -equivariant Poincaré pairing ·, ·T (see 5.6.1.1) identifies naturally to 1 ⊗ ID M ,  ID T (M) S(t ∨ ) ⊗ (M)T [dM ] −−−−−−→ S(t ∨ ) ⊗ Hom•R c (M)T , R 1⊗ID M

P ⊗α

 −−−−−−→



 P ⊗ β → α∧β M

and the right adjunction (see 5.6.1.2) to ID T (M)  S(t ∨ ) ⊗ c (M)T [dM ] −−−−−− → S(t ∨ ) ⊗ Hom•R (M)T , R  1⊗ID M

P ⊗β

 −−−−−−→

 

α∧β P ⊗ α → M

Theorem 5.6.4.1 Let T be a compact connected torus, and M an oriented T manifold of dimension dM . 1. The HT -graded morphism of complexes  ID T (M) : T (M)[dM ] −→ Hom•T T ,c (M), T is an injective quasi-isomorphism.

210

5 Equivariant Poincaré Duality

2. The morphism ID T (M) induces the ‘Poincaré duality morphism in T -equivariant cohomology’ (see 5.4.7.2-(1))  D T ,M : HT (M)[dM ] −→ Hom•HT HT ,c (M), HT If HT ,c (M) is a free HT -module, D T ,M is an isomorphism. 3. There are natural spectral sequences converging to HT (M)[dM ] ⎧ p,q ⎨ IE 2 (M) =

p,q  ExtHT HT ,c (M), HT

⎩

p

p,q

q

p+q

⇒ HT

p+q

IF 2 (M) = HT ⊗R HomR (Hc (M), R) ⇒ HT

(M)[dM ]

(M)[dM ] ,

where, in the first, q denotes the HT -graded vector space degree. 4. If, in addition, M is of finite type, then the G -graded morphism of complexes  ID T (M) : T ,c (M)[dM ] −→ Hom•HT T (M), HT is a quasi-isomorphism, and mutatis mutandis for (2) and (3). Proof The theorem is a particular case of 5.6.2.1 taking G := T . Notice however that since we have the identification ID T (M) = 1 ⊗ ID M , we may also conclude using 4.3.3.1-(4).   Comment 5.6.4.2 Recall that HT ,c (M) is a free HT -module whenever M has no odd (or no even) degree compactly supported cohomology (4.3.3.1-(4c)). Clearly, though not very interesting, this is true also when T acts trivially on M, since then ι(Y ) = θ (Y ) = 0, ∀ Y ∈ t, and HT ,c (M) = HT ⊗R Hc (M).

Chapter 6

Equivariant Gysin Morphism and Euler Classes

We define the Gysin morphisms within the equivariant framework following the approach described in Sect. 2.8. Note that thanks to the equivariant de Rham comparison Theorem 4.8, we know a priori that equivariant Gysin morphisms correspond to relative Gysin morphisms over H (IBG; R) (3.5). In Sect. 6.5, we recall the definition of equivariant Euler classes as an application of Gysin morphisms and state their basic properties.

6.1 G-Equivariant Gysin Morphism 6.1.1 Equivariant Finite de Rham Type Coverings Recall that by 2.5.3.1, when G is a compact Lie Group, a G-manifold M is the union of a countable ascending chain U := {U0 ⊆ U1 ⊆ · · · } of G-stable open subsets of finite type. The following theorem, a simple corollary of the G-equivariant Poincaré duality Theorem 5.6.2.1, is the equivariant analogue to Proposition 2.6.3.1. Its proof is the same, so details are left to the reader. Theorem 6.1.1.1 Let G be a compact Lie group, and let M be an oriented Gmanifold. Then, for every filtrant cover U of M by G-stable open subsets, the natural morphism lim U ∈U G,c (U ) → G,c (M) is an isomorphism, and the −→ analogue to 2.6.3.1–(2) is a well-defined morphism of complexes:  ID G,U : (G,c (M), dG )[dM ] −→ lim U ∈U Hom•G (G (U ), G ), −D , −→ and, if the open sets in U are of finite type, then ID G,U is a quasi-isomorphism.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Arabia, Equivariant Poincaré Duality on G-Manifolds, Lecture Notes in Mathematics 2288, https://doi.org/10.1007/978-3-030-70440-7_6

211

212

6 Equivariant Gysin Morphism and Euler Classes

6.1.2 G-Equivariant Gysin Morphism for General Maps We now follow closely the steps of Sect. 2.8.2. Let f : M → N be a G-equivariant map between oriented G-manifolds. To β ∈ G,c (M) we assign by equivariant integration the G -linear form ID G,f (β) : G (N ) → G ,

α →

 M

f ∗α ∧ β .

We thus have the diagram G,c (M)[dM ]

f!

⊕

ID G,f

G,c (N)[dN ]

quasi-iso if N is of finite type

ID G,N

Hom• G

G)

G

which can be commutatively closed in derived category D(DGM(G )) when N is of finite type, since, in that case, ID G,N is a quasi-isomorphism (5.6.2.1-(4)). When N is not of finite type, we fix an equivariant cover U of N of open finite type subspaces (6.1.1), and replace ID G,N with the morphism ID G,U of Theorem 6.1.1.1. Then, the following diagram, defined as in 2.8.2, G,c (M)[dM ] ID G,f,

f!

⊕

G,c (N)[dN ] ID

(quasi-iso)

G, •

Hom

lim U ∈

G

G

G)

can be commutatively closed in D(DGM(G )) since ID G,U is a quasiisomorphism. The closing arrow in D(DGM(G )) f! := (ID G,U )−1 ◦ ID G,f,U : G,c (M)[dM ] → G,c (N )[dN ] .

(6.1)

is the equivariant Gysin morphism associated with f . It defines in cohomology the morphism of G -gm’s f! : HG,c (M)[dM ] → HG,c (N )[dN ]

(6.2)

Theorem 6.1.2.1 (And Definitions) Let G be a compact Lie group. With the above notations, the following statements are verified. 1. The equality  M

f ∗ [α] ∪ [β] =

 N

[α] ∪ f! [β]

holds for all [α] ∈ HG (N ) and [β] ∈ HG,c (M).

(6.3)

6.1 G-Equivariant Gysin Morphism

213

2. Furthermore, f! is a morphism of HG (N )-modules, i.e. the equality, called the equivariant projection formula,  f! f ∗ [α] ∪ [β] = [α] ∪ f! ([β])

(6.4)

holds for all [α] ∈ HG (N ) and [β] ∈ HG,c (M). 3. The correspondence  (−)! : G-Manor  D DGM(G )

with

M M! := G,c (M)[dM ] f  f!

is a covariant functor, which we refer to as the equivariant Gysin functor for general maps. And, mutatis mutandis replacing D DGM(G ) by GM(G ) and G,c (−) by HG,c (−). 4. Suppose that M and N are manifolds of finite type. If the pullback morphism f ∗ : HG (N) → HG (M) is an isomorphism, then the Gysin morphism f! : HG,c (M)[dM ] → HG,c (N )[dN ] is an isomorphism too. Proof (1) Immediate from the definition of the Gysin morphism. (2) Unlike the proof of the nonequivariant statement 2.6.2.1–(2), this claim is no longer a formal consequence of (1) because equivariant cohomology may have torsion elements, which are killed by equivariant integration. Instead, when N is of finite type and since then ID N is a quasi-isomorphism, we can check that the following equality holds at the cochain level, ID G,f (f ∗ (α) ∪ β) = ‘ID N (α ∪ f! (β))’ = ID G,f (β) ◦ μr (α) ,

(6.5)

where the central term is included for purely heuristic reasons and where we denote μr (α) : G (N) → G (N ) the right multiplication by α, i.e. μr (α)(−) = (−) ∪ α. The identification of the left and right-hand terms in 6.5 is then a straightforward verification from the definition of ID G,f . When N is not of finite type, we use the same arguments with ID G,f,U instead of ID G,f . (3) is clear. (4) As f ∗ : G (N ) → G (M) is a quasi-isomorphism, the induced map Hom•HG (G (N ), HG ) → Hom•HG (G (M), HG ) is also a quasi-isomorphism, after 5.4.7.2–(3). We can thus conclude, since ID G,M and ID G,N are both quasiisomorphisms.   Exercise 6.1.2.2 Prove the following enhancement of the statement 6.1.2.1–(4). If π : V → B is a vector bundle over an oriented manifold B, then π is of finite type (2.5.2.1), and π ∗ : HG (B) → HG (V ) and π! : HG,c (V )[dV ] → HG,c (B)[dB ] and both isomorphisms (Exercise 2.5.2.4–(1)).

214

6 Equivariant Gysin Morphism and Euler Classes

6.1.3 G-Equivariant Gysin Morphism for Proper Maps Following 2.8.1, let f : M → N be a proper G-equivariant map between oriented G-manifolds. To α ∈ G (M) we assign the HG -linear form on G,c (N ) defined by ID G,f (α) : β → M f ∗ β ∧ α. In this way we obtain the diagram G (M)[dM ] ID G,f

f∗

G (N)[dN ]

⊕

ID G,N (quasi-iso) G,c (N)

∨

which may be commutatively closed in D(DGM(G )) because ID G,N is a quasiisomorphism after 5.6.2.1-(1). The closing arrow in D(DGM(G )) f∗ := (ID G,N )−1 ◦ ID G,f : G (M)[dM ] → G (N )[dN ] ,

(6.6)

is the equivariant Gysin morphism associated with the proper map f . It defines in cohomology the morphism of G -gm’s f∗ : HG (M)[dM ] → HG (N )[dN ]

(6.7)

Theorem 6.1.3.1 (And Definitions) Let G be a compact Lie group. With the above notations, the following statements are verified. 1. The equality  M

f ∗ [β] ∪ [α] =

 N

[β] ∪ f∗ [α]

(6.8)

holds for all [α] ∈ HG (M) and [β] ∈ HG,c (N ). 2. Furthermore, f∗ is a morphism of HG,c (N )-modules, the equality, called the equivariant projection formula for proper maps,  f∗ f ∗ [β] ∪ [α] = [β] ∪ f∗ [α]

(6.9)

holds for all [β] ∈ HG,c (N ) and [α] ∈ HG (M). 3. The correspondence  f∗ :

G-Manor pr

 D DGM(G )

with

M  M∗ := G (M)[dM ] f  f∗

is a covariant functor, which we will refer to as the equivariant Gysin functor for proper maps. And, mutatis mutandis replacing D DGM(G ) by GM(G ) and G (−) by HG (−).

6.1 G-Equivariant Gysin Morphism

215

4. If the pullback morphism f ∗ : HG,c (N ) → HG,c (M) is an isomorphism, then the Gysin morphism f∗ : HG (M)[dM ] → HG (N)[dN ] is also an isomorphism. 5. The natural map φ(−) : HG,c (−)[d− ] → HG (−)[d− ] (2.2.3) is a homomorphism between the two equivariant Gysin functors (−)! → (−)∗ over the category G-Manor pr , i.e. we have natural commutative diagrams HG,c (M) f!

HG,c (N)

φ(M)

⊕ φ(N)

HG (M) f∗

HG (N)

Proof (1, 2, 3, and 4) Same as the proof of Theorem 6.1.2.1. (5) Immediate after definitions.  

6.1.4 Gysin Morphisms through Spectral Sequences The next theorem establishes a close connection between the nonequivariant and the equivariant Gysin morphisms. It is a basic tool for the generalization of known properties of classical Gysin morphisms within the equivariant framework. Theorem 6.1.4.1 Let G be a compact connected Lie group and f : M → N a Gequivariant map between oriented G-manifolds. There exists a natural morphism of the spectral sequences IF of Theorem 5.6.2.1–(3) converging to the Gysin morphism f! : HG,c (M)[dM ] → HG,c (N )[dN ], IF c,2 (M) = HG ⊗ Hc (M)[dM ] ⇒ HG,c (M)[dM ] ⏐ ⏐ ⏐ ⏐ 1⊗f!   f! IF c,2 (N ) = HG ⊗ Hc (N )[dN ] ⇒ HG,c (M)[dN ] and in the proper case to f∗ : HG (M)[dM ] → HG (N )[dN ], IF 2 (M) = HG ⊗ H⏐(M)[dM ] ⇒ HG (M)[d ⏐ M] ⏐ ⏐ 1⊗f∗   f∗ IF 2 (N ) = HG ⊗ H (N)[dN ] ⇒ HG (M)[dN ] In particular, an equivariant Gysin morphism is an isomorphism if the corresponding nonequivariant Gysin morphism is so. Proof Clear from the proof of Theorem 5.6.2.1 and the definition of Gysin morphisms.  

216

6 Equivariant Gysin Morphism and Euler Classes

6.2 Group Restriction and Equivariant Gysin Morphisms 6.2.1 Group Restriction and Equivariant Cohomology Since a closed subgroup H of a compact Lie group G is a compact Lie group, we have projection maps, functorial on M ∈ G-Man, between Borel constructions IEG×M νH,M

νG,M G νH,M

MH := IEG ×H M (x, m) mod H

IEG ×G M =: MG (x, m) mod G

where the reader will have observed that we are using the same space IEG as universal fiber bundle for G and H simultaneously. In particular, the classifying space for H is IBH := IEG/H (Sect. 4.9). Proposition 6.2.1.1 Let H ⊆ G be an inclusion of compact Lie groups, and let M be a G-manifold. 1. We have a Cartesian diagram of fiber bundles MH := IEG ×H M

G νH,M

[G/H ]

MG := IEG ×G M

πH,M [M]

IBH := IEG/H

[M] πG,M G νH,IB

[G/H ]

(6.10)

IBG := IEG/G

where the vertical arrows are induced by the constant map cM : M → {•}, and where fiber spaces are shown in brackets. G 2. The map νH,M : MH → MG is a morphism of fiber bundles MH

G νH,M

MG

[G/H ] [G×H M] [M]

(6.11) πG,M

G ◦π νH,IB H,M

IBG with base IBG and fibers respectively G×H M and M, as shown in brackets. Proof (1) After 3.1.4.2–(2), the diagram is Cartesian if and only if it is so locally relative to IBG. For every open subspace V ⊆ IBG which is trivializing for the canonical projection ν : IEG → → IBG, let σ : V → IEG be a section of ν defining

6.2 Group Restriction and Equivariant Gysin Morphisms

217

the slice S := σ (V ) ⊆ IEG (4.5.2). We then have ν −1 (V )  S×G, and the local view of (6.10) is given by the diagram S×G×H M

S×G×G M (6.12) S × G/G

S×G/H

which is Cartesian after Exercise 3.1.4.3. G (2) The same diagram (6.12) shows that the map νH,M ◦ πH,M : MH → IBG is a locally trivial fibration of fiber G×H M.   Definition 6.2.1.2 Let H ⊆ G be an inclusion of compact Lie groups, and let M be a G-manifold. We call group restriction morphisms and we denote ResG H : Ω(MG ; k) → Ω(MH ; k)

and

ResG H : G (M) → H (M) ,

G : MH → MG . the pullback morphisms induced by the map νH,M Thanks to 3.1.10.2–(2), these restrictions respect compact supports and therefore induce the group restriction morphisms of complexes

ResG H : Ωcv (MG ; k) → Ωcv (MH ; k)

and

ResG H : G,c (M) → H,c (M) ,

functorial on M ∈ G-Manpr .

6.2.2 Group Restriction and Integration An oriented G-manifold M, is automatically an oriented H -manifold, and, because of 6.2.1.1–(1) and 3.2.1.3–(1), we get commutative diagrams combining group restriction and integration morphisms Ωcv (MG ; R) M

Ω(IBG; R)

ResG H

⊕ ResG H

Ωcv (MH ; R) M

Ω(IBG; R)

G,c (M)

ResG H

H,c (M)

⊕

M

M

ResG H G

H

It is then natural to expect some kind of compatibility between G and H -equivariant Gysin morphism. The following theorem addresses this question. Theorem 6.2.2.1 Let H ⊆ G be an inclusion of compact Lie groups. For every G-equivariant map f : M → N between oriented G-manifolds, the following

218

6 Equivariant Gysin Morphism and Euler Classes

diagrams of Gysin morphisms are commutative. HG (M) ResG H

f∗

HG (N) ResG H

(if f is proper)

HH (M)

f∗

HH (N)

HG,c (M)

f!

HG,c (N) ResG H

ResG H

HH,c (M)

f!

(6.13)

HH,c (N)

Proof By the equivariant de Rham comparison Theorem 4.8, this is a particular case of the commutativity between Gysin functors and base change established in Proposition 3.5.2.1.  

6.3 Adjointness of Equivariant Gysin Morphisms This is the equivariant analogue to Sect. 3.6.1 where we discussed the concept of adjoint Poincaré morphisms at the level of complexes of differential forms. There, the concept was based on the property of nondegeneracy of the relative Poincaré pairing, a property which holds also for the equivariant Poincaré pairing, as established in Proposition 5.6.1.1, which allows us to introduce the same concept in the equivariant context.

6.3.1 Adjointness Property The following proposition, analogue to Proposition 3.6.1.1, can help find explicit Gysin morphisms at the level of G -dg-modules. The proof, is word for word the proof of Proposition 3.6.1.1. Proposition 6.3.1.1 Let f : M → N be an equivariant map between oriented G-manifolds. 1. A set-theoretic map f" : G,c (M)[dM ] → G,c (N )[dN ] verifying,  M

f ∗ (α) ∧ β = 

 M

α ∧ f" (β) ,

(6.14)

for all α ∈ G (N ) and β ∈ G,c (M), is automatically a morphism of G -dgmodules inducing the Gysin morphism f! in D(DGM(G , d)). Furthermore,  f" f ∗ (α) ∧ β = α ∧ f" (β) , for all α ∈ G (N ) and all β ∈ G,c (M).

(6.15)

6.3 Adjointness of Equivariant Gysin Morphisms

219

2. Let f be proper. A set-theoretic map f" : G (M)[dM ] → G (N )[dN ] verifying,  M

f ∗ (β) ∧ α = 

 M

β ∧ f" (α) ,

(6.16)

for all α ∈ G (M) and β ∈ G,c (N ), is automatically a morphism of G -dgmodules inducing the Gysin morphism f∗ in D(DGM(G , d)). Furthermore,  f" f ∗ (β) ∧ α = β ∧ f" (α) ,

(6.17)

for all α ∈ G (M) and all β ∈ G,c (N ). Remarks 6.3.1.2 1. Take care that the converse of the statements in 6.3.1.1 are not true. Indeed, since the equivariant Poincaré pairing at the level of complexes is nondegenerated (5.6.1.1), expressions 6.14 and 6.16, where they exist, characterize unique morphisms of complexes M and ϕ∗ , while in general many induce the same Gysin morphisms in cohomology.1 2. Unlike the nonequivariant case (3.6.1.2), the equivariant Gysin morphism in cohomology is generally not characterized by the equality:  M

f ∗ [α] ∪ [β] =

 N

[α] ∪ f! [β] ,

∀[α] ∈ HG (N ) , ∀[β] ∈ HG,c (M) , (6.18)

due to the presence of torsion elements. For example, the uniqueness of f! satisfying the relation 6.18, results only from the injectivity of the map:  ID G,N : HG,c (N ) −−→ Hom•HG HG (N ), HG 

 [β] −−→ [α] → [α] ∧ [β] N

a property which is not always satisfied. Indeed, let T be a torus and let N a compact oriented T -manifold without fixed points. We will see from the localization theorem (Sect. 7.5.1.1) that HT (N ) is a torsion HT -module (7.4.1). This implies that Hom•HT (HG (N ), HT ) = 0, in which case ID G,N is null, although HT (N ) = 0. By contrast, when HG (N ) and HG,c (N ) are nontorsion (free for example) conditions (6.18) et (6.8) characterize well Gysin morphisms in cohomology.

1I

thank the referee for this clarification.

220

6 Equivariant Gysin Morphism and Euler Classes

6.4 Explicit Constructions of Equivariant Gysin Morphisms Based in the adjunction existence criterion 6.3.1.1, we now review well-known explicit constructions of Gysin morphisms associated with familiar maps, just as in the nonequivariant case discussed in Sect. 3.6.

6.4.1 Equivariant Open Embedding Let M be an oriented G-manifold, and let U be a G-stable open set in M. Denote by j : U ⊆ M the injection and endow U with the induced orientation. We have the natural extension by zero j! : G,c (U ) → G,c (M) inclusion of Cartan complexes, and the obvious equality  U

j ∗ (α) ∧ β =

 M

α ∧ j! (β) ,

∀α ∈ G (M) , ∀β ∈ G,c (U ) .

The pair (j ∗ , j! ) is thus an equivariant Poincaré adjoint pair in DGM(G , d), and this gives immediately, by Proposition 6.3.1.1–(2), the identification of the equivariant Gysin morphism associated with the open restriction j ∗ : G (M) → G (U ) and the extension by zero morphism j! : G,c (U )[dU ] → G,c (M)[dM ] .

6.4.2 Equivariant Constant Map Let M be an oriented G-manifold. The constant map cM : M → {•} is ∗ :  G-equivariant, and G ({•}) = G . The pullback cM G → G (M) is the G -module structure morphism and the equivariant integration map M : G,c (M)[dM ] → G is G -linear (cf . Sect. 5.5). Hence, we have  M

∗ cM (α) ∧ β =

 M

α∧β =α∧

 M

β=

 {•}

α∧

M

β.

∗ , The pair (cM M ) is thus an equivariant Poincaré adjoint pair in DGM(G , d), and this gives, by Proposition 6.3.1.1–(2), the identification of the equivariant ∗ :  ({•}) →  (M) and the Gysin morphism associated with the pullback cM G G equivariant integration morphism: cM! =

 M

: G,c (M)[dM ] → G,c ({•}) = G .

6.4 Explicit Constructions of Equivariant Gysin Morphisms

221

6.4.3 Equivariant Projection Given two oriented G-manifolds M and N , let π : N × M → → N , denote the projection map (x, y) → x. The morphisms of integration along fibers M

: cv (N ×M)[dM ] → (N)

and

M

: c (N ×M)[dM ] → c (N ) (6.19)

which verify after Proposition 3.2.1.3–(1) M

π ∗α ∧ β = α ∧

M

∀α ∈ (N) , ∀β ∈ cv (N ×M) ,

β,

(6.20)

commute with the action of G and with g-interior products since, locally on N ×M, we have M (g · (ν ⊗ μ)) = M (g · ν) ⊗ (g · μ) = (g · ν) ⊗ M (g · μ) = (g · ν) ⊗ M μ = g · M ν ⊗ μ (6.21)    deg ν ι(X)(ν ⊗ μ) = ι(X)(ν) ⊗ μ + (−1) ν ⊗ ι(X)(μ) M M M  =1 M ι(X)(ν) ⊗ μ = ι(X) ν ⊗ M μ , where (=1 ) is justified since M ι(X)(μ) = 0. The morphisms (6.19) can then be naturally extended to Cartan complexes M

: G,cv (N ×M)[dM ] → G (N )

and

M

: G,c (N ×M)[dM ] → G,c (N ) ,

 where G,cv (N ×M) := S(g ∨ ) ⊗ cv (N ×M) G . In addition, by Fubini’s theorem, we have:  N ×M

π ∗ (α) ∧ β =



 N M

π ∗ (α) ∧ β =1

 N

α∧

M

β,

(6.22)

for all α ∈ (N ) and β ∈ cv (N×M), and where (=1 ) is justified by (6.20), which also extends naturally to the equivariant context, showing that the pair (π ∗ , M ) is an equivariant Poincaré adjoint pair in DGM(G , d). We can thus again conclude, by Proposition 6.3.1.1–(2), to the identification π! =

 M

: G,c (N×M)[dM ] → G,c (N ) ,

thus extending the case of the constant map discussed in Sect. 6.4.2.

222

6 Equivariant Gysin Morphism and Euler Classes

6.4.4 Equivariant Fiber Bundle Equivariant projections are particular cases of equivariant fiber bundles. Let (E, B, π, M) be an oriented G-equivariant fiber bundle with fiber M. Integration along fibers gives a morphism of complexes M : cv (E) → c (B) such that if ψ : E → E is an isomorphism exchanging fibers, then M ◦ ψ ∗ = ψ ∗ ◦ M , so that M is G-equivariant. On the other hand, M commutes with the interior products ι(X). Indeed, since these are local operators, it suffices (modulo partitions of unity if necessary) to verify the claim over a trivializing open subset of E, i.e. over π −1 (U ) for U s.t. π −1 U ∼ U ×M, where we are in the case of a projection already discussed in Sect. 6.4.3. (Beware that although U ×M is not the product of two G-manifolds, the infinitesimal action of G induces ‘vertical ’ and ‘horizontal ’ interior products ι(X), and equalities (6.21) remain valid.) The study of equivariant projections of Sect. 6.4.3 extends then naturally to the case of equivariant fiber bundles, and the pair (π ∗ , M ) is an equivariant Poincaré adjoint pair in DGM(G , d). Hence the identification π! =

 M

: G,c (E)[dM ] → G,c (B) .

6.4.5 Zero Section of an Equivariant Vector Bundle Let (V , B, π, Rn ) be a G-equivariant oriented vector bundle, and let σ : B → V be the zero section map, which is also a G-equivariant map.

6.4.5.1

The Equivariant Thom Class

The underlying formalism of the nonequivariant Thom class (Sects. 3.2.3 and 3.6.5) remains valid in the equivariant framework. In Proposition 3.2.3.1, we proved that integration along fibers  M

: cv (V )[n] → (B)

(6.23)

is a quasi-isomorphism, and we defined the Thom class of (V , B, π, M) as the dM unique cohomology class π ∈ Hcv such that  M

π = 1 .

The following proposition generalizes this fact in the equivariant setting.

(6.24)

6.4 Explicit Constructions of Equivariant Gysin Morphisms

223

Proposition 6.4.5.1 (and Definition) Let G be a compact Lie group and let (V , B, π, Rn ) be a G-equivariant oriented vector bundle.  1. The submodule S(g ∨ ) ⊗ cv (V ) G of G (V ) is stable under the action of Cartan’s differential dG := d + ι(X) and defines the G (B)-dg-module of Gequivariant differential forms with π -proper supports  G,cv (V ) := S(g ∨ ) ⊗ cv (V ), dG G .

(6.25)

The S(g ∨ )-linear extension of the integration along fibers (6.23) defines morphisms of G (B)-dg-modules 

(i) (ii)



Rn

: G,cv (V )[n] → G (B)

Rn

: G,c (V )[n] → G,c (B) ,

(6.26)

where (i) is a quasi-isomorphism. 2. There exists a homogeneous G-equivariant cocycle in cv (V ) of total degree n ΦG = Φ [n] + Φ [n−2] + Φ [n−4] + · · ·  G [n] ∈ n (V )G represents with Φ [n−2k] ∈ S k (g ∨ ) ⊗ n−2k cv (V ) , and where Φ cv n the Thom class of (V , B, π, R ) (see Proposition 3.2.3.1). Two such cocycles are cohomologous, and the maps σ! : G (B) −→ G,cv (V )[n] ν

σ! : G,c (B) −→ G,c (V )[n]

∗

 −→ π ν ∧ ΦG

ν

 −→ π ∗ ν ∧ ΦG

are morphisms of Cartan complexes which are right quasi-inverses to (6.26). 3. The zero section σ : B → V of the vector bundle (V , B, π, Rn ) is a proper G-equivariant map and we have σ ∗ (α) =

 Rn

α ∧ ΦG ,

∀α ∈ G (V ) .

Proof (1) As recalled in Sect. 4.4, the natural action of G on the de Rham complex ((V ), d) induces its structure of g-dg-algebra. The relations g · α = |α|, |dα| ⊆ |α| and |ι(X)(α)| ⊆ |α|, for all α ∈ (V ), then show that the subcomplexes (c (V ), d) and (cv (V ), d) are (G, g)-dg-submodules of ((V ), d). Furthermore, for reasons similar to those in Sects. 6.4.3 and 6.4.4, the maps Rn

: cv (V )[n] → (B)

and

Rn

: c (V )[n] → c (B) ,

(6.27)

are morphisms of ((B), g)-dg-modules. Consequently, and by functoriality of Cartan complexes (cf . Theorem 4.3.3.1), these morphisms (6.27) induce the mor-

224

6 Equivariant Gysin Morphism and Euler Classes

phisms of Cartan complexes in (6.26), i.e. 

(i) (ii)



Rn

: G,cv (V )[n] → G (B)

Rn

: G,c (V )[n] → G,c (B) ,

(6.28)

which are clearly also morphisms of G (B)-dg-modules. To see that (i) is a quasi-isomorphism, it will suffice, as usual, to assume G connected, in which case we can consider the induced action of Rn on the second page of the spectral sequences associated with the complexes G,cv (V ) and G (B) filtered by polynomial degrees, which is simply the map: idS(g ∨ )g ⊗ π! : S(g ∨ )g ⊗ Hcv (V ) → S(g ∨ )g ⊗ H (B) , where π! : Hcv (V ) → H (B) is an isomorphism as it is the relative Gysin morphism over H (B) associated with the pullback π ∗ : H (B) → H (V ), which is an isomorphism because π is a homotopy equivalence. (2) The fact that (i) in (6.26) is a quasi-isomorphism proves the existence of the equivariant cocycle ΦG as well as its uniqueness modulo coboundaries. There is however a constructive proof of its existence, which we now recall. Since G is compact, every Thom form in ncv (V ) can be G-averaged, allowing us to assume that there exists a Thom form Φ [n] ∈ Zncv (V )G . We then have dG (Φ [n] ) = d Φ [n] +

r

i=1 e

i

⊗ ι(ei )(Φ [n] ) =

r

i=1 e

i

⊗ ι(ei )(Φ [n] ) ,

where {ei }ri=1 and {ei }ri=1 are dual bases of g, and where d(ι(ei )(Φ [n] )) = θ (ei )(Φ [n] ) = 0 ,

∀i = 1, . . . , r ,

since Φ [n] is G-invariant. It follows that ι(ei )(Φ [n] ) ∈ Zn−1 cv (V ), and since after (6.23), we have Hcv (V )  H j −n (B) = 0 , j

∀j < n ,

(6.29)

[n] there exists ϕi ∈ n−2 cv (V ) such that d ϕi = ι(ei )(Φ ), in which case



dG Φ [n] − ri=1 ei ⊗ ϕi ∈ S 2 (g ∨ ) ⊗ n−3 cv (V ) . We then define Φ [n−2] := −



G g·

 r

i=1 e

i

 G ⊗ ϕi dg ∈ S 1 (g ∨ ) ⊗ n−2 ⊆ nG,cv (V ) , cv (V )

 so that, by construction, dG Φ [n] + Φ [n−2] = ι(X)(Φ [n−2] ) .

6.4 Explicit Constructions of Equivariant Gysin Morphisms

225

The same procedure on ι(X)(Φ [n−2] ),possible because of the vanishing condiG such that tion (6.29), leads to an element Φ [n−4] ∈ S 2 (g ∨ ) ⊗ n−4 cv (V )  dG Φ [n] + Φ [n−2] + Φ [n−4] = ι(X)(Φ [n−4] ) ∈ S 3 (g ∨ ) ⊗ n−5 cv (V ) .

The iteration of this procedure ends with a cocycle ΦG := k∈N Φ [n−2k] , which verifies, by construction, Rn ΦG = Rn Φ [n] = 1, hence the equality after (6.27) 

π ∗ (α) ∧ ΦG = α,

Rn

∀α ∈ G (B) ,

(6.30)

ending the effective construction of an equivariant Thom form. The maps σ! are morphisms of complexes since ΦG is a dG -cocycle, and the fact that they are right inverses to Rn is clear by (6.30). (2) Since π ∗ : HG (B) → HG (V ) and σ ∗ : HG (V ) → HG (B) are isomorphisms inverse to each other (6.1.2.2), we have α ∼ π ∗ (σ ∗ (α)) for all α ∈ G (V ), in which case, by (6.30), 

α ∧ ΦG = n



R

Rn

π ∗ (σ ∗ (α)) ∧ ΦG = σ ∗ (α) ,

∀α ∈ G (V ) ,  

which ends the proof of the proposition.

Proposition 6.4.5.1 can be summarized in the following equivariant analogue to diagram (3.84), p. 101, G (B)

σ∗ q.i.

G (V )

π∗ q.i.

G (B)

id G (B)

σ! q.i.

G,cv (V )[n]

π! q.i.

G (B)

(6.31)

id G,c (B)

σ!

G,c (V )[n]

π!

G,c (B)

id

where the middle row represents the relative Gysin morphisms over G (B). (Take care that the Gysin morphisms in the bottom row are generally not quasiisomorphisms, as already observed in 3.6.5.) Corollary 6.4.5.2 The equivariant Gysin morphisms associated with the zero section σ : B → V of the vector bundle (V , B, π, Rn ): 

σ! : HG,c (B))[dB ] → HG,c (V )[dV ] σ∗ : HG (B))[dB ] → HG (V )[dV ] ,

226

6 Equivariant Gysin Morphism and Euler Classes

are both induced by the morphism of complexes π ∗ (−) ∧ ΦG , where ΦG is a representative of the equivariant Thom class of (V , B, π, Rn ) (cf. Proposition 6.4.5.1–(2)). Proof By the adjointness of the equivariant Gysin morphisms 6.3.1.1, it suffices to check the equality 

σ ∗ (α) ∧ β =

B

But since

V

=

B

◦

σ ∗ (α) ∧ β =



Rn ,

 Rn

 V

α ∧ π ∗ (β) ∧ ΦG .

it is enough to check

α ∧ π ∗ (β) ∧ ΦG ,

∀α ∈ G (V ) , ∀β ∈ G (B) ,

which is obvious since, by 6.4.5.1–(3), we have  Rn

α ∧ π ∗ (β) ∧ ΦG = σ ∗ (α ∧ π ∗ (β)) = σ ∗ (α) ∧ β .

 

6.4.6 Equivariant Gysin Long Exact Sequence Let i : N → M be a closed equivariant embedding of oriented G-manifolds, and denote by j : U := M  N → M the complementary open embedding. The sequence of G -dg-modules 0

G,c (U )

j!

G,c (M)

i∗

G,c (N)

0,

where j! denotes the extension by zero morphism (6.4.1), clearly left exact, is also right exact since every compactly supported differential form of N can be extended (eventually with the help of partitions of unit) to a compactly supported differential form on M. It is a standard fact that the morphism G,c (U )

j!

G,c (M)

ι

c(j ˆ ! ) :=

G,c (M) ⊕ q.i. q

G,c (U )

j!

G,c (M)

i∗

G,c (U )[1]

(6.32)

G,c (N)

where q(x, y) = i ∗ (x), is a quasi-isomorphism of complexes (see A.1.5.7– (1)), hence an isomorphism in the derived category D DGM(G ), which allows embedding the bottom row in the top row exact triangle (see Comment A.1.6.5).

6.4 Explicit Constructions of Equivariant Gysin Morphisms

227

The duality functor IR Hom•G (−, G ) applied to (6.32) and the equivariant Poincaré duality theorem 5.6.2.1, then give rise to an isomorphism of exact triangles in D DGM(G ): G (U )[dM ]

j∗

G (M)[dM ]

IR Hom• G (c(j ˆ !

G)

(6.33) G (U )[dM ]

j∗

G (M)[dM ]

i!

G (N)[dN ]

where i! : G (N)[dN ] → G (M)[dM ] is the Gysin morphism associated with restriction i ∗ : G (M) → G (N ). It is worth noticing that local equivariant cohomology (Definition 4.10.1.1) makes its appearance in the right term of the first row as the abutment of the quasiisomorphism: G (N)[dN ] → IR Hom•G (c(j ˆ ! ), G ) = c(j ˆ ∗ )[dM −1] = G,N (M)[dM ] . We therefore have a canonical isomorphism HG (N )[dN −dM ]  HG,N (M)

(6.34)

The following proposition extends 3.7.1–(2) to the equivariant context. Proposition 6.4.6.1 The equivariant Gysin long exact sequence associated with a closed embedding i : N → M of oriented G-manifolds is the sequence i!

j∗

c

→ HG (N)[dN ] −→ HG (M)[dM ] −→ HG (U )[dU ]−→HG (N )[dN +1] , where j : U := M  N → M is the open inclusion map, j ∗ is the restriction, i! is the Gysin morphism (equivariant analogue to Proposition 3.6.6.1), and c is the Gysin morphism associated with the (proper) projection map of the fiber bundle (S (N), N, π, RdM −dN −1 ), where S (N ) denotes the sphere bundle around N defined in 3.7.1–(3.86). Proof Same as for 3.7.1–(2).

6.4.6.1

 

Exercises

1. The equivariant version of the Lefschetz fixed point theorem (cf . Sect. 3.7.2) defines the G-equivariant Lefschetz class of f by the expression LG (f ) := Gr(f )∗ (δ∗ (1)) ∈ HGdM (M) ,

228

6 Equivariant Gysin Morphism and Euler Classes

where δ : M → M×M denotes the diagonal embedding. The analog of the equivariant Lefschetz number is now a priori an equivariant cohomology class ΛG,f :=

 M

LG (f ) ∈ HG .

Prove that 

dM ResG (M) {e} LG (f ) = L(f ) ∈ H

G,f = f , where ResG {e} : HG (M) → H (M) is the group restriction (Definition 6.2.1.2). Conclude that the equivariant Lefschetz number coincides with the nonequivariant one. Deduce that if HG (M) is a torsion module (Sect. 7.4.1), then the Euler characteristic of M is zero (cf . Sect. 3.7.2–(4)). 2. Show that if f : N → M is an equivariant map between oriented Gmanifolds, then the projective limit of nonequivariant Gysin morphisms (cf . Theorem 4.7.2.2–(4))  lim f (n)! : Hc (NG (n))[dN ] → Hc (MG (n))[dM ] ←−n is well-defined and coincides with the equivariant Gysin morphism f! : HG,c (N )[dN ] → HG,c (M)[dM ] . And mutatis mutandis for f proper and the equivariant Gysin morphism f∗ : HG (N )[dN ] → HG (M)[dM ] .

6.5 Equivariant Euler Classes The reference for this section is Atiyah-Bott’s paper [7], notably §2 and §3.

6.5.1 The Nonequivariant Euler Class Let i : N ⊆ M be a closed inclusion of oriented manifolds. Denote by N a tubular neighborhood of N in M (see Sect. 3.2.3.1). As the inclusion N ⊆ N is of the same nature as the inclusion of the zero section of a vector bundle σ : B ⊆ V (cf . Sect. 3.6.5), we can define the Thom class (N, M)

6.5 Equivariant Euler Classes

229

of the pair (N, M) using the same principle, that is, by means of Gysin morphisms. We thus set: (N, M) := i∗ (1) ∈ H dM −dN (M) Definition 6.5.1.1 The Euler class Eu(N, M) is the restriction to N of the Thom class of the pair (N, M), 2 i.e. : Eu(N, M) := i ∗ i∗ (1) = (N, M) N ∈ H dM −dN (N ) .

(6.35)

6.5.2 G-Equivariant Euler Class Generalizing the concept of Euler class to the equivariant framework is straightforward thanks to the equivariant Gysin morphism formalism. Let G be a compact Lie group and let i : N ⊆ M be a closed inclusion of oriented G-manifolds. Definition 6.5.2.1 The G-equivariant Euler class EuG (N, M) is defined by the same formula (6.35): EuG (N, M) := i ∗ i∗ (1) = G (N, M) N ∈ HGdM −dN (N )

(6.36)

where now i∗ : HG (N )[dN ] → HG (M)[dM ] is the equivariant Gysin morphism. Proposition 6.5.2.2 Let H be a closed subgroup of G, then EuH (N, M) = ResG H (EuG (N, M)) . Proof Immediate after the compatibility between Gysin morphisms and group restrictions established in Theorem 6.2.2.1.   Exercise 6.5.2.3 Given oriented G-manifolds L ⊆ N ⊆ M, prove the following formula for nested equivariant Euler classes ( , p. 349) EuG (L, M) = EuG (L, N ) ∪ EuG (N, M)

2 Cf.

formula (2.19), p. 5, in Atiyah-Bott [7].

L.

(6.37)

230

6 Equivariant Gysin Morphism and Euler Classes

6.5.3 G-Equivariant Euler Class of Fixed Points In the sequel, we denote by M G the subspace of G-fixed points of M. For a discrete subspace N of M G , we have EuG (N, M) ∈ HGdM (N ) =

 b∈N

S dM /2 (g ∨ )G ,

(6.38)

and EuG (N, M) is simply the family of invariant polynomials   EuG (N, M) = EuG (b, M) b∈N ⊆ S dM /2 (g ∨ )G . Comment 6.5.3.1 Note that the exponent dM /2 in (6.38) refers to the polynomial degree, in which case EuG (N, M) = 0 when M is odd dimensional. We will see in Proposition 6.5.4.2 that this elementary observation is closely related to the fact that in odd dimensional manifolds there are no isolated T -fixed points, where T denotes a maximal torus in G. Proposition 6.5.3.2 Let G be a compact Lie group and let M be an oriented Gmanifold. If N is a finite subset of M G , then  b∈N



EuG (b, M) =

M

G (N, M) ∪ G (N, M)

and |N | =

 M

G (N, M) .

Proof The constant function 1N and a fortiori the Thom class G (N, M), both have compact supports in which case i∗ = i! : HG,c (N ) → HG,c (M). The adjointness property of Gysin morphisms then gives:  b∈N

α

b

=

 M

i! (1N ) ∪ α ,

∀α ∈ HG (M) .

The proposition results by taking α := G (N, M) and α := 1M respectively.

 

6.5.4 T -Equivariant Euler Class of Fixed Points Let T be a maximal torus of the compact connected Lie group G and denote by T  := NG (T ) the normalizer of T in G. We have T ⊆ T  ⊆ G. The universal fiber bundle IEG (cf . Sect. 4.6.2) will be used as universal fibre bundle for every closed subgroup H ⊆ G. We therefore set for every G-manifold M MH := IEG ×H M .

6.5 Equivariant Euler Classes

231

We then have a natural commutative diagram of Borel constructions: MT := IEG ×T M

p

MT := IEG ×T M

q

MG := IEG ×G M

(6.39) p

q

IBT

IBT

IBG

where p and q denote the obvious quotient maps. The Weyl group of the pair (T , G), i.e. the finite group W := NG (T )/T , acts on the right of MT by (x, m) · w := (x · w, ˜ w˜ −1 m), where w˜ is a lift of w in T  . We have therefore an identification IEG ×T  M  (IEG ×T M)/W , and p is the orbit map for this action. Consequently, the pullback p∗ : H (MT  ; R) → H (MT ; R)W is an isomorphism, as is also the morphism p∗ : HT  (M) −→ HT (M)W , 

after the equivariant de Rham Theorem 4.8.3.1. 3 The fibers of q : MT  → → MG are isomorphic to G/T  which is a rational acyclic space. Indeed, this space is the orbit space of G/T for the right action of W and we know from Leray4 that, under this action, H (G/T ) is isomorphic to the regular representation of W . In particular H (G/T  ) = H (G/T )W = R (see Sect. 8.4.1). This implies that q ∗ : H (MG ; R) → H (MT  ; R) is an isomorphism, as is, again after the equivariant de Rham Theorem, the pullback morphism q ∗ : HG (M) −→ HT  (M) . 

Summing up, we have the following canonical isomorphisms HG (M)

3 Recall

q∗

HT (M)

p∗

HT (M)W .

(6.40)

that the fact that over a field of characteristic 0 and for finite groups, the cohomology of the orbit space is the invariant subspace of the cohomology of the space, is a general result, see Grothendieck [49] Théorème 5.3.1 and Corollaire de la Proposition 5.2.3. But here the context is much simpler since the action of W is free. 4 Leray [73] Théorème 2.1, p. 103, and Lemma 27.1 in the Ph.D. thesis of Borel [13], p. 193.

232

6 Equivariant Gysin Morphism and Euler Classes

Comment 6.5.4.1 When M = {•}, we obtain a commutative diagram of ChernWeil homomorphisms S(

∨

)G

HG

Chv q∗

S( ∨ )W

⊆ p∗

HT

S( ∨ ) (6.41) HT

where Chv : S(g ∨ )G → S(t ∨ )W associates with a symmetric polynomial function on g, its restriction to the subspace t. The diagram shows that Chv is an isomorphism, a claim known in the theory of Lie algebra representations as the Chevalley isomorphism or the Chevalley’s restriction theorem. 5 It is worth noting that for each b ∈ M G the group G acts naturally on the tangent space Tb (M) through a linear representation. Now, if we endow M with a G-invariant Riemannian metric, then the exponential map exp : Tb (M) → M (cf . Sect. 4.5.3) is a G-equivariant diffeomorphism between Tb (M) and an open neighborhood of b in M, and the computation of equivariant Euler classes on fixed points may be greatly simplified by linearizing the data through the exponential map. The following proposition deals with the linear case. Proposition 6.5.4.2 Let V be a linear representation of a compact connected Lie group G. Let T be a maximal torus of G. 1. The equivariant Euler class EuT (0, V ) belongs to S(t ∨ )W and the Chevalley isomorphism Chv : S(g ∨ )G → S(t ∨ )W exchanges EuG (0, V ) and EuT (0, V ). 2. If V := V1 ⊕ V2 as G-module, then EuG (0, V ) = EuG (0, V1 ) EuG (0, V2 ). 3. Denote by C(α) the complex vector space C endowed with the representation of T corresponding to the weight α ∈ t ∨ , i.e. exp(tx)(z) = e2π itα(X) z ,

∀(t ∈ R), ∀(X ∈ Lie(T )), ∀(z ∈ C) .

 μ(α) , is the decomposition of V in irreducible If V = Rμ0 ⊕ α C(α) representations of T , where μ0 and μ(α) denote respectively the multiplicities of R and C(α) in V , then the T -equivariant Euler class at 0 ∈ V is given by the formula EuT (0, V ) = 0μ0

 α

α μ(α) .

In particular, EuG (0, V ) = 0 if and only if V T = 0. Proof (1) Results from Proposition 6.5.2.2. (2) Left to the reader. (3) Following (2), it suffices to show that EuT (0, R) = 0 and EuT (0, C(α)) = α.

5 See

Chriss-Ginzburg [31] §3.1.37 Chevalley Restriction Theorem, p. 140.

6.5 Equivariant Euler Classes

233

EuT (0, R). Since the action of T on R is trivial, we have HT (R) = HT ⊗ H (R) and EuT (0, R) = 1 ⊗ Eu(0, R) = 0, simply because H 1 (R) = 0. EuT (0, C(α)). Taking polar coordinates (ρ, θ ) ∈ R+ × [0, 2π ] in C, the nonequivariant Thom form Φ(0, C) is of the form Φ [2] = λ(ρ) ρ dρ ∧ dθ , where λ : R → R is a nonnegative bump ∞ function with compact support equal to 1 in a neighborhood of 0 and such that 0 λ(ρ) ρ dρ = 1/2π . By definition of the action of g ∈ T in C(α), we have g · Φ [2] = Φ [2] so that Φ [2] is T -invariant. We can thus construct an equivariant Thom class following the procedure described in the proof of 6.4.5.1–(2). We have  (dT Φ [2] )(X) = ι(X) λ(ρ) ρ dρ ∧ dθ ' ) = −λ(ρ)ρ ∧ ι(X)dθ = −λ(ρ)ρ ∧ L(X)(θ = −λ(ρ)ρ ∧ d (2π tα(X)) = −2π α(X) λ(ρ) ρ dρ , dt

so that Φ [0] (X) must satisfy the following two conditions Φ [0] (X) ∈ S 1 (t ∨ ) ⊗ 0c (C(α))T

and

d Φ [0] (X) = −2π α(X) λ(ρ) ρ dρ .

But then, the only possibility is Φ [0] (X) = −2π α(X)

 0

ρ

λ(ρ) ρdρ −

 +∞

 λ(ρ) ρdρ ,

0

since Φ [0] (X) is of compact support. Consequently, EuT (0, C(α)) = T (0, C(α))

0

= Φ [0] (0) = α .  

as stated.

Exercise 6.5.4.3 If G = SO(3), show that EuG = 0. Conclude that isolated G-fixed points may have a null equivariant Euler class when G is nonabelian, contrary to the abelian case. ( , p. 349) (0, R3 )

Chapter 7

Localization

We describe the behavior of de Rham Equivariant Poincaré Duality and Gysin Morphisms under the Localization Functor. In Sect. 8.6 the same topic will be addressed for equivariant cohomologie in positive characteristic.

7.1 The Localization Functor The graded ring of fractions of G = HG (Appendix D) will be denoted by QG (simply Q when G is understood). The localization functor is then defined as the base change functor QG ⊗G (−) : GM(G )  Vec(QG ) . Following Proposition D.1, for any G -module N , the HG -module QG ⊗HG N is flat and injective. Consequently, the localization functor is exact and when applied to Cartan complexes, we obtain the localized Cartan complexes 

 Q ⊗ G (M) := QG ⊗G G (M), id ⊗ dG  Q ⊗ G,c (M) := QG ⊗G G,c (M), id ⊗ dG

whose cohomologies, the localized equivariant cohomologies, respectively denoted by Q ⊗ HG (M) and Q ⊗ HG,c (M), satisfy: Q ⊗ HG (M) = QG ⊗HG HG (M) and

Q ⊗ HG,c (M) = QG ⊗HG HG,c (M) .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Arabia, Equivariant Poincaré Duality on G-Manifolds, Lecture Notes in Mathematics 2288, https://doi.org/10.1007/978-3-030-70440-7_7

235

236

7 Localization

7.2 Localized Equivariant Poincaré Duality The localized equivariant cohomology is very close to the non equivariant cohomology in that the Poincaré duality pairings are perfect. The following, analogue to the equivariant Poincaré duality Theorem 5.6.2.1, results straight forwardly from the fact that QG is a flat and injective G -module in which case: IR Hom•QG (QG ⊗ (−), QG ) = IR Hom•G ((−), QG ) = Hom•G ((−), QG ) , since the functor Hom•G ((−), QG ) is exact in DGM(G ). Theorem 7.2.1 Let G be a compact Lie group, and let M be an oriented Gmanifold of dimension dM . 1. The morphism of (nongraded) complexes  ID G,M : Q ⊗ G (M)[dM ] → Hom•QG Q ⊗ G,c (M), QG induces an isomorphism  D G,M : Q ⊗ HG (M)[dM ] → Hom•QG Q ⊗ HG,c (M), QG 2. If, in addition, M is of finite type, then the morphism of complexes  ID G,M : Q ⊗ G,c (M)[dM ] → Hom•QG Q ⊗ G (M), QG induces an isomorphism  D G,M : Q ⊗ HG,c (M)[dM ] → Hom•QG Q ⊗ HG (M), QG Exercise 7.2.2 Let M be of finite type. Prove that the torsion-freeness (cf . Sect. 5.6.3) of HG (M) (resp. HG,c (M)) is a necessary and sufficient condition for  D G,M : HG (M)[dM ] → Hom•HG HG,c (M), HG (resp. D G,M ) to be injective. Discuss the case where M is not of finite type. ( , p. 350)

7.3 Localized Equivariant Gysin Morphisms As a consequence of Theorem 7.2.1, if f : M → N is a map between oriented G-manifolds, then the localized Gysin morphisms 

f! : Q ⊗ HG,c (M) → Q ⊗ HG,c (N ) f∗ : Q ⊗ HG (M) → Q ⊗ HG (N ) ,

if f is proper,

7.4 Torsion in Equivariant Cohomology Modules

237

are uniquely determined by the adjointness equalities,  ⎧ ∗ ⎪ f [β] ∪ [α] = [β] ∪ f! [α] ⎨

⎪ ⎩



M

∗

M

f [β] ∪ [α] =



N N

[β] ∪ f∗ [α] ,

if f is proper.

as in the nonequivariant framework.

7.4 Torsion in Equivariant Cohomology Modules 7.4.1 Torsion In Sect. 5.6.3 we defined the annihilator of an element v of an HG -gm V as the ideal Ann(v) := {P ∈ HG | P · v = 0} . We say that an element v ∈ V is torsion if Ann(v) = 0, otherwise we say that v is nontorsion. An HG -gm V is a torsion module if all its elements are torsion, it is called nontorsion or torsion-free if 0 is its only torsion element. Exercise 7.4.1.1 Given an HG -gm V , let τ (V ) be the subset of its torsion elements. Show that 1. τ (V ) is a torsion module and the quotient ϕ(V ) := V /τ (V ) is torsion-free. The natural map: QG ⊗HG V → QG ⊗HG ϕ(V ) is an isomorphism. 2. QG ⊗HG V = 0 if and only if V is a torsion module. 3. Homgr0HG (V , QG ) = 0 if and only if V is torsion. ( , p. 350) 4. An inductive limit of torsion modules is a torsion module. 5. A projective limit of torsion modules may be a nontorsion module. Exercise 7.4.1.2 The annihilator of an HG -module is the ideal Ann(V ) := {P ∈ HG | P · V = 0} =

, v∈V

Ann(v) .

1. Show that if Ann(V ) = 0, then V is a torsion module, but the converse statement may fail to be true. ( , p. 350) 2. Show that if V is a unital HT -algebra, then Ann(V ) = Ann(1). 3. Let {U1 ⊆ U2 ⊆ · · · Un · · · } be anincreasing family of G-stable open subsets of a G-manifold M such that M = n Un . Suppose that HG (Un ) and HG,c (Un ) are torsion for all n ∈ N. Show that HG,c (M) is torsion, whereas HG (M) may fail to be torsion.

238

7 Localization

Fig. 7.1 The slice S(x)

4. In (3) show that {Ann(HG (Un ))}n is a decreasing sequence of ideals and that Ann(HG (M)) =

, n∈N

Ann(HG,c (Un )) .

In particular, if the set {Ann(HG (Un ))} is finite, then HG (M) is torsion.

7.5 Slices and Orbit Types 7.5.1 The General Slice Theorem In Sect. 4.5.2 we recalled the theorem of slices for free G-manifolds, but the same proof works for any G-manifold M, whether the action of G is free or not. The general slice theorem1 then claims that, for every x ∈ M there exists a (locally closed) submanifold S(x) # x, stable under the action of isotropy subgroup group Gx ⊆ G of x, such that the map G ×Gx S(x) → M ,

[(g, x)] → g · x ,

is a diffeomorphism onto a G-stable neighborhood Vx of x (Fig. 7.1) Such submanifold is called a slice. We then have HG (Vx ) = H (IEG ×G G ×Gx S(x)) = HGx (S(x)) , and HG,c (Vx ) = HGx ,c (S(x)) (exercise) ( , p. 350). As a consequence, the HG -module structures of HG (Vx ) and HG,c (Vx ) factors through the natural ring homomorphism ρx : HG → HGx . Proposition 7.5.1.1 Let T be a torus. For every point x in a T -manifold M, the following equivalences hold. 1. ρx : HT → HTx is injective if and only if x ∈ M T . The HT -modules HG (Vx ) and HG,c (Vx ) are torsion if and only if x ∈ M T 2. If x ∈ M T , then EuT (x, M) = 0 if and only if x is an isolated point of M T . 1 See

Hsiang [55] §I.3 Theorem (I.5’), pp. 11–12.

7.5 Slices and Orbit Types

239

Proof (1) If Tx = T , there exist closed subtorus H ⊆ T such that T = H × Tx and dim(H ) > 0, in which case ker(ρx ) = HH+ ⊗ HTx = 0. (2) is 6.5.4.2–(3).   Remarks 7.5.1.2 1. Proposition 7.5.1.1 is interesting in that it translates topological properties of a point in a T -manifold into algebraic properties of HT -modules, opening the way to the algebraic study of the equivariant cohomology of T -spaces. 2. Take care that in 7.5.1.1, we cannot replace T by a non abelian Lie group G, since both claims may fail. For 7.5.1.1–(1), if T is a maximal torus in G and if M = G/T , then the isotropy group of x = g[T ] ∈ M is the maximal torus Gx = gT g −1 , and ρx is the inclusion HG = (HT )W ⊆ HGx . Thus, ρx is injective although x is not a G-fixed point. A counterexample for 7.5.1.1–(2) is given in Exercise 6.5.4.3.

7.5.2 Orbit Type of T -Manifolds The torsions of the HT -modules HT ,c (M) and HT (M) play a central role in the fixed point theorem. When M T = ∅, the slice theorem and 7.5.1.1–(1) show that M can be covered by a family of T -stable open subspaces Vx where HT (Vx ) is killed by the elements of the nontrivial kernel ρx : HT → HTx . Any finite union of those subspaces will also have torsion equivariant cohomology thanks to Mayer-Vietoris sequences. We therefore conclude that if M is compact without fixed points, then HT (M) is a torsion HT -module. On the contrary, when M is not compact, the same conclusion can fail to be true (Exercise 7.4.1.2–(3)) unless we have a better control on the set of kernels KT (M) := {ker(ρx ) | x ∈ M}. For example, a sufficient condition, as shown in Exercises 7.4.1.2–(4), is the finiteness of KT (M), or, which amounts to the same, the finiteness of the set of the isotropy groups OT (M) := {Tx | x ∈ X}, usually referred to as the orbit type of the T -space M.2 Definition 7.5.2.1 A T -manifold M is said of finite orbit type if OT (M) is finite. Exercise 7.5.2.2 Show that a T -manifold M is always locally of finite orbit type. In particular, if M is compact, then it is of finite orbit type. ( , p. 351) Proposition 7.5.2.3 If M T = ∅ and M is of finite orbit type, then HT ,c (M) ⊗HT QT = HT (M) ⊗HT QT = 0 .

2 See

Hsiang [55] ch. IV §2, p. 54, for the general definition especially for non abelian groups.

240

7 Localization

Proof – Torsion of HT ,c (M). Let (U , ⊆) be the set of G-stable open subspaces U ⊆ M, such that HT ,c (U ) is torsion, partially ordered by set inclusion. The set U is non empty as it contains every slice neighborhood Vx (7.5.1) and is an inductive poset by Exercise 7.4.1.2–(3), so that the Zorn Lemma can be applied. Let U be a maximal element in U . For any y ∈ M, let Vy be a slice neighborhood of y. By the exactness of the Mayer-Vietoris sequence for compactly supported cohomology: 0 0 0 0 0 · · · → HG,c (U ∩Vy ) → HG,c (U )⊕HG,c (Vy ) → HG,c (U ∪Vy ) → HG,c (U ∩Vy )[1] → ,

0 (U ∪V ) is torsion. Then U ⊇ V , by the maximality we easily conclude that HG,c y y of U , hence U = M. – Torsion of HT (M). We cannot use the same argument as in the compactly supported case because a projective limit of torsion modules is not necessarily torsion. The finiteness assumption on the set of orbit types will now be crucial. Let I be the intersection of all the ideals ker(ρx : HT → HTx ) for x ∈ M. The finiteness of the orbit type of M ensures that I = 0. Let (U , ⊆) be the set of G-stable open subspaces U ⊆ M, such that I ⊆ Ann(HT (U )), partially ordered by set inclusion. The set U is non empty as it contains every slice neighborhood Vx (cf . Sect. 7.5.1) and it is an inductive poset by Exercise 7.4.1.2– (4), so that the Zorn Lemma can be applied. Let U be a maximal element in U . For any y ∈ M, let Vy be a slice neighborhood of y. Thanks to the exactness of the first terms of the Mayer-Vietoris sequence: 0 → HG0 (U ∪ Vy ) → HG0 (U ) ⊕ HG0 (Vy ) → HG0 (U ∩ Vy ) → , we easily see that 1 ∈ HG0 (U ∪ Vy ) is killed by I . Then I ⊆ Ann(HG (U ∪ Vy )) by 7.4.1.2–(2) and U ⊇ Vy , by the maximality of U , hence U = M.  

7.6 Localized Gysin Morphisms Given a T -manifold M and a closed subgroup H ⊆ T , the fixed point set M H := {x ∈ M | h · x = x ∀h ∈ H } is a submanifold whose connected components (not necessarily of equal dimensions) are stable under the action of T , and, in addition, orientable if M is so.3 Terminology A homomorphism of HT -modules α : L → L is called an isomorphism modulo torsion if its kernel and cokernel are both torsion HT -modules,

recall the reason: under the action of H , the tangent spaces Tx (M) for x ∈ M H split as the direct sum of Tx (M H ) and a sum of H -irreducible two dimensional representations C(α) (Propositions 6.5.4.2–(3)), canonically oriented by their character. Therefore, the orientation of Tx (M H ) determines that of Tx (M) and vice versa.

3 We

7.6 Localized Gysin Morphisms

241

i.e. if the following induced homomorphism of QT -modules is an isomorphism α ⊗HT id : QT ⊗HT L → QT ⊗HT L . Proposition 7.6.1 Let M be an oriented T -manifold of finite orbit type. For any H closed subgroup of T , denote by ιH : M H → M the set inclusion. The following morphisms of HT -gm 4 are isomorphisms modulo torsion. ι

H∗

Gysin morphisms

ιH ! : HT ,c (M H )[dM H ] → HT ,c (M)[dM ] 

Restriction morphisms

: HT (M H )[dM H ] → HT (M)[dM ]

ι∗H : HT ,c (M) → HT ,c (M H ) ι∗H : HT (M) → HT (M H )

Proof The kernel and cokernel of the restriction ι∗H : HT ,c (M) → HT ,c (M H ) lay within HT ,c (U ), where U := M  M H . Now, as the isotropy groups of the points of U are strict subgroups of T , there are no T -fixed points, i.e. U T = ∅, and we can conclude that HT ,c (U ) is an HT -torsion module by Proposition 7.5.2.3. In particular, any submodule of HT ,c (U ), viz. the kernel and the cokernel of ι∗H , is a torsion HT -module. By duality the same is true for ιH ∗ : HT (M H ) → HT (M). The other restriction ι∗H : HT (M) → HT (M H ) is a little more tricky as its kernel and cokernel lay within HT ,U (X) which we have not yet proved to be an HT torsion module. For that, recall that since we have short exact sequences of local section functors over open subspaces 0 → ΓU1 ∩U1 (−) −→ ΓU1 (−) ⊕ ΓU2 (−) −→ ΓU1 ∪U2 (−) → 0 where ΓU (−) denotes the kernel of restriction Γ (M, −) → Γ (M  U, −), we may follow Mayer-Vietoris procedure to approach HT ,U (X) by successively adding slice open sets Vx ⊆ U (cf . Sect. 7.5.1). In this way, to show that HT ,U (M) is a torsion module, we need only show that each HT ,Vx (M) is so. Now, this HT -module occurs in the long exact sequence → HT ,Vx (M) → H (M) → HT (M  Vx ) → where M  Vx is T -equivariantly homotopic to M  T ·x since the slice S(x) is a submanifold of M. Hence HT ,Vx (M)  HT ,T ·x (M)  HT (T ·x) = HTx , proving that HT ,Vx (M) is a torsion HT -module.  

4 As

the submanifold M H need be neither connected nor equidimensional the shift indication in a notation as HT (M H )[dM H ] must be understood component-wise.

242

7 Localization

7.7 The Localization Formula 7.7.1 Inversibility of Euler Classes Let M be an oriented T -manifold of dimension dM and of finite orbit type. As already mentioned, a connected component F of M T is an oriented submanifold of M of some dimension dF . Let (F , F, π, RdM −dF ) be a tubular neighborhood of F in M (Sect. 3.2.3.1) and denote by ι : F → F its zero section. Since F is a retract of F , the Gysin morphism ι! : Q ⊗ HT ,c (F )[dF ] → Q ⊗ HT ,c (F )[dM ] is an isomorphism. On the other hand, Proposition 7.5.2.3 applied to F  F in the long exact sequence → Q ⊗ HT ,c (F  F ) → Q ⊗ HT ,c (F ) → Q ⊗ HT ,c (F ) → shows that the pullback i∗

Q ⊗ HT ,c (F ) −→ Q ⊗ HT ,c (F ) 

(7.1)

is an isomorphism (note that this is almost never the case before localization). We therefore have isomorphisms Q

HT ,c (F )[dF ]

ι!

Q

HT ,c (F )[dM ]

ι∗

Q

HT ,c (F )[dM ] ,

(7.2)

EuT (F,F )

whose composition is the multiplication by the equivariant Euler class EuT (F, F ) (Corollary 6.4.5.2 and Sect. 6.5.2.1). This class is in fact invertible in the ring Q ⊗ HT (F ). Indeed, applying equivariant Poincaré duality to (7.1), the corresponding equivariant Gysin morphism i∗ : Q ⊗ HT (F )[dF ] −→ Q ⊗ HT (F )[dM ] , 

ι∗ (α) = π ∗ (α) ∪ [ΦT (F, F )]

with ΦT (F, F ) the Thom class (Sects. 6.4.5.1 and 6.4.6.1), is an isomorphism too. If we then compose ι∗ with ι∗ : Q ⊗ HT (F )[dM ] → Q ⊗ HT (F )[dM ], yet an isomorphism, we get the isomorphism ι∗ ◦ ι∗ : Q ⊗ HT (F )[dF ] −→ Q ⊗  HT (F )[dM ] , given by the multiplication by the equivariant Euler class EuT (F, F ) (Definition 6.5.2.1). Therefore, EuT (F, F ) is invertible in Q ⊗ HT (F ), where we have: 1 1 = ∈ Q ⊗ HT (F ) . EuT (F, F ) EuT (F, M) Then, applying F to the middle term in (7.2), and thanks to the fundamental property of Thom class (Proposition 6.4.5.1–(3)), we get the equality  F

β=

 F

β F , EuT (F, M)

∀β ∈ Q ⊗ HT ,c (F ) ,

which is the essential ingredient in the proof of the following statement.

(7.3)

7.7 The Localization Formula

243

Proposition 7.7.1 (Localization Formula) Let M be an oriented T -manifold of dimension dM and of finite orbit type. Then, denoting by F the set of connected components F of M T , we have: 1. The following equality known as the localization formula  M

β=



 F ∈F

F

β F , EuT (F, M)

(7.4)

holds for all β ∈ Q ⊗ HT ,c (M). 2. The localized equivariant Poincaré pairing  ·, · T ,M : Q ⊗ HT (M) × Q ⊗ HT ,c (M) → QT , is perfect and the formula  α, β T ,M =

  F ∈F

F

α Fβ F , EuT (F, M)

holds for all α ∈ Q ⊗ HT (M) and β ∈ Q ⊗ HT ,c (M). 3. If M is compact of positive dimension, then 0=

  F ∈F

F

1 . EuT (F, M)

Proof (1) Let ι : M T → MT be a tubular neighborhood of M T in M. By applying Proposition 7.5.2.3 to the left-hand term in the long exact sequence → Q ⊗ HT ,c (X  M T ) → Q ⊗ HT ,c (X) → Q ⊗ HT ,c (M T ) → , for X = M, MT , we immediately deduce that the following extension by zero morphism, is an isomorphism, %

ι!

F

Q ⊗ HT ,c (F ) = Q ⊗ HT ,c (MT ) −→ Q ⊗ HT ,c (M) . 



Therefore, M = F F , and statement (1) follows from (7.3). (2) is obvious after (1). (3) results from (7.4) with β := 1 ∈ HT0,c (M).

 

The following corollary was originally proved independently by Atiyah-Bott [7] and Berline-Vergne [9].

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7 Localization

Corollary 7.7.2 Let M be an oriented T -manifold of dimension dM , of finite orbit type and such that M T is a discrete subspace of M. Then 1. For all β ∈ HT ,c (M) the following ‘localization formula’ is satisfied:  M

β=

 x∈M T

β x . EuT (x, M)

2. The localized equivariant Poincaré pairing  ·, · T ,M : Q ⊗ HT (M) × Q ⊗ HT ,c (M) → QT , is perfect and given by the formula 

 α, β T ,M =

x∈M T

α xβ x . EuT (x, M)

3. If M is compact of positive dimension 0=

 x∈M T

1 . EuT (x, M)

Proof Particular case of Proposition 7.7.1.

 

Chapter 8

Changing the Coefficients Field

The constructions and results described in the previous chapters are available with coefficients in a field k of positive characteristic prime to the cardinality of G/G0 .1 In this section, we prove equivariant Poincaré duality and define Gysin morphisms over an arbitrary field k as an application of Grothendieck-Verdier duality. Interested readers can learn the basis of this theory by consulting Iversen [58], Borel [16] or Kashiwara-Schapira [61], and obviously the article of Verdier [92], on the topics of sheaf theory, derived categories and Grothendieck-Verdier duality. We also recommend recent articles by Allday-Franz-Puppe [3–5] which lead to new results and conjectures regarding Equivariant Cohomology.

8.1 Comments about Notations Given a map f : M → N between manifolds, the notations f∗ and f! for Gysin morphisms:  f∗ : (M)[dM ] → (N )[dN ] , (f proper) (8.1) f! : c (M)[dM ] → c (N )[dN ] , can be confused with identical notations in sheaf theory, which traditionally denote the functors of direct image and direct image with proper supports  f∗ : Sh(M; k) → Sh(N ; k) , f! : Sh(M; k) → Sh(N ; k) . where Sh(−; k) denotes the category of sheaves of k-vector spaces on (−). 1 In

fact, on any ring, but the increase of technicalities that would impose is not warranted by the advantages of the resulting generality, so we prefer to limit ourselves to fields. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Arabia, Equivariant Poincaré Duality on G-Manifolds, Lecture Notes in Mathematics 2288, https://doi.org/10.1007/978-3-030-70440-7_8

245

246

8 Changing the Coefficients Field

The notations are nevertheless consistent and their meaning should be clear from the context. Indeed, a central chapter in Grothendieck-Verdier’s duality is concerned with the extension of the familiar concept of integration along fibers (Sect. 3.6.2) beyond locally trivial fibrations. The result, for manifolds, is a canonical morphism in derived category of sheaves2 f! (or M )[dM ] → (or N )[dN ] , where or denotes the orientation sheaf. Then, if c denotes the constant map, applying the derived functors IR c∗ and IR c! gives natural morphisms 

f∗ : IR cM∗ (or M )[dM ] → IR cN ∗ (or N )[dN ] ,

(f proper)

f! : IR cM! (or M )[dM ] → IR cN ! (or N )[dN ] ,

(8.2)

which happen to be the Gysin morphisms (8.1). Note the consistency in (8.2) in how the signs {∗, !} are used in Gysin morphisms and in direct images of sheaves.

8.1.1 Preliminaries By (topological) space we mean a Hausdorff, local contractible and paracompact space3 (see also Sect. B.1), for example manifolds, open subspaces of CWcomplexes, and in particular, the universal fiver bundle IEG and classifying space IBG := IEG/G of a compact Lie group G (Sect. 4.6). The category of spaces and continuous maps will be denoted by Top. For a space X, we denote by Sh(X) := Sh(X; k),

C(X) := C(Sh(X; k),

K(X) := K(Sh(X; k),

D(X) := D(Sh(X; k),

respectively the categories of sheaves of k-vector spaces on X, its category of complexes, its homotopy category and its derived category.

a locally compact space M, the theory defines the dualizing complex of sheaves ω •M (k), an object in the derived category of sheaves D+ (M; k). The analogue to integration along fibers then appears as a canonical morphism IR f! : ω •M (k) → ω •N (k) in D+ (N ; k). In the case of a manifold, the complex ω •M (k) coincides in D+ (M; k) with the orientation sheaf or M [dM ]. Details on these subjects can be found in Kashiwara-Schapira [62] chap. III Poincaré-Verdier duality, p. 139, and in Iversen [58] chap. VI. Poincaré duality with general coefficients, p. 289. 3 Recall that for these spaces Alexander-Spanier, Singular, Cech, ˇ and Sheaf cohomologies are isomorphic, see Bredon [20], ch. III, Comparison with other cohomology theories. 2 For

8.2 Sheafification of Cartan Models over Arbitrary Fields

247

The word ‘cohomology’ stands for cohomology of sheaves and the notation H (X; k) is a shortcut for H (X; k X ), where k X denotes the constant sheaf on X with fiber k.

8.2 Sheafification of Cartan Models over Arbitrary Fields We denote by G a compact Lie group4 and we shrink notations IE := IEG ,

IB := IBG and

HG := H (IB; k) .

As recalled in Sect. 4.7, the Borel construction is a functor from the category of G-spaces to the category of spaces based on IB, X

f G-equiv.

Y

XG

fG

π

•

YG π

IB

The analogues of the ordinary and the compactly supported equivariant cohomology are the ordinary and the π -properly supported cohomologies over k, of the Borel construction,5 i.e. we set: HG (X; k) := H (XG ; k)

and

HG,c (X; k) := Hπ -cv (XG ; k) ,

both endowed with the structure of HG -modules induced by the projections π . In the language of sheaf cohomology, this amounts to setting:6 HG (X; k) := IR Γ (IB, IR π∗ k XG )

and

HG,c (X; k) := IR Γ (IB, IR π! k XG ) .

where k XG denotes the constant sheaf on XG .

Convention When no confusion is likely to arise, we will omit the field k in the notations. For example, HG (X), HG,c (X) will stand for HG (X; k), HG,c (X; k). These rewritings provide clues to the replacements needed in order to transpose the work on equivariant Poincaré duality in the framework of sheaf cohomology over the coefficients field k.

4 Later

we will require that the cardinality of G/G0 be prime to char(k). 5 H ( ) for cohomology with compact vertical supports, after Bott-Tu [18] p. 61. cv − 6 For a continuous map f : X → X  , the notations f , f : Sh(X; k)  Sh(X  ; k) denote ∗ !

the functors of direct image and of direct image with proper supports, respectively. One then denotes by IR f∗ , IR f! : D+ (X; k)  D+ (X ; k) the corresponding derived functors.

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8 Changing the Coefficients Field

P-1. The complexes G (X) and G,c (X) in DGM(G ) should respectively be replaced by the complexes IR π∗ k XG and IR π! k XG in D+ (IB). P-2. The graded algebra G := S(g ∨ )G should be replaced by some Γ (IB, −)acyclic resolution of the constant sheaf k IB . For a space X, we choose the complex of sheaves Ω (XG ) := Ω (XG ; k) of Alexander-Spanier cochains of X as resolution of its constant sheaf k X ,7 

d0

d1

0 → k XG −→ Ω 0 (XG ; k) −→ Ω 1 (XG ; k) −→ · · · Although there are many other possible choices, this one has the advantage of being familiar to topologists and useful in the theory of sheaves. Among its properties, we note the following. a. The sheaves Ω i (X) are Γ (X; −)-acyclic (B.6.3.4–(1)), for every family of supports  (see B.6.2). This has convenient implications. – The complex of global Alexander-Spanier cochains supported in : (Ω (X), d∗ ) := Γ (X; (Ω • (X), d∗ )) ,

(8.3)

computes the cohomology of X with supports in . We will be concerned mainly with  being the families of closed subspaces of X, of compact subspaces of X, of closed subspaces of Y ⊆ X, which respectively compute H (X), Hc (X) and HY (X). – For any continuous map f : X → X , the natural morphisms 

π∗ (Ω i (X)) → IR π∗ (Ω i (X)) π! (Ω i (X)) → IR π! (Ω i (X)) ,

(8.4)

are isomorphisms in D+ (X ).8 – The sheaves π∗ (Ω i (X)) and π! (Ω i (X)) are Γ (X , −)-acyclic, which implies that the natural morphism of complexes 

(X) := Γ (X ; π∗ Ω (X)) → IR Γ (X ; π∗ Ω (X)) cv (X) := Γ (X ; π! Ω (X)) → IR Γ (X ; π! Ω (X)) .

(8.5)

are isomorphisms in D+ (Vec(k)). b. The cup product of Alexander-Spanier cochains endows (Ω • (X), d∗ ) of a structure of sheaves of differential graded algebras.

7 Godement

[46] §2.5, example 2.5.2, p. 134, and in §3.7, example 3.7.1, p. 157. [61], ch. II.2.5 Sheaves on locally compact spaces, p. 102.

8 Kashiwara-Schapira

8.2 Sheafification of Cartan Models over Arbitrary Fields

249

c. The correspondence X  Ω (X) has a functorial behavior on the category Top, in the sense that, given a continuous map f : X → X , the maps fi# : Ω i (X ) → f∗ Ω i (X) ,

ζ → ζ ◦ f ,

∀i ∈ N ,

where ζ denotes a section of Ω i (X ), define a morphism of differential graded algebras f # : Ω (X ) → f∗ Ω (X). For every continuous map g : X → X , we then have (g ◦ f )# = g∗ (f # ) ◦ g # . The usual functoriality behavior of the correspondence X  H (X), respectively of X  Hc (X) for proper maps, then follows. d. In the specific case of a G-space X, the map π # : Ω (IB) → π∗ Ω (XG ) is compatible with the underlying differential graded algebra structures, and π∗ Ω (XG ) appears as an Ω (IB)-dg-algebra. Similarly, the subcomplex π! Ω (XG ) ⊆ π∗ Ω (XG ) is also stabilized by the action of π # and is therefore an Ω (IB)-sub-dg-module of π∗ Ω (XG ). At this point, the reader will have noticed the parallel with the preliminaries of Chap. 5 for the category DGM(G ) (Sect. 5.1.2). To emphasize the similarities further, we introduce the notations: Ω G := Ω (IB) ,

Ω G (X) := π∗ Ω (XG ) ,

Ω G,c (X) := π! Ω (XG ) ,

and denote by DGM(Ω G ) the category of sheaves of differential graded modules over Ω G . This allows us to rephrase things in a more convenient language, which we set out in more detail in Sect. 8.5, formulas (8.34) and (8.35). – The correspondence X  Ω G (X) is a contravariant functor of the category G- Top of G-spaces and equivariant maps to the category DGM(Ω G ) X

f G-equiv.

•

Y

Ω G (Y ) π#

π∗ (fG# )

ΩG

Ω G (X) π#

 – The correspondence X  Ω G,c (X) ⊆ Ω G (X) is a contravariant functorial inclusion from the category G- Toppr of G-spaces and equivariant

250

8 Changing the Coefficients Field

proper maps to the category DGM(Ω G )

X

proper f G-equiv.

Ω G,c (Y ) Y Ω G (Y )

{• }

π#

π! (fG# ) π∗ (fG# )

ΩG

Ω G,c (X) Ω G (X) π#

e. After P-2-(a), the sheaves Ω iG,c (X) and Ω iG (X) are acyclic for the functors Γ (X, −), Γc (X; −) and ΓZ (X, −), in which case,

if

⎧ ⎫ ΩG (X) := Γ (X; Ω •G (X)) ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ i • G,c (X) := Γc (X; Ω G (X)) ⎪ ⎪ ⎪ ⎪ ⎩ i ⎭ G,Z (X) := ΓZ (X; Ω •G (X))

then

⎧ ⎫ • ⎪ HGi (X) = hi (ΩG (X)) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ i i • HG,c (X) = h (G,c (X)) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ H i (X) = hi (• (X)) ⎪ ⎭ G,Z G,Z

P-3. The obvious candidate to replace the duality functor IR Hom•G (−, G ) (cf . Sect. 5.2.3) should be the functor (−)  IR Hom •Ω G (−, Ω G ) but since we are considering D DGM(Ω G ) within D+ (IB), where k IB is isomorphic to Ω G , we can achieve the same functor by instead considering (−)  IR Hom •k (−, k IB ) . Indeed, for M, N ∈ DGM(Ω G ), the natural morphisms of sheaves

Hom Ω G (M, N) → Hom Ω G (Ω G ⊗k M, N) → Hom k (M, N) are isomorphisms in D+ (IB). Proof The first, induced by the morphism Ω G ⊗k M → M, ω ⊗ m → ωm is a quasi-isomorphism in DGM(Ω G ), as germ analysis shows, while the second is classical (almost algebraic), already isomorphism at sheaves level.  

8.2.1 Dictionary To summarize, we are exchanging:

8.2 Sheafification of Cartan Models over Arbitrary Fields

251

Real de Rham cohomology ↔ Sheaf cohomology over k G := S(g ∨ )G ↔ Ω G ← k IB q.i.

D(DGM(G )) ↔ D+ (IB) G (X) ↔ Ω G (X) := π∗ Ω (XG ) ← IR π∗ k XG q.i.

G,c (X) ↔ Ω G,c (X) := π! Ω (XG ) ← IR π! k XG q.i.

IR Hom•G (−, G ) ↔ IR Hom •k (−, k IB ) Note however that, unlike G , the differentials in Ω G and in ΩG := Γ (IB; Ω G ) are generally not zero.

8.2.2 Reformulation of The Poincaré Adjunctions Under the current substitutions, the sheafification of the equivariant Poincaré adjunctions (Sect. 5.6) for an oriented G-manifold M of dimension dM : ⎧ ⎨ (i)

 ID G,M : G (M)[dM ] → IR Hom•G G,c (M), G  • ⎩ (ii) ID  G,M : G,c (M)[dM ] → IR HomG G (M), G ,

(Sect. 5.6.1.1) (Sect. 5.6.1.2)

become the morphisms in the derived category of sheaves D+ (IB) ⎧  ⎨ (I) ID G,M : Ω G (M)[dM ] → IR Hom •k Ω G,c (M), k IB  • ⎩ (II) ID  G,M : Ω G,c (M)[dM ] → IR Hom k Ω G (M), k IB .

(8.6)

Notice that, by Grothendieck-Verdier duality, the last term in (I) verifies IR Hom •k (IR π! k XG , k IB ) = IR π∗ IR Hom •k (k X , π ! k IB ) = IR π∗ (π ! k IB ) , so that (I) is simply a morphism in D+ (IB): (I)

ID G,M : IR π∗ (π −1 k IB )[dM ] → IR π∗ (π ! k IB ) ,

(8.7)

On the other hand, (II) is the dual of (I) composed with the natural morphism (II)

 Ω G,c (M) → IR Hom •k IR Hom •k (Ω G,c (M), k IB ), k IB

(8.8)

252

8 Changing the Coefficients Field

so that proving that (I) and (II) are isomorphisms in D+ (IB) is equivalent to proving that (I) and (II) are too. The next proposition establishes that this is the case in a slightly more general situation. Proposition 8.2.2.1 Let (E, IB, π, M) be an oriented fiber bundle where M is an equidimensional manifold of dimension dM .9 1. The Poincaré adjunction applied to the morphism IR π! (k E ) → IR π∗ (k E ), defines a natural morphism of complexes π −1 k IB [dM ] → π ! k IB ,

(8.9)

which is an isomorphism in D+ (E; k). 2. If dimk Hc (M; k) < +∞, the natural morphism  IR π! k E → IR Hom •k IR Hom •k (IR π! k E , k IB ), k IB

(8.10)

is an isomorphism in D+ (IB). Proof (1) We must first construct the morphism. Grothendieck-Verdier duality canonically identifies  IR π∗ IR Hom •k (π −1 k IB )[dM ], π ! k IB =  = IR Hom •k IR π! (π −1 k IB )[dM ], k IB ,

(8.11)

where π −1 k IB is the constant sheaf k E , and where the complex of sheaves IR π! (k E ) admits a simple description over a trivializing open subset U ⊆ IB since π : E → IB is a locally trivial fibration. Indeed, the open subset U˜ := π −1 (U ) is simply U × M and since k U˜ = k U  k M ,10 we get: IR π! (k U˜ ) = (id  IR cM! )(k U  k M ) = k U ⊗ IR cM! (k M )

9 As shown in Exercise 3.1.1.1, the total space E of a locally trivial fibration π : E → IB with non-connected fiber manifold M, is a disjoint union of open subspaces Ei on which the restrictions πi := π Ei are locally trivial fibrations with equidimensional fibers. The hypothesis on equidimensionality in the proposition is therefore not really restrictive. The hypothesis regarding orientability is related to the comment 3.1.7.3 and means that we choose an atlas of E made of trivializations U × M, where U is open in IB, and such that the transition maps induce oriented isomorphisms on M. Recall that IB = lim-ind IBG(n) (4.6), and that when G is connected the spaces IBG(n) are simply connected, in which case such fiberoriented atlases always exist for M orientable (Corollary 3.1.7.2–(3c)). 10 Recall that if p : Z × Z → Z is the canonical projection, and if F is a sheaf in Z , then, the i 1 2 i i i sheaf F1  F2 is, by definition, F1  F2 := p1−1 F1 ⊗ p2−1 F2 .

8.2 Sheafification of Cartan Models over Arbitrary Fields

253

where cM : M → {•} is the constant map. As a consequence, the restriction of complex of sheaves, at the right-hand side of (8.11), to U verifies  IR Hom •k IR π! (π −1 k IB )[dM ], k IB

=  = IR Hom •k k IB  IR cM! (kM )[dM ], k IB  = k IB ⊗ Hom•k IR cM! (k M )[dM ]; k , U

(8.12)

where the reader will have noticed that, in the third line, instead of IR Hom•k , we wrote Hom•k , the reason being that the functor Hom•k (−; k) is already exact on the category Vec(k) of k-vector spaces. Now, if 0 → k E → Ω ∗ (E) is the resolution by Alexander-Spanier cochains (see Sect. 8.2-(P-2)), the complex of k-vector spaces IR cM! (k E ) = Γc (M,  ∗E ) , computes the compactly supported cohomology Hc (M; k). On the other hand, an orientation of M is a choice of a generator ζi ∈ HcdM (Mi ) on each connected component Mi of M, and, as such, it determines a unique linear form taking the value 1 on each ζi , which is the familiar integration operator over M,  M

: Hc (M ; k)[dM ] → k .

(8.13)

At this point, recall that since the category D+ (Vec(k)) is split, i.e. a complex is isomorphic to its cohomology, we have Hom•k (IR cM! (k M )[dM ]; k) = Hom•k (Hc (M)[dM ]; k) .

(8.14)

Therefore, (8.13) determines a 0-cycle of the left-hand term in (8.14), hence in (8.12) and in (8.11). In this way, taking global sections over IB and 0-cohomology, we get a well-defined element in    H 0 Γ IB; IR π∗ IR Hom •k π −1 k IB [dM ], π ! k IB = MorD+ (IB) π −1 k IB [dM ], π ! k IB ,

which is the morphism π −1 k IB [dM ] → π ! k IB (8.9) stated in (1).11 To show that (8.9) is an isomorphism in D+ (E; k), we need only show that the induced morphism at the germs of those complexes at each x ∈ E are quasiisomorphisms, and, for convenience, to restrict ourselves to the open subsets U˜ , as

procedure glues together the individual Poincaré adjunctions on the fibers π −1 (x), for all x ∈ IB, in a coherent way over the whole base space IB, which is possible by having a coherent choice of orientations for these fibers. This is what we meant, in 8.2.2.1-(1), by Poincaré adjunction applied to the morphism IR π! (k E ) → IR π∗ (k E ).

11 The

254

8 Changing the Coefficients Field

they cover E. In that case, for x := (b, m) ∈ U˜ := U × M, we have the following identification at the level of germs: 

π −1 k IB [dM ]

x

     ! → π ! k IB x = k M [dM ] m → cM k m = ω M,m

(8.15)

where ω M := ω M (k) denotes the dualizing complex on M (fn. (2 ), p. 246), i.e. the complex of sheaves defined by the complex of presheaf on M ω M : V → Hom•k (IR cV ! k V ; k) = Hom•k (Hc (V ); k) . Likewise, the morphism in the right-hand side of (8.15) is induced by the morphisms of presheaves k M → ω M [−dM ]: Γ (V ; k) k

Γ (V ;

M )[−dM ]

Hom•k (Hc (V ; k); k)[−dM ]

(8.16)

which assigns to 1 the integration V relative to the orientation of V , induced by the orientation of the fiber bundle E. Now, as M is a manifold, a point m ∈ M has a basis of open neighborhoods V homeomorphic to RdM , in which case Hc (V ; k) is concentrated in degree dM , with HcdM (V ; k) = k. As a consequence, the second row in (8.16) is an isomorphism for such V ’s, which implies that the morphisms in (8.15) are quasi-isomorphisms for all x ∈ E, ending the proof of (1). (2) A complex of sheaves of k-vector spaces F ∗ := (F ∗ , d∗ ) on a topological space IB is said to be cohomologically bounded if its cohomology sheaves Hi (F∗ , d∗ ) = 0 vanish for |i| big. It is said to be perfect if its cohomology sheaves Hi (F∗ , d∗ ) are locally trivial sheaves of finite rank, i.e. if for all i ∈ Z, the space IB can be covered n(V ) by open subsets V such that Hi (F∗ , d∗ ) V ∼ k V , for some n(V ) ∈ N. Lemma If F ∗ is cohomologically bounded and perfect, the natural morphism  ι : F ∗ → IR Hom •k IR Hom •k (F ∗ , k IB ), k IB , is an isomorphism in D+ (IB). Proof of the Lemma First, notice that since

Hom k (knV , k V ) ∼ Hom k (kV , k V )n ∼ k nV , we immediately see that if L is a locally trivial sheaf of finite local rank, then the functor Hom k (L, −) is exact on the category of sheaves. As a consequence, IR Hom k (L, k IB ) = Hom k (L, k IB ) ,

8.2 Sheafification of Cartan Models over Arbitrary Fields

255

is again a locally trivial sheaf of finite ranks, so that ι : L → IR Hom k (IR Hom k (L, kIB ), k IB ) = Hom k (Hom k (L, kIB ), k IB ) (8.17) is an isomorphism in D+ (IB). Next, if 0 → F 0 → F 1 → · · · → F → 0 is a perfect complex of sheaves, the truncated complex τ< (F ∗ ) is still perfect and we have a small exact sequence of perfect complexes of sheaves 0 → τ< (F ∗ ) → F ∗ → H (F ) → 0 , giving rise in D+ (IB, k) to the morphism of exact triangles12 )

∗

))∨∨

ι ∗ ∨∨ )

∗

(

τ ι

(τ (

∗

(

∗

H (

) (8.18)

ι

(H (

∗

))∨∨

where (−)∨∨ := IR Hom k (IR Hom k (−, kIB ), k IB ). In (8.18) the right-hand vertical arrow is an isomorphism, since (8.17) is too. Hence, we can conclude that the central arrow is an isomorphism if and only if the left-hand arrow is also. But since the truncated complex τ< (F ∗ ) has less than

terms, we can assume by induction on this number, that this is indeed the case. Therefore, the central arrow in (8.18) is an isomorphism in D+ (IB). To finish, we need only recall that in D+ (IB), a cohomologically bounded complex is always isomorphic, after a shift of subscripts, to a bounded complex as in the previous case, which ends the proof of the lemma.  The proof of (2) is now reduced to proving that the complex IR π! k E is cohomologically bounded and perfect. This is already the case over a trivializing open subset U ⊆ IB. Indeed, we already explained in the poof of (1) that one has in D+ (U ; k) IR π! k E

U

 k U ⊗ Hc (M; k) ,

which gives the perfection property as it is a local property. It also shows that the sheaves Hi (IR π! k E ) vanish for i ∈ [[0, dM ]], which is also a local property. But, it is because bounds in [[0, dM ]] are independent of the trivializing open subset U , that in the end explains that IR π! k E is globally cohomologically bounded and that we can apply the lemma and finish the proposition’s proof.   Now we have all we need to extend the validity of the equivariant duality to any field of coefficients k.

is the subcomplex G ∗ ⊆ F ∗ , such that G i := F i for i < , G := im(d −1 ) and G i := 0 for all i > . Notice that, by construction, Hi (G ∗ ) = Hi (F ∗ ) if i < and Hi (G ∗ ) = 0 otherwise.

12 This

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8 Changing the Coefficients Field

8.3 Equivariant Poincaré Duality over Arbitrary Fields 8.3.1 The Equivariant Duality Theorem The following theorem is the analogue to the equivariant Poincaré duality Theorem 5.6.2.1. Notice its similarity with the relative Poincaré duality Theorem 3.3.3.1, which is unsurprising since both theorems use the same underlying formalism of Grothendieck-Verdier Duality. For the same reason it is worth emphasizing that all the results in Chap. 3 apply, with the obvious changes to the current framework of sheaf cohomology. Recall the notations : Ω G (M) := π∗ Ω (MG ) ,

Ω G,c (M) := π! Ω (MG )

and

HG := H (IB) .

Theorem 8.3.1.1 (Equivariant Poincaré Duality) Let G be a compact Lie group such that |G/G0 | is prime to char(k), and let M be an oriented G-manifold of dimension dM . Then, 1. The sheafification of the left Poincaré adjunction (8.6)-(I)  ID G,M : Ω G (M)[dM ] −→ IR Hom •(Ω G ,d) Ω G,c (M), Ω G

(8.19)

is an isomorphism in D DGM(Ω G ). 2. Applying IR Γ (IB; −) to (8.19) we get a quasi-isomorphism  D G,M : ΩG (M)[dM ] −→ IR Hom•(ΩG ,d) ΩG,c (M), ΩG

(8.20)

which is the left Poincaré adjunction isomorphism in D(DGM(ΩG )). 3. If, in addition, M is of finite type, then the sheafification of the right Poincaré adjunction (8.6)-(II)  ID G,M : Ω G,c (M)[dM ] → IR Hom •(Ω G ,d) Ω G (M), Ω G

(8.21)

is an isomorphism in D DGM(Ω G ), and the analogue to (2) also holds. Proof (1) Results, after the reformulation of the Poincaré adjunction in Sect. 8.2.2, from proposition 8.2.2.1-(1) applied to the fiber bundle (MG , IB, π, M). (2) Follows from (1), since by Proposition B.9.1.1-(1), we have  IR Hom•(ΩG ,d) (−, −) = IR Γ B, IR Hom •(Ω G ,d) (−, −) . (3) Results from Proposition 8.2.2.1-(2) applied to the fiber bundle (MG , IB, π, M), and the same arguments as for (1,2).  

8.4 Formality of IBG over Arbitrary Fields

257

Comment 8.3.1.2 Notice an important difference between Theorems 5.6.2.1-(1) and 8.3.1.1-(2). While the first states the quasi-isomorphy for the G -dual:  G (M)[dM ] −→ Hom•G G,c (M), G ,

(8.22)

the second states the quasi-isomorphy only for the derived G -dual:  ΩG (M)[dM ] −→ IR Hom•(ΩG ,d) ΩG,c (M), ΩG .

(8.23)

The reason being that (8.22) concerns equivariant de Rham complexes, and because G,c (M) is a free G -graded module (Eq. (5.73), p. 206), while (8.23), concerns Alexander-Spanier cochains complexes and we do not know if ΩG (M) and ΩG,c (M) are K-projective (ΩG , d)-graded modules (cf . fn. (24 ), p. 295), especially in positive characteristic, which prevents us from making the same conclusion.

8.4 Formality of IBG over Arbitrary Fields As recalled in the review on Cartan’s 1950 Seminar (4.1.1), formality of the classifying space IBG over the real numbers was highlighted by Cartan (4.1.1.3) as a by product of the Cartan-Weil’s methods. However, the same methods do not apply over fields of positive characteristic, and the question arises if IBG can be formal in those cases. The question was approached by Borel in a different setting in his Ph.D. thesis13 although the result is not explicitly stated.

8.4.1 The Integral Cohomology of G/T The key ingredient in the description of H (IBG; k) is the fact that H (G/T ; Z) is the regular representation of the Weyl group W (G), which we now recall. Notice that we already used this result in Sect. 6.5.4 when comparing the cohomologies of IBG and IBT . Given a compact Lie group G denote by G0 the connected component of e ∈ G, and let T be a maximal torus of G (hence of G0 ). Lemma 8.4.1.1 The Weyl groups W (G) := NG (T )/T , W (G0 ) := NG0 (T )/T are finite groups.14 We have W (G0 )  W (G), a an exact sequence of groups: → G/G0 → {1} . {1} → W (G0 ) → W (G) → 13 Borel

(8.24)

[13] Chapitre VII. Cohomologie entière et mod p de quelques espaces homogènes, more precisely §29. Le quotient d’un groupe compact par un tore maximal, p. 197. 14 N (T ) denotes the normalizer in G of T , i.e. the group of g ∈ G such that gT g −1 = T . G

258

8 Changing the Coefficients Field

Proof We have W (G0 )  W (G) because G0  G and NG0 (T ) = G0 ∩ NG (T ). We therefore get a natural injection W (G)/W (G0 ) ⊆ G/G0 , and claim that it is in fact an equality. To see this, we use the well-know result that for compact connected Lie groups the Weyl group W (G0 ) acts transitively on the set of maximal tori of G0 . Hence, given g ∈ G, there exists h ∈ G0 such that gT g −1 = hT h−1 , in which case h−1 g ∈ NG (T ), and g ∈ NG (T ) G0 , or, equivalently, W (G)/W (G0 ) = G/G0 . The sequence (8.24) is therefore exact. The Weyl group W (G0 ) is well-known to be finite and G/G0 is finite because G is compact.   Proposition 8.4.1.2 If the cardinality of the Weyl group W (G) is nonzero in k, then the space G/NG (T ) is k-weakly contractible.15 Proof The group W (G) acts naturally on G0 /T and on G/G0 and the obvious exact sequence of G0 -spaces G0 /T → G/T → → G/G0 is compatible with this action. Consequently, (G0 /T )/W (G) = (G/T )/W (G), which implies that   G/NG (T ) = G0 /T /W (G) = G0 /NG0 (T ) /(G/G0 ) , after Lemma 8.4.1.1. But then, to prove the proposition we need only establish the k-weakly contractibility of G0 /NG0 (T ) = (G0 /T )/W (G0 ). To see how, we recall that G0 , being connected, the Bruhat decomposition decomposes the manifold G0 /T as union of even dimensional Schubert cells indexed by the Weyl group W (G0 ), which immediately shows that the integral cohomology of G0 /T satisfies H (G0 /T ; Z) = Z[W (G0 )] , in which case Leray’s theorem stating that the W (G0 )-module H (G0 /T ; R) is the regular representation of G0 (cf . fn. (4 ), p. 231) extends to integral coefficients. Therefore H (G0 /NG0 (T ); k) =1 H (G0 /T ; k)W (G0 ) = k[W (G0 )]W (G0 ) = k , where (=1 ) is justified by the fact that |W (G0 )| is invertible in k.

 

Theorem 8.4.1.3 Let G be a compact Lie group. If the cardinality of the Weyl group W (G) is nonzero in k, then the classifying space IBG is a k-formal space. In particular, the classifying space IBT of torus T is a k-formal space for every k. Proof Let T be a maximal torus in G0 and let T  := NG0 (T ). The locally trivial fibration p : IBT → → IBG factors as the composition of three locally trivial

15 Recall that a topological space is said to be k-weakly contractible is its k-Betti numbers are those

of the singleton {•} (cf . fn. (31 ), p. 146).

8.4 Formality of IBG over Arbitrary Fields

259

fibrations q

r

s

[G0 /T ]

[G/G0 ]

IBG0 −−−→ IBG , IBT −−−→ IBT  −−−→  [W (G0 )]

where fibers are shown in brackets. – The fibration s : IBG0 → IBG. This is the orbit map of the action of the finite group G/G0 on IBG0 , and since |G/G0 |/|W (G)| after Lemma 8.4.1.1, the pullback s ∗ : Ω(IBG; k)−→Ω(IBG0 ; k)G/G0 q.i.

(8.25)

is a morphism of dg-algebras and a quasi-isomorphism. – The fibration r : IBT  → IBG0 . Since G0 is connected, the space IBG0 is simply connected and the sheaves Hi (IR r∗ k IBT  ) are constant and equal to k IBG0 , for i = 0, and 0, for i > 0. Whence, k IBG0  IR p∗ k IBT  in D(IBG0 ), and the pullback r ∗ : Ω(IBG0 ; k)−→Ω(IBT  ; k) q.i.

(8.26)

is a morphism of dg-algebras and a quasi-isomorphism. – The fibration q : IBT → IBT  . This is the orbit map of the action of W (G0 ) on IBT , hence the pullback q ∗ : Ω(IBT  ; k)−→Ω(IBT ; k)W (G0 ) , q.i.

(8.27)

which is also is a morphism of dg-algebras and a quasi-isomorphism. Summing up (8.25), (8.26), and (8.27), the pullback associated with the fibration p : IBT → → IBG is a morphism of dg-algebras and a quasi-isomorphism: p∗ : Ω(IBG; k)−→Ω(IBT ; k)W (G) . q.i.

(8.28)

We now recall how to prove that IBT is k-formal for every field k, in a way that keeps track of the action of the Weyl group W (G). Let X(T ) denote the group of irreducible characters of T , i.e. the set of group (continuous) homomorphisms Hom(T , Cu ), where Cu denotes the unit circle in C. We recall that X(T ) is isomorphic to the free Z-module Zdim T . For every non trivial character α ∈ X(T ), denote by C(α) the corresponding representation of T , i.e. t · z := α(t) z, for all t ∈ T and z ∈ C(α). Then, for every n ∈ N, the group T acts on the unit sphere S2n−1 (α) ⊆ C(α)n , well-known to be (2n−1)-connected, and we define S∞ (α) := lim n∈N S2n−1 (α) , −→ where the inductive limit is relative to the usual inclusions S2n−1 (α) ⊆ S2n+1 (α). It is easily seen that S∞ (α) is a CW-complex and a contractible space.

260

8 Changing the Coefficients Field

The Weyl group W (G) acts by group automorphisms on T , inducing and right action on X(T ) by α · w := α ◦w. Let B ⊆ X(T ) denote a W (G) stable finite subset of generators of the group X(T ). The representation of T on C(B) :=

 α∈B

C(α) ,

is then faithful, and the induced action on the spaces IET (B, n) :=

 α∈B

S(α, n)

and

IET (B) :=

 α∈B

S(α, ∞)

is free. Since IET (B) is a CW-complex and a contractible space, it can be used as universal fiber bundle for T . We then define   P(α, n) and IBT (B) := P(α, ∞) , IBT (B, n) := α∈B

α∈B

where P(α, n) := S2n−1 (α)/T is the projective space Pn−1 (C) of complex dimension (n−1). But the projective spaces Pn (C) have a canonical well-known structure of CW-complex with even dimensional cells, compatible with the inclusions Pn−1 (C) ⊆ Pn (C), hence inducing a CW-complex structure on IBT (B) := lim-ind IBT (B, n), always with even dimensional cells. The dg-algebra associated with the CW-structure of IBT (B) is then canonically isomorphic to its cohomology H (IBT ; k), which gives a morphism of dg-algebras  : H (IBT (B); k)−→Ω(IBT (B); k) ,

(8.29)

q.i.

that is a quasi-isomorphism and establishes the formality of IBT . Now, since B is stable under the action of W (G), this group acts naturally on IBT (B) and the morphism  is W (G)-equivariant and hence induce a morphism of dg-algebras and quasi-isomorphism  : H (IBT (B); k)W (G) −→Ω(IBT (B); k)W (G) , q.i.

(8.30)

where H (IBT (B); k)W (G) = H (IBT (B)/W (G); k) = H (IBG; k) , since |W (G)| is invertible in k, and also after (8.28). The composition of (8.28) and (8.30) gives the zig-zag p∗

p∗

Ω(IBG; k) −−→ Ω(IBT ; k)W (G) ←→ Ω(IBT (B); k)W (G) ←−− H (IBG; k) , q.i.

q.i.

which establishes the k-formality of IBG.

q.i.

 

8.5 Equivariant Gysin Morphisms over Arbitrary Fields

261

Corollary 8.4.1.4 (Enhanced Equivariant Poincaré Duality) Let G be a compact Lie group such that the cardinality of the Weyl group W (G) is invertible in k. Let M be an orientable G-manifold of dimension dM . Then 1. The left Poincaré adjunction isomorphism 8.3.1.1-(2) in D DGM(ΩG ): DB,M : G (M)[dM ]  IR Hom•(ΩG ,d) (ΩG,c (M), ΩG ) ,

(8.31)

induces a convergent spectral sequence: p,q

IE 2

q

p+q+dM

:= hp IR Hom•HG (HG,c (M), HG ) ⇒ HG

(M) .

2. If dimproj (HG,c (M)) ≤ 1 as HG -graded module, then the Poincaré adjunction induces an isomorphism in D DGM(HG ): DB,M : HG (M)[dM ]  IR Hom•HG (HG,c (M), HG ) ,

(8.32)

and the spectral sequence (IE r , dr ) in (1) degenerates, i.e. dr = 0 for r ≥ 2. Furthermore, if HG,c (M) is a projective HG -graded module, then (8.32) induces an isomorphism of HG -graded modules  HG (M)[dM ]  Hom•HG HG,c (M), HG . 3. If M is of finite type, the statements (1) and (2) remain true even if we swap the terms ΩG (M) ↔ ΩG,c (M) and HG (M) ↔ HG,c (M). Proof Since IBG is a k-formal space, the statements are immediate applications of Corollary A.3.3.2.   Comment 8.4.1.5 When G is a torus T , we have W (G) = {1} and Theorem 8.4.1.4 is valid over any field k. Moreover, when G is the circle group S1 , IBG = P∞ (C) and the ring H (IBG; k) is isomorphic to the polynomial algebra k[X] which is of homological dimension 1. In that case, the condition dimproj (HG,c (M)) ≤ 1 in 8.4.1.4-(2) is automatically verified and can be omitted.

8.5 Equivariant Gysin Morphisms over Arbitrary Fields Now that we have proved Poincaré duality over a field k of characteristic prime to |W (G)|, we can mimic the method used to introduce Gysin morphisms in de Rham cohomology and achieve the same result over k. In 8.2-(P-2d), we explained how an equivariant map f : X → Y between Gspaces defines a morphism in D+ (YG ) f # : Ω (YG ) → f∗ Ω (XG ) inducing the usual pullback f ∗ : HG (Y ) → HG (X).

(8.33)

262

8 Changing the Coefficients Field

Applying the functor πY ! to (8.33), we get a morphism in D+ (IB) πY ! (f # ) : πY ! (Ω (YG )) → πY ! (f∗ Ω (XG )) = πX! (Ω (XG )) where one recognizes the complexes Ω G (X) and Ω G (Y ). We can thus write: πY ! (f # ) : Ω G (Y ) → Ω G (X) .

(8.34)

If f is, in addition, a proper map, we have f! = f∗ and the same ideas that apply to πY ∗ , now lead to a morphism in D+ (YG ) πY ∗ (f # ) : Ω G,c (Y ) → Ω G,c (X) ,

(8.35)

inducing the usual pullback f ∗ : HG,c (Y ) → HG,c (X). In the next sections, M and N are equidimensional manifolds of dimensions dM and dN , orientable over a field k of characteristic prime to |W (G)|.

8.5.1 Gysin Morphism for General Maps We apply the duality functor (−)∨ := IR Hom •k (−, k IB ) to (8.34), and consider the diagram in D+ (IB) Ω G (M)∨

(πY ! (f # ))∨

ID G,N

ID G,M

Ω G,c (M)[dM ]

Ω G (N)∨

f!

(8.36)

Ω G,c (N)[dN ]

where the right-hand vertical arrow is an isomorphism when N is of finite type, after Theorem 8.3.1.1–(3). Following Sects. 2.8.2 and 6.1.2, we can proceed to defining the Gysin morphism for a general map in D+ (IB) as the unique morphism f! closing the diagram (8.36). The cohomology of global sections then gives the Gysin morphism for a general map in equivariant cohomology over k: f! : HG,c (M; k)[dM ] → HG,c (N ; k)[dN ] . The case where N is not of finite type is dealt with as in Sect. 6.1.2, taking limits over filtrant covers of finite type open subsets of N .

8.6 The Localization Formula over Arbitrary Fields

263

8.5.2 Gysin Morphism for Proper Maps The underlying idea is the same as in the previous case, but we now apply the duality functor (−)∨ to (8.35). We then obtain the diagram Ω G,c (M)∨

(πY ∗ (f # ))∨

ID G,N

ID G,M

Ω G (M)[dM ]

Ω G,c (N)∨

f∗

Ω G (N)[dN ]

where both vertical arrows are isomorphisms in D+ (IB), after 8.3.1.1-(1). The Gysin morphism for proper maps in D+ (IB) is then defined, as in Sects. 2.8.2 and 6.1.2, as the only morphism f∗ that closes the diagram. The cohomology of global sections then gives the Gysin morphism for proper maps in equivariant cohomology over k: f∗ : HG (M; k)[dM ] → HG (N ; k)[dN ] .

8.6 The Localization Formula over Arbitrary Fields Following Sect. 8.4, we know that when |W (G)| is invertible in k, the classifying space IBG is k-formal, and H (IBG; k) = H (IBT ; k)W (G)  k[X1 , . . . , Xr ]W (G) , where r := dimR (T ). We can then set QG and QT the respective rings of graded fractions of HG and HT and follow word-by-word Chap. 7 on Localization, in which case all the results up to the Localization Formula 7.7.1 remain true under the only additional hypothesis that the cardinality of W (G) is invertible in k.

Appendix A

Basics on Derived Categories

This compendium of basic ideas aims at a quick overview of the theory of derived categories and derived functors. For more thorough study of the subject, we encourage the reader to consult the profuse literature on the subject, for example: • • • • • • •

R. Hartshorne, Residues and Duality (chap. I) . . . . . . . . . . . . . . . . . . . [51] B. Keller, Derived categories and their uses . . . . . . . . . . . . . . . . . . . . . [65] B. Iversen. Cohomology of Sheaves (chap. XI) . . . . . . . . . . . . . . . . . . . [58] M. Kashiwara, P. Schapira, Categories and sheaves (chap. 13) . . . . . . . . . [62] Stacks Project, Differential Graded Algebra . . . . . . . . . . . . . . . . . . . . . [87] J.-L. Verdier, Des catégories dérivées des catégories abéliennes . . . . . . . . [93] C. Weibel, An introduction to homological algebra (chap. 10) . . . . . . . . . [95]

A.1 Categories of Complexes The aim of Algebraic Topology is, as the name suggests, the algebraization of topology. This entails, among other ideas, attaching to a topological space complexes whose cohomology encodes topological properties. Although the main interest of this discipline lies in the interaction between algebra and topology, the theme of complexes or, more precisely, of differential graded algebras and modules has undergone significant development, to the point of becoming a subject of study in its own right. We recall now some very basic definitions and results on the subject that so far in this book we have given little attention. We will confine to complexes of cochains.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Arabia, Equivariant Poincaré Duality on G-Manifolds, Lecture Notes in Mathematics 2288, https://doi.org/10.1007/978-3-030-70440-7

265

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A Basics on Derived Categories

A.1.1 The Category of Complexes of an Abelian Category Given an abelian category Ab, we denote by C(Ab) the category of complexes (of cochains) of an abelian category Ab. C-1. A complex of cochains of Ab, is a graded object C ∈ Ob(AbZ ) together with a differential, i.e. with d ∈ MorAb (C, C[1]), such that d 2 = 0: d−3 d−2 d−1  d0 d1 d2 (C, d) := · · · −−→ C−2 −−→ C−1 −−→ C0 −−→ C1 −−→ C2 −−→ · · · .

C-2. A morphism of complexes f ∈ MorC(Ab) ((C, d), (C  , d  )) is a morphism of graded objects f ∈ MorAbZ (C, C  ) such that f ◦ d = d  ◦ f . C-3. The cohomology functor is the covariant (additive) functor h : C(A)  AbZ ,

(A.1)

which associates with (C, d), its cohomology, i.e. the graded object   h(C, d) := ker(di )/im(di−1 ) i∈Z ∈ AbZ . C-4. A complex (C, d) is said to be acyclic if h(C, d) = 0. C-5. For m ∈ Z, the category Ab is embedded in C(Ab) as the full subcategory of complexes concentrated in (degree) m, i.e. complexes (C, d) with C i = 0 for all i = m and (hence) d = 0. The embedding is given by the functor [m] : Ab  C(Ab) , which associates with X ∈ Ob(Ab), the complex X[0] concentrated in m, where X[m]m := X, and likewise for f ∈ MorAb (X, X ), where f [m]m := f . C-6. For every m ∈ Z, the shift functor ‘[m]’ [m] : C(Ab) → C(Ab) ,

(A.2)

associates with (C, d) the complex (C, d)[m] := (V , D), where Vi := Ci+m and Di := (−1)m di+m , and it associates with α : (C, d) → (C  , d  ), the morphism α[m] : (C, d)[m] → (C  , d  )[m], where α[m]i = αi+m .

A

Basics on Derived Categories

267

A.1.2 Extending Additive Functors from Ab to C(Ab) Let F : Ab → Ab be an additive functor between abelian categories.1 Applying F to a complex (C, d) ∈ C(Ab), term to term, results in a graded object F (C, d) := (F (C), F (d)), with F (C)i = F (Ci ) and F (d)i := F (di ): (C d) := Ci

di

Ci+1

F

F(C d) := F(Ci )

F (d i )

F(Ci+1 )

The fact that F is additive implies that F (C, d) is also a complex since F (di+1 ) ◦ F (di ) = F (di+1 ◦ di ) = F (0) = 0 . If f ∈ MorC(Ab) ((C, d), (C  , d  )), we then denote by F (f ) : F (C, d) → F (C  , d  ) the morphism of graded objects F (f )i := F (fi ), which is clearly a morphism of complexes of C(Ab ). We thus extend the definition of F from Ab to C(Ab), which we denote by the same letter F : C(Ab)  C(Ab ). The extension is clearly compatible with the shift functors [m], making commutative the diagram of functors

(A.3)

A.1.3 The Mapping Cone A fundamental operation in categories of complexes, is the mapping cone associated with a morphism of complexes.2 Given a morphism α : (C, d) → (C  , d  ) in C(Ab), consider, for all i ∈ Z, the morphism $i in Ab defined by

1 We

recall that in an abelian category Ab the sets of morphisms MorAb (X, Y ) are abelian groups for all X, Y ∈ Ob(Ab). A covariant functor between abelian categories F : Ab → Ab is said to be additive if the maps FX,Y : MorAb (X, Y ) → MorAb (F (X), F (Y )) are homomorphisms of groups for all X, Y ∈ Ob(Ab); and likewise for contravariant functors. The functor is then said to be exact, left exact or right exact, if it transforms every short exact sequence of Ab in respectively an exact, left exact or right exact sequence of Ab , and this regardless of whether F is covariant or contravariant. 2 Useful references to consult are Weibel [95] §1.5 Mapping Cones and Cylinders, p. 18, and Kashiwara-Schapira [62] ch. 11.1 Differential Objects and Mapping Cones, p. 270.

268

A Basics on Derived Categories $i

 Ci ⊕ Ci+1 −−−→ Ci+1 ⊕ Ci+2

(x  , x)  −−−→

(di x  + αi+1 (x), −di+1 (x))

(A.4)

One has $i+1 ◦ $i = 0, whence a complex, called the cone of α, c(α) ˆ := (C  ⊕ C[1], $)

(A.5)

We then consider the morphisms of complexes p2

ι1

(C  , d  ) −−→ (C  ⊕ C[1], $) (x  , 0) x   −−→

(C  ⊕ C[1], $) −−→ (C, d)[1]  −−→ x (x  , x)

(A.6)

giving rise to a triangle (Definition A.1.5.4) of morphisms of complexes: α

(C, d)

(C , d )

[1] ι1

p2

(ˆ equivalently denoted by p2

ι1

α

(C, d) −−→ (C  , d  ) −−→ c(α) ˆ −−→

(A.7)

[1]

where the notation ‘[1]’ recalls that degrees are increased by 1.3

A.1.4 Homotopies A morphism of complexes f : (C, d) → (C  , d  ) is said to be homotopic to zero if there is a family of morphisms hi ∈ MorAb (Ci , C  i−1 ), such that  ◦ hi + hi+1 ◦ di fi = di−1

···

Ci−1 fi−1

···

3 Beware

C

i −1

hi di−1

(A.8)

Ci

di

Ci+1

fi

hi+1

fi+1

C

i

C

i+1

that the compositions ι1 ◦ α and α ◦ p2 are not equal to 0.

··· ···

A

Basics on Derived Categories

A.1.4.1

269

Terminology and Notations

K-1. Two morphisms of complexes f, g : C → C  are said to be homotopic, and we write f ∼ g, if the difference f − g is homotopic to zero. K-2. A complex C is said to be homotopic to zero, and we write C ∼ 0, if the identity morphism idC is homotopic to zero. K-3. A morphism f : C → C  is said to be an isomorphism up to homotopy, if there exists a morphism g : C  → C such that g ◦ f ∼ idC and f ◦ g ∼ idC  . We then say that C and C  are homotopy-equivalent, and write C ∼ C  . K-4. The relation ‘∼’, is an equivalence relation compatible with the additive structure of C(Ab), i.e. if f ∼ f  and g ∼ g  then (f + g) ∼ (f  + g  ). As a consequence, the sets of homotopy-equivalence classes: Hot•Ab (C, C  ) := Hom•Ab (C, C  ) ∼ .

(A.9)

is an abelian group. K-5. The relation ‘∼’ is also compatible with the composition of morphisms of complexes, i.e. if f ∼ f  et g ∼ g  then (f ◦ g) ∼ (f  ◦ g  ), whenever the compositions make sense. The following proposition summarizes well-known properties of the definitions and constructions that we have introduced so far. Proposition A.1.4.1 Let Ab be an abelian category. 1. The category of complexes C(Ab) is abelian. α

β

2. Let (0 → A − →B − → C → 0) be an exact sequence in C(Ab). There exists a connecting morphism c : h(C) → h(A)[1] such that the following long sequence of cohomology is exact: h(α)

h(β)

c

· · · −→ h(A) −−−→ h(B) −−−→ h(C) −−−→ h(A)[1] −→ · · · α

ι

p

3. Let A −→ B −→ c(α) ˆ −→A[1] be a mapping cone in C(Ab). a. The following long sequence of cohomology is exact: h(α)

h(ι)

h(p)

· · · −→ h(A) −−−→ h(B) −−−→ h(c(α)) ˆ −−−→ h(A)[1] −→ · · · b. α is a quasi-isomorphism if and only if h(c(α)) ˆ = 0. c. α is an isomorphism modulo homotopy if and only if c(α) ˆ ∼ 0. Sketch of Proof (1) and (2) are well-known standard facts. (3a) Results, for example, from Lemma 1.5.3 in Weibel [95] (p. 19). Rather than giving details, we encourage the reader to prove the claim himself by straightforward verification after the definitions of ι and p in formula (A.6). (3b) is immediate after (3a) and (3c) demands manual verifications which are also left to the reader. 

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Exercise A.1.4.2 Let F : Ab  Ab be an additive functor between abelian categories. Show the equivalence of the following statements. ( , p. 351) 1. F : Ab  Ab is exact. 2. F : C(Ab)  C(Ab ) preserves acyclicity. 3. F : C(Ab)  C(Ab ) preserves quasi-isomorphisms.

A.1.5 The Homotopy Category K(Ab) When interested in the homotopy of topological spaces, continuous maps which coincide up to homotopy induce homotopic morphisms of associated complexes. One is thus led to defining the homotopy category K(Ab) whose objects are the same as in C(Ab) but whose morphisms are the homotopy classes of morphisms of complexes A.1.4.1-(K-4), i.e. we set MorK(Ab) (C, C  ) := Hot0Ab (C, C  ) .

(A.10)

The category K(Ab) is additive. Furthermore, the complexes in C(Ab) which are homotopy-equivalent become isomorphic, and the morphisms in C(Ab) which are isomorphisms up to homotopy are invertible. Furthermore, the well-known fact that homotopic morphisms of complexes induce the same morphism in cohomology makes the cohomology functor h in (A.1) to factor through K(Ab). We thus have the factorization C(Ab)

ι

K(Ab)

h

AbZ ,

h

(A.11)

where ι : C(Ab)  K(Ab) denotes the functor which is the identity on objects, and which associates its homotopy class with a morphism. Exercise A.1.5.1 Show that the mapping cones of homotopic morphisms in C(Ab) are homotopy-equivalent complexes. Conclude that the mapping cone of a morphism in K(Ab) is canonically defined ( , p. 351) The following Proposition is an important foundational result. Proposition A.1.5.2 1. Given C , C  ∈ Ob(C(Ab)), there is a canonical identification    hi Hom•Ab C, C  )  MorK(Ab) C, C  [i] ,

∀i ∈ Z .

(A.12)

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271 ι

α

p

2. For C ∈ Ob(C(Ab)) and every mapping cone A −→ B −→ c(α) ˆ −→A[1] in C(Ab), the following induced long sequences (K stands for K(Ab)) are exact. ˆ −→ MorK (C, A[1]) −→ −→ MorK (C, A) −→ MorK (C, B) −→ MorK (C, c(α)) ˆ C) −→ MorK (B, C) −→ MorK (A, C) −→ MorK (A, C[1]) −→ −→ MorK (c(α), in particular the morphisms ι ◦ α, p ◦ ι, α[1] ◦ p are all homotopic to 0. Sketch of Proof (1) The isomorphism (A.12) is immediate after the definition of the complex (Hom• (−, −), D) (Sect. 2.1.8). (2) The exactness of the long sequences induced by the functors MorK (C, −) and MorK (−, C) can be proven by hand using only the definition of homotopy. This is a great exercise to do at least once in your life. The result is however generally shown as a corollary of the fact that K(Ab) is a triangulated category (Sect. A.1.5.2), which has the advantage of proving (2) in greater generality.4  A main difference between C(Ab) and K(Ab) is that the homotopy category is generally not abelian as shown in the following exercise. Exercises A.1.5.3 The aim is to show that K(Ab) is an abelian category if and only if Ab is a split category, in which case the fully faithful embedding AbZ ⊆ K(Ab) is an equivalence of categories, and K(Ab) too is a split category.5 1. Given a monomorphism α : A → B in K(Ab) and its mapping cone: p ι α →B − → c(α) ˆ − →A[1] , show that p = 0, that α admits a retraction ρ : B → A− A, and that ι is an epimorphism admitting a section σ : c(α) ˆ → B.6 If, in addition, K(Ab) is an abelian category, then the following sequence is a split sequence in K(Ab), α

0

A ρ

4 See

ι

B σ

c(α) ˆ

0,

Hartshorne [51] Proposition 1.1 (b), p. 23, or Weibel [95] Example 10.2.8, p. 377. result is valid for any triangulated category (Sect. A.1.5.2) in lieu of K(Ab). 6 In any category C a morphism α : X → X  is said to be a monomorphism, if it is left cancellable,  i.e. α ◦ β = α ◦ β  ⇒ (β = β  ), in other words, if for every Y ∈ Ob(C), the pushforward  α∗ : MorC (Y, X) → MorC (Y, X ), α∗ (β) := α  ◦ β, is injective. Dually, α : X → X is said to be an epimorphism, if it is right cancellable, i.e. β ◦ α = β  ◦ α ⇒ (β = β  ), in other words, if for every Y ∈ Ob(C), the pullback α ∗ : MorC (X , Y ) → MorC (X, Y ), α ∗ (β) := β ◦ α, is injective. If in addition C is an additive category, a kernel for α : X → X , is any monomorphism ι : K → X such that α ◦ ι = 0 and such that that every morphism β : Y → X verifying α ◦ β = 0 factors through ι (in a unique way). Dually, a cokernel for α : X → X  is any epimorphism ν : X → C such that ν ◦ α = 0 and such that every morphism β : X → Y verifying β ◦ α = 0 factors through ν (in a unique way). 5 The

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and there are isomorphisms inverse of each other, ρ⊕ι

/ (A ⊕ c(α)) ˆ

B o

h(ρ)⊕h(ι)

h(B) o

and

α+σ

/ h(A) ⊕ h(c(α)) ˆ ,

h(α)+h(σ ))

respectively in K(Ab) and AbZ . The categories K(Ab) and Ab are therefore split categories. ( , p. 351) 2. Show that if Ab is split, then the embedding AbZ ⊆ K(Ab) is an equivalence of categories and K(Ab) is a split abelian category. ( , p. 352)

A.1.5.1

Triangles and Exact Triangles

Since the category K(Ab) is generally not abelian, we are led to relegate short exact sequences in favor of mapping cones, or, more generally, sequences isomorphic in K(Ab) to mapping cones, which bring us to recall the definitions of triangles and exact triangles in the category K(Ab). Definition A.1.5.4 A triangle in K(Ab) is a sextuple (A, B, C, u, v, w) where A, B, C are complexes and u : A → B, v : B → C and w : C → A[1] are morphisms of complexes. These data can be represented by a triangular diagram, hence the name ‘triangle’, u

A w [1]

B equivalently denoted by

v

C

A

u

A

u

B B

v

C

v

C

w w [1]

A[1] , or .

A morphism of triangles from (A, B, C, u, v, w) to (A , B  , C  , u , v  , w  ) is any triple (α, β, γ ) of morphisms in K(Ab) making a commutative diagram in K(Ab) A

u

α

A

v

B

C γ

β u

B

w

v

C

A[1] α[1]

w

(A.13)

A[1]

The composition of morphisms of triangles is defined component-wise. The morphism (u, v, w) is an isomorphism if its components u, v, w are isomorphisms. The triangles of most interest to us are those giving rise to long exact sequences when applying the cohomology functor h and the functors MorK (C, −) and MorK (−, C) (Proposition A.1.5.2). These are exactly those triangles which are isomorphic, in K(Ab), to mapping cones. One is thus led to the following definition.

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Definition A.1.5.5 A triangle in K(Ab) is called exact (or distinguished) if it is isomorphic to the mapping cone of a morphism. β

α

→ B − → C → 0) is a short exact Comment A.1.5.6 Beware that if (0 → A − sequence in C(Ab), none of the triangles T := (A, B, C, α, β, 0)

and

h(T ) := (h(A), h(B), h(C), h(α), h(β), c)

where c is the connection morphism, is necessarily an exact triangle. However the triangles T remain important for us as they generate long exact sequences in cohomology. This pinpoints the fact that exact triangles do not completely replace short exact sequences, which is an issue of the category K(Ab), since we still need the category C(Ab) in order not to lose the information carried by short exact sequences. We will see that this is a lack of the category K(Ab) which the derived category D(Ab) is free from (Comment A.1.6.5), which is another reason to prefer the latter. Exercise A.1.5.7 We justify in two different ways the usually claimed fact that not every short exact sequence in C(Ab) can be embedded in an exact triangle of K(Ab). β

α

1. Given an exact sequence E := (0 → A − →B− → C → 0) in C(Ab), show that the map  : c(α) ˆ → C, defined by (x, y) := β(x) is a quasi-isomorphism of complexes. α

A

ι

B

c(α) ˆ

p

A[1]

q.i.

0

α

A

β

B

C

0

Give an example in C(Mod(Z)) where  has no homotopy inverse. ( , p. 352) p q →B− → C → 0) in Ab, show that an 2. Given an exact sequence E := (0 → A − exact triangle of the form p[0]

A[0]

/ B[0]

q[0]

/ C[0]

γ

/ A[1]

exists in K(Ab), if and only if E is split in Ab, in which case γ = 0. Hint. Use Exercise A.1.5.3-(1). ( , p. 353)

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Triangulated Categories7

We owe to Verdier8 the study of the properties of the family of exact triangles in the homotopy category K(Ab) allowing the axiomatization of the underlying abstract structure, today referred to as a triangulated category. This is an additive category C together with an automorphism [1] : C  C, the translation functor, and a collection T of triangles (defined as in A.1.5.4), called exact (or distinguished) triangles, which is required to satisfy four axioms9 TR-1. Any triangle isomorphic to a triangle in T belongs to T. Every morphism α : A → B can be imbedded in a triangle (A, B, C, α, β, γ ) ∈ T. For every A ∈ Ob(C), the triangle (A, A, 0, idA , 0, 0) belongs to T. TR-2. (A, B, C, α, β, γ ) ∈ T if and only if (B, C, A[1], β, γ , α[1]) ∈ T. TR-3. Given T := (A, B, C, u, v, w), T  := (A , B  , C  , u , v  , w  ) in T, and given morphisms α : A → A and β : B → B  such that the diagram u

A α

A

B

⊕ u

v

w

∃ γ

β

B

C

v

C

A[1] α[1]

w

A[1]

(A.14)

is commutative, there exists γ : C → C  (not necessarily unique) such that (α, β, γ ) = T → T  is a morphism of triangles. TR-4. The Octahedral Axiom. Given exact triangles (A, B, C, u, j, −) ,

(B, C, A , v, −, i) ,

(A, C, B  , v ◦ u, −, −)

there exist morphisms f : C  → B  and g : B  → A such that the triangle (C  , B  , A , f, g, j [1] ◦ i) is exact and the two other faces of the octahedron with f and g as edges, are commutative (view Fig. A.1). Readers discovering these axioms for the first time, especially TR-3 and TR-4, should bear in mind that that they are natural extensions to triangles of familiar and useful elementary properties of short exact sequences in abelian categories. For example, TR-3 is obvious if T and T  are exact sequences, in which case v and v  are cokernels.

7 The reader will find a thorough presentation of Triangulated Categories in Hartshorne [51], Chapter I §1, p. 20, Kashiwara-Schapira [62], Chapter 10, p. 241, Weibel [95], Chapter 10.2, p. 373, and obviously in Verdier [93]. 8 As reported by Illusie in the preface to Verdier [93]. 9 As presented in Hartshorne [51], Chapter I-§1, Triangulated Categories, p. 20.

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Fig. A.1 The Octahedral Axiom

The verification of the four axioms in the homotopy category K(Ab) is not difficult and leads to the following basic fact.10 Proposition A.1.5.8 1. The category K(Ab) together with the collection of exact triangles (A.1.5.5) is a triangulated category. 2. A triangle (A, B, C, ., ., .) in K(Ab) is exact if and only if the induced sequences −→ MorK (−, A) −→ MorK (−, B) −→ MorK (−, C) −→ MorK (−, A[1]) −→ −→ MorK (C, −) −→ MorK (B, −) −→ MorK (A, −) −→ MorK (A, −[1]) −→ (where K stands for K(Ab)) are long exact sequences.

A.1.6 The Derived Category D(Ab) When interested in homology, one quickly realizes that homology is much more flexible than complexes for essentially two reasons: quasi-isomorphic complexes are not always isomorphic in K(Ab) and acyclic complexes11 are not always homotopy-equivalent to zero (Exercise A.1.6.1). We owe Grothendieck for the idea of filling this gap by formally inverting the quasi-isomorphisms of K(Ab) following a procedure of localization analogue to the construction of noncommutative rings of quotients, which we now describe.12 ,13 Exercise A.1.6.1 (Counterexamples) 1. Consider in Mod(Z), the acyclic complexes, for m ≥ 0,

10 See

Hartshorne [51] Chapter I-§2, p. 25, or Weibel [95] Proposition 10.2.4, p. 376. complex (C, d) is said to be acyclic if it is exact, i.e. if it has zero cohomology. 12 Hartshorne [51] Chapter I. The Derived Category, §3 Localization of Categories p. 28. 13 Lam [71] Chapter 4. Rings of Quotients, §10A. Ore Localizations, p. 299. 11 A

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 C(m) := 0 → (m) → Z → Z/(m) → 0 . a. Show that the C(m)’s are two-by-two quasi-isomorphic and two-by-two non homotopy-equivalent. ( , p. 353) b. Show that C(m) is homotopic to 0 if and only if m = 0. 2. Show that every quasi-isomorphism in K(Ab) is an isomorphism if and only if every morphism in Ab is split.14 ( , p. 354)

A.1.6.1

Multiplicative Collection of Morphisms

A collection S of morphisms of a category C is said to be multiplicative if it contains the identity morphisms idX for all X ∈ Ob(C), if it is stable under composition of morphisms in C, and if it satisfies the following conditions: S-1. Given s ∈ S, any diagram in C of shape a commutative diagram

s

•

t

•

•

t

•

s

• • t

•

s

can be completed to

, for some s  ∈ S. And the same,

reversing arrows. S-2. Given t1 , t2 ∈ MorC (X, Y ), the following conditions are equivalent. a. There exists s ∈ S such that t1 ◦ s = t2 ◦ s. b. There exists s ∈ S such that s ◦ t1 = s ◦ t2 .

A.1.6.2

Universal Property of Localized Categories

Let C be a category, and let S be a multiplicative collection of morphisms of C. The localization of C with respect to S is a category C[S−1 ], together with a functor Q : C  C[S−1 ] such that the following conditions are satisfied. a) Q(s) is an isomorphism for every s ∈ S. b) Any functor F : C  C such that F (s) is an isomorphism for every s ∈ S factors uniquely throughQ.

(A.16)

The reader will find a constructive proof of the following existence theorem of localized categories in Hartshorne [51] (Proposition 3.1, p. 29.). Theorem A.1.6.2 If S is a multiplicative collection of morphisms in a category C, then the localized category C[S−1 ] exists and is unique up to isomorphism. In addition, if C is additive, then so is C[S−1 ].

morphism f : X → Y in an abelian category is said to be split if there are isomorphisms X ∼ K ⊕ Z and Z ⊕ L ∼ Y through which f reads as (k, z) → (z, 0).

14 A

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A.1.6.3

277

The Derived Category D(Ab) as Localization of K(Ab)

A remarkable property of the category K(Ab), which is not verified by C(Ab) (Exercise A.1.6.4), is that in K(Ab) the collection S of quasi-isomorphisms, that we want to invert, is indeed a multiplicative system. Furthermore, S is compatible with the triangulated category structure of K(Ab), which means that the following two additional conditions, obvious in K(Ab), are satisfied: S-3 if s ∈ S then s[1] ∈ S; S-4 in axiom A.1.5.2-TR-3, if α, β ∈ S then γ ∈ S. The category K(Ab)[S−1 ] therefore exists, it is the derived category of Ab. Proposition A.1.6.3 (And Definition) Let Ab be an abelian category. The family S of quasi-isomorphisms in K(Ab) is a multiplicative collection of morphisms compatible with the triangulated category of K(Ab). The derived category of Ab is then, by definition, the localized category D(Ab) := K(Ab)[S−1 ] .

(A.17)

Furthermore, the category D(Ab) is a triangulated category relative to the collection of triangles isomorphic (in D(Ab)) to cones of morphisms. Sketch of Proof See Hartshorne [51] §I, Proposition 3.2, p. 32, and Proposition 4.1, p. 35. The proof that S is multiplicative is an application of the mapping cone construction (Sect. A.1.3). This is one of the first examples where it is necessary to soften the category of complexes C(Ab) by identifying homotopic morphisms. We encourage the reader to provide his own detailed proof of this proposition without looking at ours ( , p. 354).  Exercise A.1.6.4 (A Counterexample) It is instructive to see why the collection S of quasi-isomorphisms is not multiplicative in the category C(Ab).  idX p2 ι → c(id ˆ X) − → be the mapping cone (A.1.3). Let X −→ X − 1. Show that the diagram D :=



0 ι X



cannot always be completed in

/ c(id ˆ X)

the category C(Ab) to a commutative diagram

•

0

s

ι

X

c(id ˆ X)

where s is a quasi-

isomorphism. ( , p. 356) 2. Complete the diagram D in K(Ab). Comment A.1.6.5 Coming back to Exercise A.1.5.7-(1), given the exact sequence α

β

E := (0 → A − →B− → C → 0) in C(Ab), the quasi-isomorphism (x, y) := β(x) A

α

B

ι

c(α) ˆ

p

A[1]

q.i.

0

A

α

B

β

C

0

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is invertible in D(Ad), so that we can now state that every short exact sequence in C(Ab) can be embedded in an exact triangle of D(Ab) (cf . Comment A.1.5.6). Notice that the embedding associates E with the triangle (A, B, C, α, β, γ ), where γ = p ◦  −1 is generally different from 0.

A.1.6.4

The Morphisms in the Derived Category D(Ab)

The morphisms in D(Ab) from a complex C to another C  are represented by zigzag paths of morphisms in K(Ab) of the form s1

t1

s2

sr−1

q.i.

q.i.

tr

sr

C ←−− • −−→ • ←−− • · · · • ←−− • −−→ • ←−− C  . q.i.

q.i.

(A.18)

−1 ◦ · · · ◦ s2−1 ◦ t1 ◦ s1−1 , where the arrows in the ‘wrong’ denoted by sr−1 ◦ tr ◦ sr−1 direction are quasi-isomorphisms. Two paths are said to be equivalent if applying the property A.1.6.1-(S-2) of multiplicative systems we can transform one into the other by means of a finite number of exchanging steps of the form

t1 ◦ s1−1

s2−1 ◦ t2

s1

•

•

s2 ◦ t1 = t2 ◦ s1 in K(Ab)

t2

t1

• •

(A.19)

s2

where si ∈ S et ti ∈ Mor(C). A morphism in D(Ab) from C to C  ) is then an equivalence class of paths of the form (A.18). We denote by MorD(Ab) (C, C  ) the set of such morphisms. The composition law is the concatenation of zig-zag paths.

A.1.6.5

Factorization of the Cohomology Functor

Thanks to the universal property of localizations A.1.6.2-(b), we can now add a new step to the factorization of the cohomology functor in (A.1): ι

C(Ab)

K(Ab)

D(Ab)

h

AbZ

(A.20)

h

The category D(Ab) thus appears to be the closest possible to cohomology while preserving complexes, which is where the strength of the derived category lies. Exercise A.1.6.6 Show that a morphism in D(Ab) can always be lifted in K(Ab) as two-step paths • ←− • −→ • and • −→ • ←− •. ( , p. 357) q.i.

q.i.

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279

A.1.7 The Subcategories C∗ (Ab), K∗ (Ab) and D∗ (Ab) A complex C ∈ C(Ab) is said to be bounded (resp. bounded below, bounded above) if C m = 0 for |m|  0 (resp. m  0, m  0). We denote by Cb (Ab), C+ (Ab) and C− (Ab) the full subcategories of C(Ab) whose objects are the complexes respectively bounded, bounded below and bounded above. The categories K∗ (Ab) and D∗ (Ab) for ∗ ∈ {b, +, −} are defined in the same way. The shift functors [m] (A.1.1-C-6) extend to K∗ (Ab) and D∗ (Ab), where they clearly preserve exactness of triangles, quasi-isomorphisms and acyclicity. Everything we have said since Sect. A.1.1 applies mutatis mutandis to the categories of bounded complexes.

A.2 Deriving Functors A.2.1 Extending Functors from Ab to D(Ab) Given an abelian category Ab, we have introduced the sequence of ‘extensions’ Ab

[m]

C(Ab)

ι

K(Ab)

D(Ab),

(A.21)

and now ask whether it is possible to ‘extend’ a functor F : Ab  Ab . More precisely, whether it is possible to construct a commutative diagram of functors Ab

[m]

F

Ab

[m]

C(Ab)

ι

K(Ab) ∃?

∃?

C(Ab )

ι

K(Ab )

D(Ab) ∃?

(A.22)

D(Ab )

We will consider this question only when F : Ab → Ab is an additive functor.

A.2.2 Extending Functors from C(Ab) to K(Ab) Consider the functor C(Ab)

ι

/o /o / K(Ab) ,

(A.23)

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which is the identity on complexes and which associates with a morphism of complexes its homotopy class. The map → MorK(Ab) (−, −) , ι−,− : MorC(Ab) (−, −) →

f  (f mod ∼) ,

(A.24)

is a surjective homomorphism of groups, so that the diagram C(Ab)

ι

K(Ab) ∃? G

G

C(Ab )

ι

(A.25)

K(Ab )

can be closed in a unique way to a commutative diagram of functors, if and only if the functor G : C(Ab)  C(Ab ) is compatible with homotopies, in other words such that if (f ∼ f  ), then (G(f ) ∼ G(f  )), in which case, we will denote by the same letter the induced functor G : K(Ab) → K(Ab ). Notice that when G : C(Ab) → C(Ab ) is induced by an additive functor F : Ab  Ab , as in (A.3), the homotopy condition (A.8) is automatically respected:   ◦ hi + hi+1 ◦ di ) = F (di−1 ) ◦ F (hi ) + F (hi+1 ) ◦ F (di ) , F (fi ) = F (di−1

and the (unique) extension (A.25) exists. We have thus all the arguments proving the following proposition. Proposition A.2.2.1 An additive functor of abelian categories F : Ab  Ab admits unique extensions F : C(Ab)  C(Ab ) and F : K(Ab)  K(Ab ) making commutative the diagram of functors Ab

[m]

F

Ab

C(Ab)

ι

K(Ab)

F [m]

C(Ab )

F ι

(A.26)

K(Ab ) .

And likewise for every homomorphism of additive functors (F → G) : Ab → Ab .

A.2.3 Extending Functors from K(Ab) to D(Ab) As the category D(Ab) is the localization of K(Ab) relative to the collection of quasi-isomorphisms, the universal property of localized categories A.1.6.2 tells us that the necessary and sufficient condition to complete commutatively a diagram K(Ab) G

K(Ab )

D(Ab) ∃? G

D(Ab )

(A.27)

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is that the functor G : K(Ab)  K(Ab ) preserves quasi-isomorphisms. When G is the extension of an additive functor F : Ab  Ab , this condition expresses the fact that F is exact (cf . Exercise A.1.4.2), a property that is too strong, and in practice without real interest. The difficulty of F not being exact is that in K(Ab) classes of quasi-isomorphic complexes generally include more than one homotopy class (A.1.6.1-(2)) and it is not clear a priori how to choose one of them in a functorial way. Yet, this can be done if Ab is an abelian category with enough injective or projective objects, provided that we restrict ourselves to the bounded subcategories D+ (Ab) or D− (Ab), respectively.

A.2.3.1

Projective and Injective Objects

– An object I ∈ Ob(Ab) is said to be injective if the functor HomAb (−, I ) : Ab → Mod(Z),

(A.28)

is exact, in other words, if every diagram

X

Y

can be closed to a commutative diagram

X

Y

⊕

I

I

The category Ab is said to have enough injective objects if for every X ∈ Ob(Ab), there exists a monomorphism X I where I is an injective object. In that case, for every bounded below complex C ∈ C≥ (Ab), there exist a complex I ∈ C≥ (Ab) with injective terms and a quasi-isomorphic monomorphism C I.

(A.29)

Such data constitute an injective resolution of C. By the universal property of injective objects, two injective resolutions of C are homotopy-equivalent. – An object P ∈ Ob(Ab) is said to be projective if the functor HomAb (P , −) : Ab → Mod(Z),

(A.30)

is exact, in other words, if every diagram

Y

X P

can be closed to a commutative diagram

Y

⊕

X P

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The category Ab is said to have enough projective objects if for every X∈Ob(Ab), there exists an epimorphism P → → X where P is a projective object. In that case, for every bounded above complex C ∈ C≤ (Ab), there exist a complex P ∈ C≤ (Ab) with projective terms and a quasi-isomorphic epimorphism P→ →C.

(A.31)

These data constitute a projective resolution of C. By the universal property of projective objects, two projective resolutions of C are homotopy-equivalent. The next proposition states that in a wide diversity of abelian categories there are indeed enough projective and injective objects.15 Proposition A.2.3.1 (16 ) Let A be a ring (resp. a graded unital algebra). The category Mod(A) of A-modules (resp. the category GM(A) of A-graded modules (5.1.1.1)) has enough projective and injective objects. The main interest of these resolutions lies in the following theorem. Theorem A.2.3.2 (17 ) Let Ab be an abelian category. 1. Denote by K+ I (Ab) the full subcategory of K(Ab) of bounded below complexes of injective objects in Ab. a. Complexes in K+ I (Ab) are isomorphic if and only if they are quasiisomorphic. b. If Ab has enough injective objects, the localization functor + Q : K+ I (Ab)  D (Ab)

(A.32)

is an equivalence of categories. 2. Denote by K− P (Ab) the full subcategory of K(Ab) of bounded above complexes of projective objects in Ab. a. Two complexes in K− P (Ab) are isomorphic if and only if they are quasiisomorphic. b. If Ab has enough projective objects, the localization functor − Q : K− P (Ab)  D (Ab)

(A.33)

is an equivalence of categories. We can now enhance proposition A.2.2.1 with a new extension.

15 For

a general overview on the problem of the existence of projective and injective objects, see Grothendieck [49], chapter I, §1.10, p. 135. 16 Godement [46] §1.3-4, pp. 4-7, and Weibel [95] §2.2-2, pp. 33-43, for nongraded rings. The case of graded unital algebras follows by the same arguments as for proposition 5.1.2.2-(2). 17 Hartshorne [51]. §I.4 Prop. 4.7, p. 46.

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Theorem A.2.3.3 Let F : Ab → Ab be a covariant additive functor of abelian categories. 1. If Ab has enough injective objects, then there exists a unique functor IR F : D+ (Ab)  D+ (Ab ) ,

(A.34)

called the right derived functor of F , such that, for all m ∈ Z, the following diagram is commutative. Ab

[m]

ι

F

F

Ab

C+ (Ab)

[m]

K+ (Ab)

D+ (Ab)

F

C+ (Ab )

ι

K+ (Ab )

IR F

D+ (Ab )

(A.35)

Moreover, if I : C I is an injective resolution of C ∈ Ob(D+ ) (A.29), we have F (I) = IR F (C) and a natural transformation of functors  : F → IR F .

(A.36)

2. If Ab has enough projective objects, then there exists a unique functor ILF : D− (Ab)  D− (Ab ) ,

(A.37)

called the left derived functor of F , such that, for all m ∈ Z, the following diagram is commutative. Ab

[m]

C− (Ab)

[m]

C− (Ab

F

F

Ab

ι

K −(Ab) F

)

ι

−

K (Ab )

D− (Ab) ILF

D− (Ab

)

(A.38)

Moreover, if P : P → → C is a projective resolution of C ∈ Ob(D− ) (A.31), we have ILF (C) = F (P) and a natural transformation of functors  : ILF → F .

(A.39)

Comments A.2.3.4 1. In the theorem, the word covariant has been underlined to emphasize that care must be taken with the variance of functors. By convention, G : Ab → Ab is contravariant if G : Abop → Ab is covariant (fn. (4 ), p. 14). At the same time, the exchange Ab ↔ Ab entails exchange of C± (Ab) ↔ C∓ (Ab) as well as that of projective ↔ injective objects. Hence, the right derived functor of a

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contravariant functor G is defined on D− (Ab) and not in D+ (Ab). We therefore have IR G : D− (Ab) → D+ (Ab) . If  : P → → C is a projective resolution in C− (Ab), then IR G(C) = G(P), and we get, by contravariance, a natural morphism G(C) → G(P) = IR G(C). We therefore have a natural transformation of functors  : G → IR G, where, again, the right derived functor appears at the right-hand side, as in the covariant case (A.36), justifying the choice of the notation IR. The same can be said mutatis mutandis for the left derived functor of a contravariant functor which is hence defined as ILG : D+ (Ab) → D− (Ab) , and for which we have a natural transformation  : ILG → G , where the left derived functor still appears at the left-hand side. 2. The different notations for the right and left derived functors are justified because they do not necessarily coincide. Even when Ab has enough injective and projective objects, in which case the two derived functors are defined on the category of bounded complexes Kb (Ab), they still can be different. Let us take a look at the case of the functor F := Z/(2) ⊗Z (−) : Mod(Z) → Mod(Z) , and for the complex Z[0] ∈ Cb (Ab). We have Z[0] ∈ KbP (Ab) and Z[0] ∼ (0 → Q → Q/Z → 0) ∈ KbI (Ab), hence ILF (Z[0]) = Z/(2)[0] = 0

whereas

IR F (Z[0]) = 0 .

Notation A.2.3.5 Derived functors take their values in categories of complexes, so their cohomology can be calculated. The traditional notations are: IR i F (−) := H i (IR F (−))

et

ILi F (−) := H −i (ILF (−)) .

Notice the change of sign on the degree i for left derived functors.

(A.40)

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Exercise A.2.3.6 Let Ab be an abelian category with enough injective (resp. projective) objects. According to Theorem A.2.3.3, if F : Ab → Ab is an additive covariant functor, we have commutative diagrams of functors (A.35) Ab

[m]

D+ (Ab)

[m]

D+ (Ab

IR F

F

Ab

Ab resp.

D− (Ab)

[m]

D− (Ab

ILF

F

Ab

)

[m]

(A.41) )

1. Show that if F left (resp. right) exact, then the morphisms − : F (−) → IR 0 F ((−)[0])



resp. − : IL0 F ((−)[0]) → F (−) ,

where  is the natural transformation (A.36) (resp. (A.39)), is an isomorphism. 2. What changes do we need to do to obtain the same conclusions for contravariant F ? ( , p. 357)

A.2.4 Acyclic Resolutions Although the injective (resp. projective) resolutions are sufficient to derive a left (resp. right) exact functor F , their use is limited by the fact that these resolutions are constructed by abstract means generally foreign to the working context. An alternative is given by F -acyclic objects. We will recall their definition only for left exact functors, as the case of right exact functors follow symmetrically. Definition A.2.4.1 Let F : Ab → Ab be a left exact functor. An object O ∈ Ob(Ab) is said to be F -acyclic if IR i F (O) = 0 ,

∀i > 0 .

An F -acyclic resolution of C ∈ C+ (Ab) is then any quasi-isomorphism C → O where O ∈ C+ (Ab) is a complex with F -acyclic terms. The usefulness of acyclic resolutions is justified by the following proposition. Proposition A.2.4.2 Let F : Ab → Ab be a left exact functor. For every complex O ∈ C+ (Ab) with F -acyclic terms, the augmentation (cf. Theorem A.2.3.3-(1)) (O) : F (O) → IR F (O) . is an isomorphism in D(Ab). In particular, if α : C → O is an F -acyclic resolution, then, in D(Ab), we have IR F (C) = F (O) .

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Proof The idea is to construct a bi-complex I ∗ C

C0

0

α

O

α0

C1 α1

C2 α2

0

O0

O1

O2

0

I00

I01

I02

0

I10

I11

I12

β

Tot⊕ I ∗

(A.42)

• the morphism O → I0∗ in an injective resolution of O; • for each i ∈ N  {0}, the i-th row Ii∗ is an injective resolution of the cokernel of ∗ → I∗ ; morphism between the previous two rows, either O → I0∗ or Ii−2 i−1 j • for each j ∈ N, the j -th column Oj → I is an injective resolution. The column ∗-filtration of the complex Tot⊕ I ∗ , is then clearly regular and the IE 2 page of the corresponding spectral spectral sequence is concentrated in q = 0, p,0 where we recover the complex O, i.e. IE 2 = Op . The spectral sequence is degenerated (dr = 0 for r ≥ 2), and the morphisms β : O → Tot⊕ I ∗ and β ◦ α : C → Tot⊕ I ∗ is a quasi-isomorphism. In particular, IR F (C) = F (Tot⊕ I ∗ ) = IR F (O) . j

Besides, the columns F (I ) have their cohomology concentrated in degree zero, j and F (Oj ) = h0 (F (I )), since Oj is F -acyclic. The arguments in the previous paragraph apply again showing that F (O) → F (Tot⊕ I ∗ ) is a quasi-isomorphism, which ends the proof of the proposition.   Example A.2.4.3 Alexander-Spanier cochains sheaves Ω i (B; k) over a mild topological space X (cf . Sect. B.1) are Γ (X; −)-acyclic for every family of supports . The same is true for every OX -module, where O := Ω 0 (X; k). Therefore, for every fiber bundle (E, X, π, F ), we can write ∗ (E; k), d) , IR Γ (X; π! Ω ∗ (E; k)) = Γ (X; π! Ω ∗ (E; k)) = (Ωcv

IR Γc (X; π! Ω ∗ (E; k)) = Γc (X; π! Ω ∗ (E; k)) = (Ωc∗ (E; k), d) , IR Γ (X; π∗ Ω ∗ (E; k)) = Γ (X; π∗ Ω ∗ (E; k)) = (Ω ∗ (E; k), d) . Mutatis mutandis for a manifold X, the sheaves of differential forms  iX and every OX -module, where OX :=  0X (cf . Theorem B.6.3.4-(1)).

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A.2.5 The Duality Functor on D(DGM(G )) IR Hom• (−, −) in D(GM(G )) G

A.2.5.1

In the previous sections we recalled the basis of derivation of functors on the homotopy category K∗ (Ab) induced by additive functors on Ab. A particular case of this is the functor of G -duality defined on Ab := GM(G ) (5.1.1)-(5.1): Homgr∗G (−, G ) : GM∗ (G )  GM∗ (G ) .

(A.43)

or, more generally, that of the bifunctor Homgr∗G (−, −) : GM∗ (G ) × GM∗ (G )  GM∗ (G ) .

(A.44)

which was the topic of Sect. 5.3.2. There, we showed that since the category GM(G ) has enough injective objects (Proposition 5.1.2.2), this bifunctor has a right derived functor IR Hom•G (−, −) : D(GM(G )) × D+ (GM(G ))  D(GM(G )) , (A.45) given by the formula IR Hom•G (C, C  ) := Hom•G (C, I) ,

(A.46)

where  I : C  I is any injective resolution. However, since the category GM(G ) also has enough projective objects, we can also fix a projective resolution P : P → → C and then consider the induced morphisms by  I et  P : •

Hom• ( P ,I)

HomG (C, I)

/ Hom• (P, I) o G

Hom• (P, I ) .

Hom•G (P, C)

These are also quasi-isomorphisms (see Proposition 5.4.3.1), giving us an alternative method for calculating derived functors. As a particular case of (A.45), the right derived functor of Hom•G (−, G ) : GM(G )  GM(G ) is given by the same formula (A.46), hence D

G ))

C

IR Hom• (− G

G)

D

G )) •

Hom

G

(C, I) ,

where G I is an injective resolution of G in C≥0 (GM(G )).

(A.47)

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A.2.5.2

The Categories K(DGM(G )) and D(DGM(G ))

We have so far discussed the derivation of a functor between categories of complexes, F : C(Ab)  C(Ab ), which is induced by an additive functor defined at the level of the abelian categories themselves, i.e. F : Ab → Ab . Now we turn to the case of the category DGM(G ) which is particular because, while it contains the category C(GM(G )) (via the Tot functors in Sect. A.2.5.3), the categories are generally not equivalent. In simple terms, there are G -dgm’s which are not complexes of G -gm’s. In that case, and since we will see that the derived category D(DGM(G )) does exist (Proposition A.2.5.2), the question arises of whether it is possible to extend the duality functor from D(GM(G )) to D(DGM(G )). The aim of this section is to explain that the answer is yes. Although different, the categories C(GM(G )) and DGM(G ) are very close to each other. They are both abelian with shift functors whose objects are complexes of vector spaces endowed with a structure of G -graded module. – In both we have the mapping cone construction. In C(GM(G )) it was defined in Sect. A.1.3. In DGM(G ), for a morphism α : X → X the definition is the same c(α) ˆ := (X ⊕ X[1], $)

with $(x  , x) := (d  (x  ) + α(x), −d(x)) . (A.48)

– In both we have homotopies. In C(GM(G )) it was defined in Sect. A.1.4. In DGM(G ), a morphism f : X → X is said to be homotopic to zero if ∃h ∈ MorGM(G ) (X, X [−1])

such that

f = h ◦ d + d ◦ h .

The corresponding relations of homotopic morphisms and homotopy-equivalent G -dgm’s, both noted ‘∼’, are the same as in C(GM(G )). The relation ‘∼’, is an equivalence relation compatible with the additive structure of DGM(G ) and the set of equivalence classes: HotDGM(G ) (X, X ) := MorDGM(G ) (X, X )

-

∼

.

(A.49)

is an abelian group. Definition A.2.5.1 The category K(DGM(G )) is the category whose objects are G -dgm’s, and whose morphisms are the homotopy classes of morphisms of G dgm’s (A.49). We denote by ι : DGM(G )  K(DGM(G )) the functor which is the identity on G -dgm’s and which associates with a morphism its homotopy class. The following proposition summarizes basic properties of these concepts.

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Proposition A.2.5.2 (And Definition) 1. The functors Tot∗ : C(GM(G ))  DGM(G ) introduced in Sect. 5.3.1 preserve cones, homotopies and quasi-isomorphisms. 2. The collection S of quasi-isomorphisms of K(DGM(G )) constitute a multiplicative collection of morphisms (A.1.6.1). The derived category of G differential graded modules is, by definition: D(DGM(G )) := K(DGM(G ))[S−1 ] . 3. The Tot∗ functors are well-defined in the homotopy and derived categories and the following diagram of functors is commutative. D

G ))

IR Hom• (− G

G)

D

Tot∗

D(DGM

G )) Tot∗

∃?

G ))

DDGM

G)

Sketch of Proof Statements (1,3) follow by straightforward verifications. For (2), the proof of the same result for K(Ab) (proof of Proposition A.1.6.3, page 354) which is based on the mapping cone construction, can be easily adapted to K(DGM(G )). Details are left to the reader. 

A.2.5.3

Extending Duality from D(GM(G )) to D(DGM(G ))

Now that we have the Tot∗ functors Tot∗ : D(GM(G ))  D(DGM(G )) , we can ask if the duality functor IR Hom•G (−, G ) can be extended to the derived category D(DGM(G )) to complete a commutative diagram D

G ))

IR Hom• (− G

Tot∗

D(DGM

G)

D

G ))

(A.50)

Tot∗ G ))

∃?

DDGM

G)

It turns out we can, as we see in the following proposition.

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Theorem A.2.5.3 For every finite18 injective resolution  : G I in the category C(GM(G )), the functor Hom•G (−, Tot I) : K(DGM(G ))  K(DGM(G )) ,

(A.51)

preserves quasi-isomorphisms and is independent of the resolution , inducing a canonical functor IR Hom•G (−, G ) : D(DGM(G ))  D(DGM(G )) ,

(A.52)

which makes commutative the following diagram D

G ))

G)

IR Hom• (− G

G)

G

D

G ))

(A.53)

Tot

Tot⊕

D(DGM

IR Hom• (−

G ))

DDGM

G) .

In addition, if we restrict the first term to Db (GM(G )), then the Tot functor on the second column can be replaced by Tot⊕ . Proof Since the functor Hom•G (−, Tot I) is well-defined in the category DGM(G ) and preserves homotopies, the functor (A.51) is also well-defined. The uniqueness relative to the choice of the resolution  is due the fact that two such resolutions are homotopic. To finish, the equality Tot IR Hom•G (−, G )  IR Hom•G (Tot⊕ (−), G ) is a consequence of Proposition 5.3.2.1, which gives the identification Tot Hom•G (−, I)  Hom•G (Tot⊕ (−), Tot I) . We still have to show that the functor IR Hom•G (−, G ) is well-defined on the entire category D(DGM(G )). In other words, that (A.51) preserves quasiisomorphisms or, equivalently, preserves acyclicity. As usual, as G is of finite homological dimension, we can choose an injective resolution of G in GM(G ) of finite length: G I := (0 → I0 → I1 → · · · → Ir → 0) .

18 The

theorem is true without this restriction. Cf. Hartshorne [51] §I.6. Lemma 6.2, p. 64.

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We have the exact sequence of complexes in C(G ) I>0 := (0

0

I1

:= (0

I0

I1

I0 [0] := (0

I0

0

I

···

Ir−1

Ir

0)

···

Ir−1

Ir

0)

···

0

0

0)

which is split if we forget the (horizontal) differentials. This implies that the a priori left exact short sequence of functors 0 → Hom•G (−, Tot I>0 ) → Hom•G (−, Tot I) → Hom•G (−, I0 [0]) → 0 , is also right exact. Then, an easy induction shows that if we can assume that for every injective module J in GM(G ), the functor Hom•G (−, J [0]) preserves acyclicity, then the same will be true for Hom•G (−, Tot I). Let us show that Hom•G (−, J [0]) preserves acyclicity. Let (X, d) be an acyclic G -dgm. A p-cocycle in Hom•G (X, J [0]) is a graded morphism α : X → J [p] of G -dgm’s, i.e. we have the commutative diagram X[−1]

d[−1]

d

X

X[1]

α

J [p]

0 We can therefore factor d through

X[−1]

d[−1] h[−1]

0

X imd[−1]

as d = ι ◦ ν d X imd[−1]

X α

0

α

ι

X[1]

h

J [p]

0

where ι is injective since X is acyclic and the extension h of α  exists since J is an injective G -gm. We can then conclude that we have α = h ◦ d = (−1)p D(h) , where D is the differential in Hom•G (X, J [0]), because the differential in J [p] is zero (Sect. 5.2.2-(5.5)). The G -dgm Hom•G (X, J [0]) is therefore acyclic.

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The possibility of substitution of Tot by Tot⊕ , is an application of the last statement in Proposition 5.3.2.1, justified by the fact that one can always choose the injective resolution of G to belong to Cb (GM(G )) after 5.1.2.2-(3).   Exercises A.2.5.4 1. Rewrite the last part of the Proof of Theorem A.2.5.3 without using the fact that G is of finite homological dimension. 2. Following the procedure of Sect. A.2.5, define the left derived functor of the bifunctor (− ⊗G −)• (5.3.3-(5.23), 5.2.2.1-(5.6)), on the derived categories D− (GM(G )) and D− (DGM(G )). Generalize Proposition 5.3.3.1 in order to construct the commutative diagram

(A.54)

A.3 DG-Modules over DG-Algebras19 A.3.1 K-Injective (A, d)-Differential Graded Modules We begin recalling that if (A, d) is a differential graded algebra (dg-algebra in short), the category DGM(A, d) of (A, d)-dg-modules is such that for each M ∈ DGM(A, d) there exist a quasi-isomorphism M → I in DGM(A, d), where I ∈ DGM(A, d) is an injective A-graded module such that if N ∈ DGM(A, d) is acyclic, then HomK(DGM(A,d)) (N, I ) = 0.

(A.55)

This last property being essential to derive functors on DGM(A, d) (Sect. A.2.3). It is important to emphasize that simply being injective as A-graded module is not sufficient for I to warrant the vanishing of HomK(DGM(A,d)) (N, I ) for all acyclic (A, d)-dg-modules N . This enhanced property entails some technicalities we will omit, instead referring interested readers to the excellent account given in the Stack Project.

19 We refer to Stacks Project [87, 90] for a thorough introduction to the derived category of dg-modules over dg-algebras. In particular sections: 24.26, The derived category, and 24.30 Equivalences of derived categories.

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The (A, d)-dg-modules I which are injective as A-graded modules and satisfy the vanishing condition (A.55) are called K-injective.20 In a number of parts of this book we have made use of the following theorem. Theorem A.3.1.1 (21 ) Let (C, O) be a ringed site. Let (A, d) be a sheaf of differential graded algebras on (C, O). For every differential graded A-module M there exists a quasi-isomorphism M → I where I is an A-graded injective module and a K-injective (A, d)-differential graded module. Moreover, the construction is functorial in M ∈ DGM(A, d). Corollary A.3.1.2 Let X be a topological space and let (, d) be a sheaf of differential graded algebras over a field k (of arbitrary characteristic). Denote by DGM(, d) the category of -differential graded modules. For every M ∈ DGM(, d) there exists a quasi-isomorphism M → I where • I is an Ω -graded injective module; • I is a complex of injective sheaves of k-vector spaces; • I is an K-injective -differential graded module. Moreover, the construction is functorial in M ∈ DGM(, d). Proof After Theorem A.3.1.1, we need now only justify that I is a complex of injective sheaves, which follows easily from the usual identity  Hom• (−, I) = Hom•  ⊗k X (−), I and the fact that k X →  is a flat extension of sheaves of rings, which is clear at the stalks level, since k is a field.  

A.3.2 Formality of DGA’s Following DGMS,22 we recall the concept of formality for dga’s. Definition A.3.2.1 A dg-algebra (A, d) is said to be formal if there exists a zig-zag diagram (A, d)

(A2 , d)

(A2 , d) (A1 , d)

(A3 , d )

···

(An , d) H (A )

of quasi-isomorphic homomorphisms of dg-algebras joining (A, d) to its cohomology H (A) := h(A, d), considered as dg-algebra with zero differential. 20 For

the existence of K-injectives, see Stacks Project [87], ch. 22 Differential Graded Algebra. §21 I -resolutions, Lemma 21.4, p. 27. 21 See Stacks Project [90] Theorem 24.25.13 22 Deligne-Griffiths-Morgan-Sullivan [36] §4. Formality of Differential Algebras, p. 260.

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The interest of formal dga’s lies in the following equivalence of categories. Proposition A.3.2.2 Let ρ : (A, d) → (B, d) be a homomorphism of dga’s. Consider the following functors: • resρ (−) : DGM(B, d)  DGM(A, d) the restriction functor which endows a (B, d)-dgm with the structure of (A, d)-dgm through the homomorphism ρ. • (B, d) ⊗(A,d) (−) : DGM(A, d)  DGM(B, d) the base change functor. 1. The functors (B, d) ⊗(A,d) (−) and resρ (−) are adjoints of each other, i.e. the following natural map is bijective:   Hom•(A,d) (B, d) ⊗(A,d) (−), − → Hom•(A,d) −, resρ (−) . 2. If ρ is a quasi-isomorphism, then the derived functors  resρ (−) : DDGM(B, d)  DDGM(A, d) L (−) : DDGM(A, d)  DDGM(B, d) (B, d) ⊗I(A,d)

are equivalences of categories inverse of each other. In particular, the adjunction map  L (− ) (−) → resρ (B, d) ⊗I(A,d)

(A.56)

is a quasi-isomorphism. Sketch of Proof (23 ) (1) is the well-known adjunction for graded modules, it then suffices to check compatibility of differentials which is straightforward. (2) Since resρ (−) does not modify the dgm’s, it trivially preserves quasi-isomorphisms and acyclicity, hence tautologically defining a functor in derived categories, in which L case, the derived functor (B, d) ⊗I(A,d) (−) is its left adjoint. The conclusion then follows by proving that the adjunction morphism (A.56) is a quasi-isomorphism. But this is clear since ρ : (A, d) → (B, d) is an isomorphism in D(DGM(A, d)), so that, by definition of derived functors, the morphism L L (−) → (B, d) ⊗I(A,d) (−) ρ ⊗ id : (A, d) ⊗I(A,d)

is also a quasi-isomorphism, and we can conclude since the morphism (−) → (A, d) ⊗(A,d) (−) ,

x → 1 ⊗ x ,

is an (obvious) isomorphism.

23 Stacks

Project [90], §24.30 Equivalences of derived categories.



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A.3.3 Formality of DGM’s We now consider a dga A := (A, d) such that d = 0, in which case, the category DGM(A) has the particularity of containing both, an A-dg-module (M, d) and its cohomology H (M) := h(M, d), considered as a dg-module with zero differential. We can therefore consider the property of formality within the category DGM(A) and say that (M, d) is formal in DGM(A) if there exists a zig-zag of quasiisomorphic morphisms in DGM(A) joining (M, d) to H (M) (M, d)

LLL &

xqqq

(M2 , d)

(M1 , d)

MMM &

xqqq

(M2 , d)

···

(Mn , d)

(M3 , d)

LLL % H (M)

The following proposition gives a sufficient condition for formality of dgm’s which is relevant in equivariant cohomology. Proposition A.3.3.1 Let A be an anticommutative (graded commutative) graded algebra. 1. If (M, d) ∈ DGM(A) is such that dimproj (H (M)) ≤ 1 as A-gm, then (M, d) is formal in DGM(A). Moreover, there is a canonical bijection   IsoDDGM(A) (M, d), H (M)  Ext1A H (M), H (M) , where Iso denotes the set of quasi-isomorphisms ϕ : (M, d) → H (M) inducing the identity in cohomology. In particular, if H (M) is projective, then   Iso

DDGM(A)

  (M, d), H (M)  = 1 .

2. If A is hereditary, i.e. if dimcoh (A) ≤ 1, then every (M, d) ∈ DGM(A) is formal. Proof (1) Since K-projective resolutions exist in DGM(A),24 we can set (M, d) → (M1 , d) to be such a resolution and assume in the sequel M to be a projective Agraded module. Lemma 1. If dimproj (H (M)) ≤ 1, then B := dM and Z := ker(d) are projective. Proof of Lemma 1. The long exact sequence of ExtiA functors associated with the short exact sequence of A-gm’s 0 → B → Z → H (M) → 0 ,

24 This

(A.57)

is the dual notion of the K-injective resolutions of Sect. A.3.1. For the definition and existence of K-projective resolutions, see Stacks Project [87], ch. 22 Differential Graded Algebra. §20 P -resolutions, Lemma 20.2, p. 24.

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where we denote B := dM and Z := ker(d), gives rise to canonical maps 

(i) Ext1A (Z, −) → → Ext1A (B, −) ,

surjective,

(ii) ExtiA (Z, −)  ExtiA (B, −) ,

isomorphic ∀i > 1 .

(A.58)

since Exti (H (M), −) = 0 for all i > 1. In the same vein, the long exact sequence of ExtiA functors associated with the short exact sequence of A-gm’s d

0 → Z −→ M −→ B[1] → 0 ,

(A.59)

gives bijections ExtiA (Z, −)  Exti+1 A (B[1], −) ,

∀i ≥ 1 ,

i+1 since ExtiA (M, −) = 0, for all i ≥ 1. But Exti+1 A (B[1], −) = ExtA (Z[1], −) by (A.58)-(ii), so that we can write

ExtiA (Z, −) = Exti+1 A (Z[1], −) ,

∀i ≥ 1 ,

whence, for fixed i > 0, we have i+n ExtiA (Z, −) = Exti+1 A (Z[1], −) = · · · = ExtA (Z[n], −),

∀n > 0 ,

and, since A is of finite homological dimension, we can conclude that ExtiA (Z, −) = 0,

∀i ≥ 1 ,

ExtiA (B, −) = 0,

∀i ≥ 1 .

and also, by (A.58), that

 We have thus proved that Z and B are projective A-graded modules. By Lemma 1, the sequence (A.59) splits in GM(A) and d : M → B[1] admits sections. Given σ : B[1] → M such that d ◦ σ = idB[1] , we consider the map:

ϕσ : (M, d) → H (M) defined by

ϕσ (x) := [x − σ (d(x))] .

Lemma 2. ϕσ is a quasi-isomorphism of A-dgm such that ϕσ (z) = [z], ∀z ∈ Z. Proof of Lemma 2. Indeed, ϕσ is a morphism of A-gm’s verifying ϕσ (z) = [z] by definition. It is also a morphism of dgm’s since we have dϕσ (x) = dx − d(σ (dx)) = 0 = [dx] = ϕσ (dx) .

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The restriction of the induced map h(ϕσ ) : h(M, d) → H (M) to Z clearly coincides with the canonical projection Z → → H (M), immediately implying that h(ϕσ ) is an isomorphism.  The first part of (1) is thus proved. For its second part, we consider the map    Ξ : σ : B[1] → M | d ◦ σ = idB[1] → IsoKDGM(A) (M, d), H (M)  σ → ϕσ := (idM − σ ◦ d)(−) (A.60) Lemma 3. The map Ξ is a surjective A-affine map. Proof of Lemma 3. For the surjectivity of Ξ , let ϕ : (M, d) → H (M) be a quasiisomorphic morphism of A-dgm’s such that ϕ(z) = [z], for all z ∈ Z. The map ϕ is a surjective morphism of A-gm’s, and, since M is projective, can be lifted to a morphism ϕ˜ : M → Z, which necessarily verifies (ϕ(z) ˜ − z) ∈ B, for all z ∈ Z. δ˜ δ

0

ϕ˜

M ϕ

⊆

Z

B

H (M)

0

Hence, we have the morphism of A-gm’s δ : Z → B, δ(z) := ϕ(z) ˜ − z , which we can extend to a morphism of A-dgm’s δ˜ : M → B, since the short exact sequence ˜ : M → Z is a (A.59) is split. As a consequence, the morphism ϕ˜  := (ϕ˜ − δ) projector of M onto Z, and ker(idM − ϕ˜  ) = Z = ker(d). Therefore, (idM − ϕ˜  )(m) = σ (dm) ,

∀m ∈ M ,

for some unique σ : B[1] → M, which is clearly a section of d : M → B[1], such  that ϕσ := idM − σ ◦ d = ϕ˜  induces ϕ : (M, d) → H (M), as expected. Applying the functor HomA (B[1], −) to the split sequence (A.59), the set of sections of d : M → B[1] is identified to the vector space HomA (B[1], Z): 0 → HomA (B[1], Z) →HomA (B[1], M)

→HomA (B[1], B[1])

→0

→idB[1]

σ

The map Ξ is therefore defined on HomA (B[1], Z) by the A-affine formula HomA (B[1], Z) # η

→

Ξ (η) :=

 idM − (σ0 + η) ◦ d (−) ,

where σ0 denotes some fixed section of d. As a consequence, we see that Ξ (η1 ) ∼ Ξ (η2 )

⇔

 (η2 − η1 ) ◦ d (−) ∼ 0 ,

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where ‘∼’ stands for ‘homotopic’. But, to ask (η2 − η1 ) ◦ d : M → H (M) to be homotopic to zero is equivalent to asking that h : M[1] → H (M) exists such that (η2 − η1 ) ◦ d = h ◦ d, in other words, such that h extends (η2 − η1 ) : B[1] → H (M) to M[1], or, equivalently, to Z[1], since Z[1] is an additive factor of M[1]. As a consequence, the image of Ξ is in bijection with the cokernel of the restriction map HomA (Z[1], H (M)) → HomA (B[1], H (M)). This cokernel appears in the long exact sequence of δ-functors {ExtiA (−, H (M))}i∈Z applied to 0 → B → Z → H (M) → 0 (A.57), where we see the terms HomA (Z[1], H (M)) → HomA (B[1], H (M)) → Ext1A (H (M)[1], H (M)) → 0 , since Z is a projective A-gm, hence proving (1). (2) By (1), since if A is hereditary, then dimproj M ≤ 1 for all M ∈ GM(A).

 

Corollary A.3.3.2 Let (A, d) be a formal dga such that the algebra H (A) := h(A, d) is of finite homological dimension. Let (M, d) be an (A, d)-dgm and denote H (M) := h(M, d). 1. There exists a convergent spectral sequence ⎧ p,q ⎨ IE 2 := hp IR Hom•H (A) (H q (M), H (A)) ⇓ ⎩ hp+q IR Hom•(A,d) ((M, d), (A, d)) 2. If dimproj (H (M)) ≤ 1, then there exists an isomorphism in DDGM(H (A)): ξ(M,A)

IR Hom•(A,d) ((M, d), (A, d)) −−−−→ IR Hom•H (A) (H (M), H (A)) 

functorial on (M, d) ∈ DGM(A, d), unique when H (M) is projective, which makes commutative the following diagram:

h0 IR Hom•(A,d) ((M, d), (A, d))

h0 (ξ(M,A)) 

/ h0 IR Hom• H (A) (H (M), H (A)) / HomH (A) (H (M), H (A))

Hot(A,d) ((M, d), (A, d)) ξ (M,A)

where ξ (M, A) is the morphism which associates the induced morphism in cohomology with a morphism of complexes (Corollary 5.4.7.2-(1)). Furthermore, the spectral sequence (IE r , dr ) in (1) degenerates for r ≥ 2.

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Proof A sequence (A, d) ↔ (A1 , d) ↔ · · · ↔ (Ar , d) of quasi-isomorphic morphisms of dga’a gives rise, after Proposition A.3.2.2, to a sequence of equivalences of categories from DDGM(A0 , d) to DDGM(Ar , d) which, given a (A, d)-dgm M, generate a sequence M ↔ M1 ↔ · · · ↔ Mr of quasi-isomorphic dgm’s. When (A, d) is formal and (Ar , d) = (H (A), 0), we therefore have IR Hom•(A,d) (M, N )  IR Hom•H (A) (Mr , Nr ) , where, in the right-hand side term, H (A) is simply a graded algebra (with no differential) whose action is compatible with the differentials in Mr and Nr . (1) Let N := (A, d). Then Nr = H (A) and we have the quasi-isomorphism IR Hom•(A,d) (M, N )  IR Hom•H (A) (Mr , H (A)) .

(A.61)

Let H (A) → I be an injective resolution in GM(H (A)), which we take of finite length since H (A) is of finite homological dimension. The H (A)dgm Tot⊕ I is then K-injective in DGM(H (A)).25 Indeed, if (N, d) is acyclic, then Hom• ((N, d), Tot⊕ I ) is acyclic too, since its cohomology is the abutment of the spectral sequence associated with the decreasing -filtration of Hom• ((N, d), Tot⊕ I ) (regular since finite), whose first page is p,q

IE 1

= Hom•H (A) (hq (N, d), Ip ) = 0 .

We can therefore write IR Hom•(A,d) (M, (A, d)) = Hom•H (A) (Mr , Tot⊕ I ) , and the same -filtration gives a convergent spectral sequence with second page p,q

IE 2

= hp Hom•H (A) (H q (Mr ), I ) = hp IR Hom•H (A) (H q (Mr ), H (A))

converging to hp+q Hom•H (A) (Mr , Tot⊕ I ) = hp+q IR Hom•(A,d) (M, (A, d)) as expected. (2) Since H (Mr ) = H (M), the H (A)-dgm Mr is formal by Proposition A.3.3.1(1), and we can exchange M ↔ H (M) in A.61. The canonicity of the isomorphism when H (M) is projective is due to Proposition A.3.3.1-(2).  

25 Stacks

Project [90] 24.25.

Appendix B

Sheaves of Differential Graded Algebras

The purpose of this appendix is to justify several auxiliary results used in various places in the book and which concern derived functors in categories of differential graded modules over sheaves of differential graded algebras, in particular for the sheaves of algebras of de Rham differential forms  B , and that of AlexanderSpanier cochains Ω B;k . We prove the following statement which plays an central role in Relative Poincaré Duality (cf . Sect. 3.3.1, especially Proposition 3.3.2.1). Theorem (B.9.1.3) Let (E, B, π, F ) be a fiber bundle of manifolds. There exist canonical isomorphisms in the derived category DVec(k): IR Hom•R B (π!  E ,  B ) •

↑

IR HomD( B ,d) (π!  E ,  B ) ↓ IR Hom•D((B),d) (cv (E), (B)) .

B.1 Mild Topological Spaces Although the book concerns mainly manifolds or inductive limits of such, as are the classifying spaces, the results remain valid for more general topological spaces. Among the topological properties needed, the most crucial are the separateness, the local contractibility, and the paracompactness and perfect normality, all well-

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Arabia, Equivariant Poincaré Duality on G-Manifolds, Lecture Notes in Mathematics 2288, https://doi.org/10.1007/978-3-030-70440-7

301

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known properties of open subspaces of CW-complexes.1 ,2 The topological spaces in this appendix will therefore be locally contractible Hausdorff topological spaces such that all of its locally closed subspaces are paracompact. We will call such spaces mild spaces. Manifolds and, much more generally, open subspaces of CWcomplexes are mild spaces.3

B.2 The Sheaf of Functions OX We will consider the following sheaves of rings of functions on a mild space X: • The sheaf OX of continuous real valued functions; • The sheaf OX of differentiable real valued functions, when X is a manifold. This is the sheaf  0X of real de Rham differential forms of degree 0. • The sheaf OX the sheaf all functions (continuous or not) with values in a field k. This is the sheaf Ω 0X,k of Alexander-Spanier cochains of degree 0, with coefficients in k. A statement concerning OX -(graded)-modules without any other precision, will be simultaneously valid in all these cases. Conventions • By ‘k ’ we denote a field. Its meaning varies according to the context. In statements concerning de Rham complexes, it will be k := R, otherwise it is an arbitrary field. In this sense, the sheaf OX is a sheaf of k-vector spaces. The word ‘sheaf ’ will implicitly mean ‘sheaf of k-vector spaces ’. • Rings and algebras have a multiplicative identity. • Graded algebras are graded by the set of integers Z and are tacitly assumed anticommutative (graded commutative). • In differential graded objects, differentials are of degree +1. • Modules over rings and algebras are tacitly left modules. • Acronyms for ‘graded algebra’, ‘graded module’, ‘differential graded algebra ’, ‘differential graded module ’ are respectively ‘ga’, ‘gm ’, ‘dga’, ‘dgm ’.

1 These

properties are thoroughly studied in Lundell-Weingram book [74]. Local contractibility in Theorem 6.6 (p. 67). Paracompactness and perfect normality in Theorems 4.2 (p. 54) and 4.3 (p. 55), and the extension of these to all subspaces, in App. I, Theorem 10, p. 205. 2 A topological space X is ‘normal’ if for every pair of disjoint closed subspaces F , F , there exist 1 2 disjoint neighborhoods Vi ⊇ Fi , and it is ‘perfectly normal’ if, in addition, there exist continuous functions f : B → R≥0 verifying f V1 = 1 and f V2 = 0. 3 Notice that we excluded the properties of being locally compact and of being countable at infinity. The first, since we need our conclusions to be applied to classification spaces which are generally not locally compact. The second, simply because we do not need it. Recall however that a CWcomplex is countable at infinity if and only if it is locally finite, which entails that it is metrizable, after Lundell-Weingram [74] Proposition 3.8, p. 52.

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303

• The notations F(U ) or 0(U ; F) equivalently denote the sets of sections of a sheaf of k-vector spaces F ∈ Sh(X; k), over subspace U ⊆ X. • A graded sheaf M is a family of sheaves M := {Mi }i∈Z . The sheaf Mi is called the i’th homogeneous component of M. It is usual to denote for a graded sheaf M and an open subspace U ⊆ X: % Mi (U ) , M(U ) = Γ (U ; M) := i∈Z

viewed as graded vector space.4 • As graded sheaves, the sheaves OX are homogeneous of degree 0. • An OX -graded algebra is a family A := {Ai }i∈Z of OX -modules, together with a family of OX -bilinear morphisms {· : Ai ⊗ Aj → Ai+j }i,k∈Z , such that the triple (A, 0, +, · ) verifies the axioms of a algebras. • If A is an OX -ga, then GM(A) denotes the category of A-gm’s. If (A, d) is an OX -dga, then DGM(A, d) denotes the category of (A, d)-dgm’s.

B.3 Global Lifting of Germs on OX -modules The following proposition establishes that in a mild space X, the germs of sections of an OX -module are germs of global sections. We will see that despite the apparent shallowness of this property, its consequences are far-reaching. Proposition B.3.1 Let X be a mild space. 1. For every x ∈ X and every open neighborhood U # x, there exists f ∈ OX (X), such that the sets {f = 1} ⊆ {f = 0} =: |f | are neighborhoods of x which are closed in X and contained in U . 2. For every OX -module M and every x ∈ X, the map M(X) → Mx which associates with s ∈ M(X) its germ sx ∈ Mx at x, is a surjective map. In particular, for every t ∈ Mx , there exists a morphism of OX -modules ξt : OX → M verifying (ξt (1))x = t. 4 Notice that it would be a mistake to denote M as the sheaf direct-sum-of-sheaves



i i∈Z M , since, while the natural map i∈Z Γ (U ; M ) → Γ (U ; i∈Z M ) is always injective, it is generally not surjective (cf . Godement [46] §2.7, p. 136, or Bredon [20] §I.5, p. 19.)



i



i

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B Sheaves of Differential Graded Algebras

Proof (1) The sets {x} and X  U are disjoint closed subspaces of X. We then distinguish three cases: a. X is a mild space and OX is the sheaf of continuous real valued functions. Then, X being perfectly normal, there exists a continuous positive function f0 : X → R verifying f0 (x) = 1 and f0 XU = 0, and we define f := sup{1, 2 f0 }, so that the sets {f = 1} and |f | = |f0 | are closed neighborhoods of x in X, contained in U . b. X is a manifold and OX is the sheaf of differentiable real valued functions. We can assume U relatively compact and contained in the domain of some chart of X, which leads us to prove (1) for x ∈ U ⊆ Rn . Then, proceeding as in Godement [46] (p. 158), we ‘regularize’ the continuous function f defined in case (a), by setting f˜(y) := Rn f (y − z) λ(z) dz, where λ is a positive differential function with compact support and integral equal to 1. Choosing λ with sufficiently small support, one then easily checks that the sets {f˜ = 1} and |f˜| are again closed neighborhoods of x in X, contained in U . c. X is a mild space and OX is the sheaf of all functions with values in k. For x ∈ X, we choose a continuous real valued function g : X → R, as in case (a), so that Z := {g = 1} is a neighborhood of x which is closed in B and is contained in U . Then, the indicator function f := 1Z : X → k, defined by 1Z (y) = 1 if y ∈ Z, and 1Z (y) = 0 otherwise, answers the question. (2) Let x ∈ X. Given sx ∈ Mx , let U ⊆ X be an open neighborhood of x such that sx is the germ at x of a section s ∈ M(U ). Then, if f is a function satisfying the conditions in (1) for the pair (x, U ), the section f s ∈ M(U ) verifies (f s)x = ((f s) {f =1} )x = sx , and, furthermore, since f s is zero in X  |f |, it extends by zero in a unique global section s˜ ∈ Γ (X; M) still verifying the equality s˜x = sx , as announced.   Exercise B.3.2 Recall that a sheaf F on a topological space X is said to be soft if for every closed subspace S ⊆ X the restriction map F(X) → F(S) is surjective. Prove that OX -modules are soft on a mild space X. ( , p. 357) Comment B.3.3 Softness, -softness, c-softness, flabbiness, fineness, etc., are properties related to different ways of lifting local sections to the whole space. A sheaf with one of these properties is Γ (X; −)-acyclic (Sect. A.2.4), something which is established following a standard procedure.5 In the sequel, we will prove the same acyclicity results for OX -modules on mild spaces following a somewhat different approach.

5 Look

at ‘soft sheaves’ in Bredon [20], Godement [46], Iversen [58], Kashiwara-Schapira [61], Weibel [95].

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305

B.4 OX -Graded Algebras Let A be an OX -ga and consider an A-gm M endowed with the induced structure of OX -gm through the action of A. One of the most relevant consequences of the existence of global lifts of the germs of M, is that it can be realized as a quotient of a free A-gm. More precisely, the morphism of sheaves %  ξs : A[s] → → M, (B.1) ξ := x∈X; s∈Mx

x∈X; s∈Mx

where s is homogeneous of degree [s], and ξs : A[s] → M is the morphism defined in Proposition B.3.1-(2), is a surjection of A-gm’s, since it is so, by construction, at the stalks level at each homogeneous component. The procedure can obviously be iterated on ker(ξ ) since it is also an A-gm. We get in this way a free presentation of M as A-gm, i.e. a right exact sequence in the category GM(A): A" → A" → M → 0 ,

(B.2)

where ‘A" ’ indicates a direct sum of A[−]’s, which it is not necessary to specify. Terminology The existence of a free presentation (B.2) is referred to as the quasicoherence (or A-quasi-coherence) of the A-gm M. Proposition B.4.1 Let A be an OX -graded algebra on a mild space X. 1. An A-gm is quasi-coherent. 2. If ν : A" → → M is a surjective morphism of OX -gm’s, then the following morphism of global section is surjective too:6 ν(X) : Γ (X; A" ) → → Γ (X; M) . 3. The functor Γ (X; −) = GM(A) → GM(Γ (X; A))

(B.3)

is exact. In particular, if A" → A" → M → 0 is a presentation of an A-gm M, then the following induced sequence of Γ (X; A)-gm’s is exact: Γ (X; A" ) → Γ (X; A" ) → Γ (X; M) → 0 .

(B.4)

recalled in the fn. (4 ), p. 303, the natural map Γ (X; A)" → Γ (X; A" ) is injective but " " generally not surjective for infinite sums, contrary to the map between stalks (A  x ) → (A )x which is always bijective. In particular, the natural morphism A⊗( i∈I Mi ) → i∈I (A⊗Mi ) is an isomorphism (cf . Godement [46] §2.7, p. 136, or Bredon [20] §I.5, p. 19.) 6 As

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Proof (1) Already justified in the preliminary remarks. (2) It suffices to restrict to homogeneous components. A section s ∈ M[s] (X) determines a germ sx ∈ M[s] x for every x ∈ X. Let tx ∈ (A[s]" )x be such that νx (tx ) = sx . Applying Proposition B.3.1-(2), we can choose, for each x ∈ X, a section t˜x ∈ Γ (X; A[s]" ) and an open neighborhood Vx # x such that ν(Vx )(t˜x ) = s Vx . Since X is paracompact, there exists a partition

of unity {φx }x∈X ⊆ OX (X) subordinate to the cover {Vx }x∈X of X. The sum t := x φx t˜x is then well-defined in Γ (X; A[s]" ), and verifies, by construction, ν(X)(t) =

 x

φx ν(X)(t˜x ) =

 x

φx s = s .

(3) Since Γ (X; −) is left exact, we need only to prove it respects surjectivity. Given a surjective morphism of A-gm’s ν : M → → N, we compose it with a surjection ξ : A" → → M (viz. (B.1)), and conclude, using (2), that Γ (X; ν ◦ ξ ), and hence Γ (X; ν), are surjective. The rest of (3) follows straightforwardly.  

B.5 Localization Functor for OX -GA’s B.5.1 The Isomorphism Hom•A (A, −)  Γ (Y ; −) Let A be a sheaf of graded algebras on a topological space Y . Given F ∈ GM(A), a homogeneous global section s ∈ F[s] (Y ) uniquely determines a morphism of Agm’s of degree [s], from A to F, by setting, for every open subspace U ⊆ Y , A(U ) # t → t s

U

∈ F(U ) .

In this way we get a canonical map Γ (Y ; F) → Hom[s] A (A; F) which is functorial in F, and gives rise to a homomorphism of functors  Γ (Y ; −) → Hom•A (A; −) : GM(A)  GM(A(Y )) ,

(B.5)

which is easily seen to be an isomorphism with inverse the map which associates with a morphism ν : A → F of degree d, the section ν(Y )(1) ∈ Fd (Y ). Comment B.5.1.1 Beware that although Γ (Y ; −) and Hom•A (A, −) coincide in the category GM(A), they are not equal functors since they first is defined in the whole category GM(k X ). In particular, one must be careful when comparing the derived functors IR Γ (Y ; −) and IR Hom•A (A; −) (Theorem B.6.1.1).

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B.5.2 Right Adjoint to Hom•A (A, −) B.5.2.1

Localization Presheaf Functor

Let A be a sheaf of graded algebras on a topological space Y . Given M ∈ GM(A(Y )), the correspondence which assigns to an inclusion of open subspaces V ⊆ U ⊆ Y the morphism ρVU ⊗idM

A(U ) ⊗A(Y ) M −−−−−−→ A(V ) ⊗A(Y ) M

(B.6)

where ρVU : A(U ) → A(V ) is the restriction map, clearly defines a presheaf of A-gm’s on Y which we denote by A ⊗ M. The construction is functorial on M. The resulting functor A ⊗ (−) = GM(A(Y ))  Modpresh. (A)

(B.7)

is the A-localization presheaf functor. It verifies, by construction, the adjunction equality: Hom•A (A ⊗ M, N) = Hom•A(Y ) (M, N(Y )) ,

(B.8)

since a morphism of presheaves ν : A⊗M → N is, by (B.6), a family of morphisms {ν(U ) : A(U ) ⊗ M → N(U )} making commutative the diagrams M

ν(Y )

⊕ A(U ) ⊗ M

ν(U )

N(Y ) ρUY

N(U )

This means that one has the equalities ν(U )(a ⊗ m) = a ν(U )(1 ⊗ m) = a ρUY (ν(Y )(m)) , showing that ν is completely determined by ν(Y ). Definition B.5.2.1 Let A be a sheaf of graded algebras on a topological space Y . The A-localization functor, A ⊗ (−) = GM(A(Y ))  GM(A) .

(B.9)

is the composition of the localization presheaf functor A ⊗ (−) (B.7), followed by the exact functor ‘associated sheaf to a presheaf ’.

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Proposition B.5.2.2 (Localization Functor) The natural map Hom•A (A ⊗ M, N) → ν →

Hom•A(Y ) (M, N(Y ))

(B.10)

ν(Y )

is an isomorphism, functorial in M ∈ GM(A(Y )) and N ∈ GM(A). The functors 

Γ (Y ; −) = Hom•A (A, −) : GM(A)  GM(A(Y )) A ⊗ (−) : GM(A(Y ))  GM(A)

constitute a pair of adjoint functors. Proof Everything follows from the adjunction equality (B.8) and the universal property of the functor ‘associated sheaf to a presheaf ’, by which the natural map Hom• (A ⊗ (−), −) → Hom• (A ⊗ (−), −) is an isomorphism. The map (B.10) is then the composition of equalities Hom•A (A ⊗ M, N) −→ Hom•A (A ⊗ M, N) = Hom•A(Y ) (M, N(Y )) . (=)

 

B.5.3 The Localization Functor for OX -GA’s We now come back to the case of a mild space X and a sheaf A of OX -ga’s. Under these assumptions, we show that for every A-gm M, the stalks Mx for x ∈ X, are determined in terms of the A(X)-gm M(X) alone. Proposition B.5.3.1 Let A be an OX -ga on a mild space X. For x ∈ X, denote by Sx the multiplicative system of functions in OX (X) equal to 1 in a neighborhood of x. 1. Let M ∈ GM(A). For every x ∈ X, the germ map M(X) → Mx factors through the localization map M(X) → Sx−1 M(X) in an isomorphism of Sx−1 A(X)-gm’s ξx (M) : Sx−1 M(X) → Mx , which is functorial in M ∈ GM(A). 2. The localization functor A ⊗ (−) : GM(A(X))  GM(A) is an exact functor.

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309

3. Let M ∈ GM(A). The natural morphism of sheaves of A-gm’s A ⊗ M(X) → M is an isomorphism. More precisely, the localization functor A ⊗ (−) is a left inverse of the global section functor Γ (X; −). Proof (1) The A(X)-gm Mx is an Sx−1 A(B)-gm so that we have a canonical factorization M(B)

Mx ξx (M)

Sx−1 M(B) where ξx (M) : Sx−1 M(B) → Mx is a morphism of Sx−1 A(B)-gm’s. The kernel of the canonical map M(B) → Sx−1 M(B) is the submodule of m ∈ M(B) annihilated by some f ∈ Sx , hence of m’s which vanish on a neighborhood of x. But this is exactly the kernel of the map M(B) → Mx , so that ξx (M) is injective. Up to this point we have not yet used the fact that A is an OX -module on a mild space. Indeed, these assumptions are needed only to establish the surjectivity of ξx (M), which is then an obvious consequence of the surjectivity of the germ morphism M(B) → Mx stated in Proposition B.3.1-(2). (2) Given an exact sequence of A(X)-gm’s 0 → L → M → N → 0, the sequence of localized A-gm’s 0→A⊗L→A⊗M →A⊗N →0

(B.11)

is exact, if it is so at each of its stalks. But, we have seen in (1) that we have the identifications (A ⊗ K)x = Ax ⊗ K = Sx−1 K ,

(B.12)

which are functorial in K ∈ GM(A(X)). Therefore, (B.11) is exact if and only if the sequence 0 → Sx−1 L → Sx−1 M → Sx−1 N → 0 is exact. And this is indeed the case since localization by Sx−1 is an exact functor. (3) The morphism is induced by the morphism of presheaves A ⊗ M(X) → M, which associates with a ⊗ m ∈ A(U ) ⊗ M(X) the section a ρUX (m) ∈ M(U ). As usual, to see that it is an isomorphism, it suffices to check it at stalks level, and thus show that, for all x ∈ X, the map Ax ⊗ M(X) → Mx is an isomorphism. But then, equality (B.12) leads us again to the natural map Sx−1 M(X) → Mx , which we already showed is an isomorphism in (1).  

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B.6 Equivalences of Some Derived Functors in D GM(A) B.6.1 An Equivalence of Categories Proposition B.5.3.1-(3), stating that the functor Γ (X; −) : GM(A)  GM(A(X)) admits a left inverse, has as first consequence that the image by Γ (X; −) of objects and morphisms constitutes a subcategory of GM(A(X)). The following theorem states important consequences of this fact. Theorem B.6.1.1 Let A be an OX -algebra on a mild space X. 1. The functor Γ (X; −) = Hom•A (A, −) : GM(A)  GM(A(X)) is exact and fully faithful. It is an equivalence of categories between GM(A) and the full essential image of Γ (X; −), denoted by Γ (X; GM(A)).7 2. An injective A-gm I is an injective sheaf of k-vector spaces. 3. There is a canonical isomorphism of functors  IR Γ (X; −) → IR Hom•A (A; −) : D GM(A)  D GM(A(X)) . 4. If I is an injective A-gm, then I(X) is an injective A(X)-gm. 5. For all M, N ∈ GM(A), there is a canonical isomorphism in D GM(A(X)) IR Hom•A (M, N) → IR Hom•A(X) (M(X), N(X)) functorial in M and N. Proof (1) Exactness is stated in Proposition B.4.1-(3). The adjunction B.5.2.2B.10 and the isomorphism A ⊗ M(X) → M in Proposition B.5.3.1-(3), give the isomorphisms: Hom•A (M, N)  Hom•A (A ⊗ M(X), N)  Hom•A(X) (M(X), N(X)) ,

(B.13)

proving fully faithfulness. The fact that Γ (X; −) : GM(A)  Γ (X; GM(A)) is an equivalence of categories is then straightforward. (2) The sheaf A considered with its natural structure of sheaf of k-vector spaces is a flat k X -gm. Indeed, flatness is a property of stalks, and (k X )x , being the field k, the stalk Ax is trivially (k X )x -flat. In particular, the base change functor A ⊗k (−) : GM(k X )  GM(A) , a functor between categories F : C  C , the full essential image of F is the smallest full subcategory of C containing F (Ob(C)) and stable under isomorphisms in C .

7 Given

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311

is exact. As a consequence, the functor Hom•k X (−, I) : GM(k X )  Vec(k) , is the composition of the two exact functors A ⊗k (−) and Hom•A (−, I), since Hom•k X (−, I) = Hom•A (A ⊗k (−), I) , and is therefore exact. Hence, the sheaf of vector spaces I is injective. (3) let M ∈ GM(A). An injective resolution M → I in GM(A) is automatically an injective resolution in GM(k X ) after (2), hence IR Γ (X; M) = Γ (X; I ) =1 Hom•A (A, I ) = IR Hom•A (A; M) , where (=1 ) was justified in Sect. B.5.1. (4) In the same way as in (2), the adjunction equality B.5.2.2-(B.10) Hom•A(X) (−, I(X)) = Hom•A (A ⊗ (−), I) , expresses the fact that the left-hand functor is the composition of the localization functor A ⊗ (−) which is exact (Proposition B.5.3.1-(1)), and the functor Hom•A (−, I) which is also exact. Hence, the A(X)-gm I(X) is injective. (5) Let N → I be an injective resolution in GM(A), then IR Hom•A (M, M) =1 Hom•A (M, I ) =2 Hom•A(X) (M(X), I (X)) =3 IR Hom•A(X) (M(X), N(X)) where (=1 ) is by definition of derived functor, (=2 ) is (B.13) and (=3 ) again by definition of derived functor, since N(X) → I (X) is an injective resolution after (1) and (2).  

B.6.2 Family of Supports8 This is the name given to a collection  of subspaces on a topological space Y , verifying the following conditions: -1. every S ∈  is a closed subset of Y ; -2. if F is closed in Y and F ⊆ S ∈ , then F ∈ ; -3. if S1 , S2 ∈ , then S1 ∪ S2 ∈ . 8 Bredon [20] §I.6 Supports (p. 21), or Godement [46] §II.2.5 (p. 133) and §II.3.2 Espaces paracompacts (p. 150).

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The family is called paracompactifying if, in addition, -4. every S ∈  is a paracompact space; -5. every S ∈  has a neighborhood in Y belonging to . If F is a sheaf of vector spaces on Y , the set   Γ (Y ; F) := σ ∈ Γ (Y ; F) | |σ | ∈  .

(B.14)

is called the set of global sections with support in . Familiar equivalent notations for Γ (Y ; −), for particular families , are • Γ (Y, −), for the paracompactifying family of closed subspaces of Y , • Γc (Y, −), for the paracompactifying family of compact subspaces of Y , • ΓZ (Y, −), for the family of closed subspaces of a closed subspace Z ⊆ Y . The set Γ (Y ; F) is vector subspace of Γ (Y ; F), and if ν : F → G is a morphism of sheaves of vector spaces, then ν(Y )(Γ (Y ; F)) ⊆ Γ (Y ; G), and we have an induced homomorphism Γ (Y ; ν) : Γ (Y ; F) → Γ (Y ; G). In this way, we get a left exact functor Γ (Y ; −) : GM(k Y )  Vec(k) , its derived functor IR Γ (Y ; −) : D+ GM(k Y )  DVec(k) , and the corresponding cohomology functor H (Y ; −) := h IR Γ (Y ; −) H (Y ; −) := D+ GM(k Y )  Vec(k)N . Definition B.6.2.1 A sheaf I ∈ GM(k Y ) is said to be Γ (Y ; −)-acyclic is Hi (Y ; I) = 0 ,

∀i > 0 .

Recall that if F → I is an acyclic resolution (Sect. A.2.4), then IR Γ (Y ; F) = Γ (Y ; I ) .

B.6.3 The functor Γ in GM(A(X)) Let X be a mild space. We are interested in two cases O-1.  is a family of supports of X, and OX is the sheaf Ω 0X,k of all functions with values in the field k.

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O-2.  is a paracompactifying family of X, and a. OX is the sheaf of continuous real valued functions; b. X is a manifold and OX is the sheaf of differentiable real valued functions. Lemma B.6.3.1 In both cases O-1 and O-2, the set of functions OΦ (X) := Γ (X; OX ) verifies the following properties. 1. The set OΦ (X) is an ideal in O(X) such that  is the set of closed subsets S ⊆ X verifying S ⊆ {φ = 1} for some φ ∈ OΦ (X), in symbols:  S ∈  ⇔ S is closed in X and (∃φ ∈ OΦ (X))(S ⊆ {φ = 1}) . 2. For every OX -gm M, we have Γ (X; M) = {m ∈ M(X) | ∃φ ∈ OΦ (X) s.t. φ m = m}.

(B.15)

Proof (1) The implication ‘⇐’ is the condition B.6.2-(-2), since S ⊆ |φ|. For the converse, let S ∈ . We distinguish the two cases: Case O-1. Take φ := 1S : X → {0, 1} ⊆ k, the indicator function of S. Case O-2. By B.6.2-(B.6.2), there exists a neighborhood T ⊇ S in . Denote by T ◦ the interior of T in X. Since X is perfectly normal, there exists φ : X → R continuous (resp. differential) such that S ⊆ {φ = 1} and |φ| ⊆ T ◦ . Hence, |φ| ⊆ T ∈  and φ ∈ OΦ (X), as needed. (2) The inclusion ‘⊇’ is clear since, if m = φ m, then |m| ⊆ |φ| ∈ . For the converse, we apply (1). If m ∈ M(X) verifies |m| ∈ , we choose φ ∈ OΦ (X) such that |m| ⊆ {φ = 1}, in which case φ m = m, since φ(x) = 1, for all x such that mx = 0.   Definition B.6.3.2 Let A be an OX -ga. Inspired by equality B.15, we define for M ∈ GM(A(X)), Γ (M) := {m ∈ M | ∃φ ∈ OΦ (X) s.t. φ m = m} .

(B.16)

This is a sub-A(X)-gm of M. If ν : M → N is a morphism of A(X)-gm’s, we have ν(Γ M) ⊆ Γ N , and then denote by Γ ν : Γ M → 0 N the induced morphism. In this way, we get an additive functor Γ : GM(A(X))  GM(A(X)) . Lemma B.6.3.3 The functor Γ is exact.

(B.17)

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Proof Given an exact sequence of A(X)-gm’s 0 → L → M → N → 0, it is immediate that the sequence 0 → Γ L → Γ M → 0 N is exact. We thus need only justify that if ν : M → → N is a surjective morphism of A(X)-gm’s, then Γ ν is surjective. Indeed, an element n ∈ Γ N verifies φ n = n for some φ ∈ OΦ (X), and if n = ν(m), we have n = φ n = ν(φ m). This leads us to check that φ m ∈ 0 M. But, since |φ| ∈ , we can apply Lemma B.6.3.1-(1) and state that there exists φ  ∈ OΦ (X) verifying |φ| ⊆ {φ  = 1}, in which case φ  φ m = φ m, and φ m ∈ 0 M by definition (B.16), as announced.   Theorem B.6.3.4 (Acyclicity of OX -Modules) Let X be a mild space. 1. In both cases B.6.3-(O-1,O-2), every OX -gm is Γ (X; −)-acyclic. 2. Let (E, X, π, F ) be a fiber bundle of mild spaces. a. For every i ∈ N, the OX -gm’s π! Ω iE,k and π∗ Ω iE k are Γ (X; −)-acyclic for every family of supports . b. There are canonical identifications in D(X; k): 

IR Γ (X; π! Ω iE,k ) = Γ (X; π! Ω iE,k ) IR Γ (X; π∗ Ω iE,k ) = Γ (X; π∗ Ω iE,k ) .

3. Let (E, X, π, F ) be a fiber bundle of manifolds. a. For every i ∈ N, the OX -gm’s π!  iE and π∗  iE are Γ (X; −)-acyclic for every paracompactifying family of supports . b. There are canonical identifications in D(X; R): 

IR Γ (X; π!  iE ) = Γ (X; π!  iE ) IR Γ (X; π∗  iE ) = Γ (X; π∗  iE )

Proof (1) On the category GM(A), the functor Γ (X; −) coincides with the composition of the functors Hom•A (A, −) and Γ , both exact respectively by Theorem B.6.1.1-(1) and Lemma B.6.3.3. Therefore, likewise Theorem B.6.1.1(2), if M → I is an injective resolution in GM(A), we get IR Γ (X; M) = Γ (X; I ) = Γ (X; M) since injective graded modules in GM(A) are injective in GM(k X ). We therefore have, for all i > 0: hi IR Γ (X; M) = hi Γ Γ (X; I ) =1 hi Γ Γ (X; M) = hi Γ (X; M) = 0 , (=1 ) since Γ (X; −) is exact (Theorem B.6.1.1-(1)). (2) The complexes π! Ω E,k and π! Ω E,k are graded modules over Ω X,k (Sect. 8.2), hence over over OX := Ω 0X,k . We can then conclude, by (1), that they are Γ (X; −)-acyclic. The last part of (2), is a straightforward application of the general fact concerning complexes of acyclic objects of Proposition A.2.4.2.

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(3) The complexes π!  E and π!  E are graded modules over  y (Sect. 3.1.10.2). The proof is then same as for (2) with appropriate modifications.  

B.7 OX -Differential Graded Algebras We now extend the validity of Theorems B.6.1.1 and B.6.3.4 about OX -graded algebras A and their modules, to the same statements about OX -dga’s (A, d) and their dgm’s. As the reader will notice, the approach is the same, with only heavier notations . . .

B.7.1 Localization Functor for OX -DGA’s The localization presheaf functor defined in B.5.2.1-(B.7) for any sheaf of graded algebras A on a topological space Y , i.e. the functor A ⊗ (−) = GM(A(Y ))  Modpresh. (A) can be modified in order to incorporate differential structures. This is done in the restriction formula (B.6), where for V ⊆ U ⊆ X, we now define: ρVU ⊗idM

(A(U ), d) ⊗A(X) (M, d) −−−−−−→ (A(V ), d) ⊗A(X) (M, d)

(B.18)

where ρVU : A(U ) → A(V ) is always the restriction map, but where the tensor product is now the tensor product in categories of dgm’s. We get in this way, exactly as in Sect. B.5.2.1 for A-gm’s, a presheaf of (A, d)dgm’s on Y , which we denote by (A, d) ⊗ (M, d). The construction is functorial on (M, d), and the resulting functor (A, d) ⊗ (−) = DGM((A, d)(X))  DGMpresh. (A, d)

(B.19)

is the (A, d)-localization presheaf functor. It verifies, by construction, the adjunction equality: Hom•A ((A, d) ⊗ (M, d), (N, d)) = Hom•A(X) ((M, d), (N, d)(Y )) ,

(B.20)

since a morphism of presheaves ν : (A, d) ⊗ (M, d) → (N, d) is, by (B.18), a family of morphisms 

 ν(U ) : A(U ) ⊗A(X) (M, d) → (N, d)(U ) U ⊆X ,

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making commutative the diagrams (M, d)

(N, d)(Y )

ν(Y )

ρUY

(A, d)(U )

A(X)

(M, d)

ν(U )

(N, d)(U ) ,

Which means that ν(U )(a ⊗ m) = a ν(U )(1 ⊗ m) = a ρUY (ν(Y )(m)) , showing that ν is uniquely determined by ν(Y ). Definition B.7.1.1 Let (A, d) be a sheaf of differential graded algebras on a topological space Y . The (A, d)-localization functor, (A, d) ⊗ (−) = DGM((A, d)(Y ))  DGM(A, d) .

(B.21)

is the composition of the localization presheaf functor (A, d)⊗(−) (B.19), followed by the exact functor ‘associated sheaf to a presheaf ’. The following is the analogue to Proposition B.5.2.2. Proposition B.7.1.2 The natural map Hom•A ((A, d) ⊗ M, N) → Hom•A(Y ) (M, N(Y )) ν → ν(Y )

(B.22)

is an isomorphism, functorial in M ∈ DGM((A, d)(Y )) and N ∈ DGM(A, d). The functors 

Γ (Y ; −) = Hom•A ((A, d), −) : DGM(A, d)  DGM((A, d)(Y )) (A, d) ⊗ (−) : DGM((A, d)(Y ))  DGM(A, d)

constitute a pair of adjoint functors. Proof Same as for Proposition B.5.2.2.

 

B.8 The Localization Functor for OX -DGA’s As in Sect. B.5.3, we come back to the case of a mild space X and a OX -dga (A, d). Under these assumptions, we show that for every (A, d)-dgm (M, d), the stalks (M, d)x for x ∈ X, are determined in terms of the (A, d)(X)-dgm (M, d)(X) alone. The following proposition is the exact analogue to Proposition B.5.3.1. Proposition B.8.1 Let (A, d) be an OX -dga on a mild space X.

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For x ∈ X, denote by Sx the multiplicative system of functions in OX (X) equal to 1 in a neighborhood of x. 1. Let (M, d) ∈ DGM(A, d). For x ∈ X, the germ map (M, d)(X) → (M, d)x is a morphism of (A, d)(X)-dgm’s which factors through the localization map (M, d)(X) → Sx−1 (M, d)(X) in a isomorphism of (A, d)(X)-dgm’s ξx (M, d) : Sx−1 (M, d)(X) → (M, d)x , which is functorial in (M, d). 2. The localization functor (A, d) ⊗ (−) : DGM((A, d)(X))  DGM(A, d) is an exact functor, i.e. respects acyclicity. 3. Let (M, d) ∈ DGM(A, d). The natural map of (A, d)-modules (A, d) ⊗ (M, d)(X) → (M, d) is an isomorphism functorial in (M, d). In other words, the localization functor (A, d) ⊗ (−) is a left inverse of the global section functor Γ (X; −). Proof Same as for Proposition B.5.3.1 with just two new details to make clear. First, the extension of the dgm structure of an (A, d)(X)-dgm M to the localized module Sx−1 M, which is given by the well-known Leibniz rule for fractions d

m s

=

dm m − 2 d(s) , s s

for all m ∈ M and s ∈ Sx . Second, in (2), the definition of exact functors between dgm’s categories, which requires that acyclic dgm’s must remain acyclic. In our case, if (M, d) is acyclic, we have to show that the sheaf of dgm’s (A, d) ⊗ (M, d) is acyclic. This concerns only stalks, so that we are lead to check that (A, d)x ⊗ (M, d) = Sx−1 (M, d) is acyclic. A germ at x ∈ X of a fraction m s coincides with 1x ⊗ m since sx = 1x . Hence, d( m ) = 0 if and only if d m = 0, in which case m = d m for some m ∈ M since s x (M, d) is acyclic, but then d(1x ⊗ m ) = 1x ⊗ m.  

B.9 Equivalences of Derived Functors in DDGM(A, d) B.9.1 K-Injective Differential Graded Modules In Sect. A.3, we reviewed the concept of the derived functor of an additive functor F : KDGM(A, d) → KDGM(B, d) between homotopy categories of categories of dgm’s over dga’s on a ringed space.

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The definition is based on the Existence Theorem A.3.1.1, which states that for every M ∈ DGM(A, d), there exists a quasi-isomorphism M → I where I is injective as A-gm and is K-injective as A-dgm. In that case the derived functor which we denote by IR F : DDGM(A, d) → DDGM(B, d) , is the (B, d)-dgm IR F (M) := F (I) .

(B.23)

In this definition, an (A, d)-dgm I is said to be K-injective (Sect. A.3.1), if, for every acyclic (A, d)-dgm N, one has Hom•K(A,d) (N, I) = 0 , where ‘K(A, d) ’ is an abbreviation of the notation ‘KDGM(A, d) ’ of the homotopy category of the category of DGM(A, d). Proposition B.9.1.1 Let (A, d) be an OX -dga on a mild space X. 1. For all M, N ∈ DGM(A, d), the canonical map  IR Γ X; IR Hom •D(A,d) (M, N) → IR Hom•D(A,d) (M, N) is an isomorphism in DDGM(A(X), d). 2. If the natural homomorphism of sheaves of algebras ι : k X → (A, d) is a quasiisomorphism, the induced homomorphism of bifunctors 

IR Hom •D(A,d) (−, −) → IR Hom •k (−, −) : DDGM(A, d)2  DDGM(k X , d) ,

is an isomorphism. Proof (1) Let N → I be a quasi-isomorphism with I injective as A-gm, and Kinjective as (A, d)-dgm. We have, by (B.23), IR Hom •D(A,d) (M, N) = Hom •A (M, I) , were we claim that Hom •A (M, I) is a complex of injective sheaves. Indeed, this is equivalent to the fact that the functor  Homk X −, Hom •A (M, I) : GM(k X ) → GM(k) is exact, which follows immediately from the well-known identification   Homk X −, Hom •A (M, I) = HomA − ⊗k X M, I)

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which shows that the left-hand functor is the composition of (−) ⊗R X I, exact since I is R X )-flat, followed by HomA (−, I) exact since I is an injective A-gm. We therefore have:   IR Γ X; IR Hom •D(A,d) (M, N) = IR Γ X; Hom •A (M, I)  =1 Γ X; Hom •A (M, I) = Hom•A (M, I) = IR Hom•A (M, N) where (=1 ) is justified by the injectivity of Hom •A (M, I). (2) Let (I, d) ∈ DGM(A, d) be K-injective and such that I is an injective Agm. Since I is an injective k X -gm after Theorem B.6.1.1-(2), we have a natural adjunction morphism of exact functors,   Hom•k X −, (I, d) → Hom•A (A, d) ⊗k X (−), (I, d) which is an isomorphism of k-dgm’s since it respects differentials and since it is already an isomorphism at the underlying level of graded modules. As a consequence, we get a natural isomorphism of bifunctors  IR Hom•k X (−, −) → IR Hom•D(A,d) (A, d) ⊗k X (−), − which leads us to prove that the morphism of bifunctors on DGM(A)  IR Hom•(D(A),d) (−, −) → IR Hom•(D(A),d) A ⊗k X (−), − , induced by the morphisms of A-dgm’s A ⊗k X (M, d) → (M, d) ,

a ⊗ m → a m ,

is an isomorphism. This is indeed the case, since it suffices to look at the level of stalks, where the morphism (A, d)x ⊗k (F, d)x → (F, d)x is easily seen to a quasiisomorphism after the well-known Künneth’s theorem.   The following theorem extends the validity of Theorem B.6.1.1 to dgm’s. Theorem B.9.1.2 Let (A, d) be an OX -dga on a mild space X. 1. The global section functor  Γ (X; −) =: DGM(A, d) → DGM A(X), d is fully faithful. 2. If (I, d) is a K-injective object in DGM(A, d), then (I, d)(X) is a K-injective object in DGM(A(C), d).

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3. For all M, N ∈ DGM(A, d), the canonical map IR Hom•D(A,d) (M, N) → IR Hom•D(A(X),d) (M(X), N(X)) is an isomorphism in DDGM(A(X), d). Proof Same as for Theorem B.6.1.1, with few new justifications needed. (1) Immediate after Proposition B.8.1-(2,3) (2) By the adjunction property (Proposition B.7.1.2), we can identify functors Hom•A(X) (−, (I, d)(X)) = Hom•A ((A, d) ⊗ (−), (I, d)) ,

(B.24)

and conclude that the left-hand functor in (B.24) respects acyclicity, since A ⊗ (−) do respects acyclicity by Proposition B.8.1-(2), and the same for Hom•A (−, (I, d)) since (I, d) is K-injective. Hence, (I, d)(X) is K-injective in DGM((A, d)(X)). (3) Follows straightforwardly by simply applying the definition of derived functors in DGM(A, d) as recalled in Sect. A.3. let N → I be a quasi-isomorphism where I is an injective A-gm and a K-injective A-dgm. Then IR Hom•D(A,d) (M, N) =1 Hom•A (M, I) =2 Hom•A(X) (M(X), I(X)) =3 IR Hom•D(A(X),d) (M(X), N(X)) (=1 ) by definition of derived functor, (=2 ) by (1), and (=3 ) by definition of derived functor since (I, d)(X) is an injective A(X)-gm after Theorem B.6.1.1-(4), and a K-injective (A, d)(X)-dgm after (2).   Corollary B.9.1.3 1. Let (E, X, π, F ) be a fiber bundle of mild spaces. a. The following morphisms induced by the augmentation  : k X → Ω X;k IR Hom •D(Ω X,k ,d) (π! Ω E,k , Ω X,k ) → IR Hom •k X (π! Ω E,k , Ω X,k ) ↑ IR Hom •k X (π! Ω E,k , k X ) are isomorphisms in D(X; k). b. The morphisms IR Hom•k X (π! Ω E,k , k X ) ↑ • IR HomD(Ω X,k ,d) (π! Ω E,k , Ω X,k ) ↓ IR Hom•D((X,k),d) (Ωcv (E, k), (X, k)) are isomorphisms in DVec(k).

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321

2. Let (E, X, π, F ) be a fiber bundle of manifolds. a. The following morphisms induced by the augmentation  : R X →  X IR Hom •D( X ,d) (π!  E ,  X ) → IR Hom •R X (π!  E ,  X ) ↑ • IR Hom R X (π!  E,k , k X ) are isomorphisms in D(X; R). b. The morphisms IR Hom•k B (π!  E ,  B ) ↑ IR Hom•D( B ,d) (π!  E ,  B ) ↓ IR Hom•D((B),d) (Ωcv (E), (B)) are isomorphisms in DVec(R). Proof Immediate consequences of Theorem B.9.1.1-(2) and Proposition B.9.1.2(3), since X is locally contractible and the morphism k X → Ω X,k (resp. R X →  X for manifolds) is a quasi-isomorphism.  

Appendix C

Cartan’s Theorem for g-dg-Ideals

In Cartan [27] §4, given a g-dg-algebra E := (E, d, θ, ι), an element ω ∈ E, is called basic if it is killed by every g-interior product and every g-derivation, i.e. ι(X)(ω) = θ (X)(ω) = 0 ,

∀X ∈ g .

The same definition clearly makes sense for any g-dg-ideal K ⊆ E. This is the name we give to any dg-ideal (K, d) ⊆ (E, d) stable under the g-interior products ι(X), for all X ∈ g. The fact that θ (X) = d ◦ ι(X) + ι(X) ◦ d (Sect. 4.2.3) warrants that such K’s are also stable under the action of g-derivations. We can therefore consider the sub-E bas -module K bas . When E is a g-dg-algebra admitting an algebraic connection f : W (g) → E and K ⊆ E is a proper g-dg-ideal, the composition of f with the canonical surjection E→ → E  := E/K is a connection for E  , which naturally leads us to consider the following diagram which extends diagram (4.2) in Sect. 4.1.1.1 K

W( ) ⊗ K

K

E

W( ) ⊗ E

E

E

W( ) ⊗ E

E

(C.1)

where, we recall, i(ω) := 1 ⊗ ω and f(α ⊗ ω) := f (α)ω.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Arabia, Equivariant Poincaré Duality on G-Manifolds, Lecture Notes in Mathematics 2288, https://doi.org/10.1007/978-3-030-70440-7

323

C Cartan’s Theorem for g-dg-Ideals

324

The problem with this diagram is that when applying the functor (−)bas , we get the diagram of Cartan-Weil morphisms (W ( ) ⊗ K)bas

K bas E bas q.i.

E ,bas q.i.

(W ( ) ⊗ E)bas (W ( ) ⊗ E )bas

K bas E bas

(C.2)

q.i.

E ,bas q.i.

where, although in the second and third rows, Cartan-Weil morphisms are isomorphisms after Cartan’s theorem 4.1.1.1, we cannot conclude the same for the first row since columns are generally not exact. The way to avoid this issue is to replace the cokernel of the inclusion κ : K ⊆ E by its mapping cone. Indeed, the cone in question : E  := c(κ) ˆ := (E ⊕ K[1], $) , is equipped with a structure of g-dg-algebra by defining a multiplication by (x, y) · (x, y  ) := (xx  , xy  + yx  + yy  ) , and an action of g-interior products by ι(X)(x, y) := (ι(X)(x), ι(X)(y)) ,

∀X ∈ g .

These settings entail that g-derivations are given by the formula θ (X)(x, y) := (θ (X)(x), θ (X)(y)) ,

∀X ∈ g ,

and the verification of the axioms of g-dg-algebras in Sect. 4.2.3 is almost immediate. More importantly, an algebraic connection f : W (g) → E composed with the inclusion ι : E → E ⊕ K[1] clearly gives an algebraic connection for E  so that Cartan’s theorem applies to E  . If we now reconsider diagram (C.2) under these new data, we see that while rows remain the same, the columns have become mapping cones since c(κ) ˆ bas = c(κ ˆ bas ) ,

C Cartan’s Theorem for g-dg-Ideals

325

which is easily checked. But then, applying the cohomology functor, we get the diagram of exact columns hi−1 (E bas )

hi−1 ((W ( ) ⊗ E)bas )

hi−1 (E bas )

hi−1 (E ,bas )

hi−1 ((W ( ) ⊗ E )bas )

hi−1 (E ,bas )

hi (K bas )

hi ((W ( ) ⊗ K)bas )

hi (K bas )

hi (E bas )

hi ((W ( ) ⊗ E)bas )

hi (E bas )

hi (E ,bas )

hi ((W ( ) ⊗ E )bas )

hi (E ,bas )

(C.3)

where we can apply the Five Lemma and conclude that Cartan-Weil morphisms in the third (middle) row are indeed isomorphisms. Hence, the following enhancement of Cartan’s theorem. C.1 Proposition If E ∈ DGA(g) admits algebraic connections, then, for every gdg-ideal K ⊆ E, the Cartan-Weil morphism i : K bas → (W (g) ⊗ K)bas is a quasi-isomorphism. In particular, the Cartan-Weil morphism  H (f) : H (W (g) ⊗ K)bas → H (K bas ) is an isomorphism too, inverse of H (i), hence independent of the connection.

Appendix D

Graded Ring of Fractions

Let A := A0 ⊕ A1 ⊕ · · · be a graded ring and denote by S the multiplicative system generated by the nonzero graded elements of A. The graded ring of fractions of A is, by definition, the ring QA := S −1 A.1 The following fact has been mentioned several times (Exercise 5.1.2.3 and Sect. 7.1). D.1 Proposition The graded ring QA is such that, for every A-graded module N , the tensor product QA ⊗A N is a flat injective A-graded module. Proof QA ⊗ N is flat. For any graded ideal I of A, one has the long exact sequence: 0 → TorA 1 (QA , A/I ) −→ QA ⊗ I −→ QA −→ QA ⊗ (A/I ) → 0

(D.1)

where A/I is a torsion graded A-module. The annihilators of the elements of A/I are graded ideals, generated, as such, by invertible elements of QA . Therefore TorA 1 (QA , A/I ) = 0 ,

∀i ∈ N ,

and we have from (D.1) the equality QA ⊗ I = QA from which, we deduce QA ⊗ I ⊗ N = QA ⊗ N for any A-graded module N . The ideal criterion of flatness applies, and the Agraded module QA ⊗ N is flat.

ring QA is the zero ring is A has zero divisors, something that never occurs for A := HG , since HG is an integral domain.

1 The

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Arabia, Equivariant Poincaré Duality on G-Manifolds, Lecture Notes in Mathematics 2288, https://doi.org/10.1007/978-3-030-70440-7

327

328

D Graded Ring of Fractions

QA ⊗ N is injective. Let α : M1 ⊆ M2 be a graded inclusion of graded Amodules. We must show that any morphism λ : M1 → QA ⊗ N of graded Amodules can be extended to M2 . α ⊆

M1 λ

M2 λ

QA ⊗ N Otherwise, the Zorn Lemma will lead us to assume that M2  M1 and that λ may not be further extended. In particular, A · m ∩ M1 = 0 for any homogeneous m ∈ M2 \ M1 , hence the quotient M2 /M1 is a torsion module. One then has Q A ⊗ M1 = QA ⊗ M2 , and a contradiction arises as a consequence of the diagram HomgrA (M2 , QA ⊗ N)

HomgrA (M1 , QA ⊗ N)

∼ =

HomgrQA (QA ⊗ M2 , QA ⊗ N)

∼ = (=)

HomgrQA (QA ⊗ M1 , QA ⊗ N)

where the horizontal arrows are induced by the inclusion M1 ⊆ M2 and the vertical arrows are the well-known canonical natural isomorphisms.  

Appendix E

Hints and Solutions to Exercises

Ch. 2. Nonequivariant Background 1. Exercise 2.1.5.2–(3) (p. 12). In both cases, ⊕ and , the coordinates λi of λ are linear forms determined by their action in the coordinate subspaces V m . If all these linear forms vanish, then λ = 0 on each V m , in which case λ = 0. 2. Exercise 2.1.5.2 (p. 12). – In the case of the functor ⊕, we have to compare the two sets 

Homk (⊕V , ⊕W ) = i Homk (Vi , ⊕W ) Homgr∗k (V , W ) = i j Homk (Vi , Wj ) = i Homk (Vi , W ) ,

which are clearly equal if and only if V = 0 or ⊕W = W . Hence, V ,W is surjective if and only if V = 0 or W is bounded. – In the case of the functor , the sets to compare are 

Homk (V , W ) = j Homk (V , Wj ) Homgr∗k (V , W ) = i j Homk (Vi , Wj ) = j Homk (⊕V , Wj ) ,

which are equal if and only if W = 0 or ⊕V = V . Hence, V ,W is surjective if and only if V is bounded or W = 0. – In both cases, the announced equivalence (4a) and nonequivalence (4b) of categories is proved. 3. Exercise 2.1.6.1 (p. 13). Given a di : Xi → Xi+1 in a split category Ab, we can assume that Xi = Zi ⊕ Bi+1 , that Xi+1 = Bi+1 ⊕ Ni+1 and that d(z, b) = (b, 0). Then, di−1 : Xi−1 → Xi , such that di ◦ di−1 = 0, factors through the inclusion Zi ⊆ Xi , and we can assume that Xi−1 = Zi−1 ⊕ Bi , that Zi = Bi ⊕ Hi and that © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Arabia, Equivariant Poincaré Duality on G-Manifolds, Lecture Notes in Mathematics 2288, https://doi.org/10.1007/978-3-030-70440-7

329

330

E Hints and Solutions to Exercises

di−1 (z, b) = (b, 0). As a consequence, the complex (X, d) ∈ C(Ab) is isomorphic to a split complex: di−1

di

di+1

Xi−2 · · · ∼

Xi+1 ∼

Xi ∼

Xi−1 ∼

···

di−2

∼

· · · Xi−2 Bi−1

Bi

Bi+1

Bi+2

Bi+3

Hi−2

Hi−1

Hi

Hi+1

Hi+2

Bi−2

Bi−1

Bi

Bi+1

Bi+2

···

(E.1) In which case, the cohomology of h(X, d) is canonically isomorphic to the graded object {Hi }i∈Z ∈ AbZ . Now, an additive functor F : Ab → Ab applied to (E.1) respects the decompositions in direct sums, so that it tautologically satisfies the equality h(F (V , d)) = {F (Hi )}) = F h(V , d) proving the exactness of F . Conversely, assume that every additive functor on Ab is exact. Then if ι : X Y is a monomorphism in Ab, we consider the short exact sequence ι

ν

0 → X −−→ Y −−→ K → 0 , where ν is a cokernel for ι, to which we apply the (exact) functor HomAb (K, −). We obtain the short exact sequence ι

ν

0 → HomAb (K, X −−→)HomAb (K, Y −−→) HomAb (K, K → 0) . Hence, there exists a morphism ρ : K → Y such that ν ◦ ρ = idK . But then the morphism X ⊕ K → Y , (x, k) → x + ρ(k) is an isomorphism. We have thus shown that every subobject d’un object in Ab admits a complement. The category Ab is therefore a split category. The incomplete basis theorem shows that Vec(k) and GV(k) are both split categories, and any additive functor defined in these categories is exact.

E Hints and Solutions to Exercises

331

4. Exercise 2.1.8.2 (p. 15). We have  D (v  ⊗ λ) (v) = d  (λ(v)v  ) − (−1)[v]+[λ] λ(dv)v  = λ(v)d  (v  ) + (−1)[v] (Dλ)(v)v  = (d  (v  ) ⊗ λ)(v) + ((−1)[v] v  ⊗ (Dλ))(v)  =  $(v  ⊗ λ) (v) 5. Exercise 2.1.8.3 (p. 15). We treat only the case of the tensor product. The map 2 : (V ⊗ W )[s + t] → V [s] ⊗ W [t] , which is the identity over each component V a ⊗ W b , is an obvious isomorphism of graded vector spaces. Given v ∈ V homogeneous, the notation v[s] ∈ V [s] should recall that the degree of v[s] is [v] − s. We then have:   2 $[s + t](v ⊗ w) = (−1)s+t 2 dv ⊗ w + (−1)[v] v ⊗ dw

(E.2) = d[s]v[s] ⊗ (−1)t w[t] + (−1)[v]+s v[s] ⊗ d[t]w[t]

while  $ 2(v ⊗ w) = d[s]v[s] ⊗ w[t] + (−1)[v]−s v[s] ⊗ d[t]w[t] ,

(E.3)

which shows that 2 does not respect the differentials. More precisely, the issue lies in the t-shift, otherwise, if t = 0, then 2 is compatible, so that we can state that the map 2s : (V ⊗ W )[s] → V [s] ⊗ W ,

(v ⊗ w)[s] → v[s] ⊗ w ,

identifies both complexes. It is now useful to recall that the following (anticommutative) transposition τ :V ⊗W →W ⊗V ,

τ (v ⊗ w) := (−1)[v][w] w ⊗ v ,

dictated by the Koszul sign rule (Sect. 2.1.9), is an isomorphism of complexes.

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E Hints and Solutions to Exercises

Indeed, it is clearly bijective and it verifies:   τ $(v ⊗ x) = τ dv ⊗ w + (−1)[v] v ⊗ dw = (−1)([v]+1)[w] w ⊗ dv + (−1)[v] (−1)[w+1][v] dw ⊗ v  = (−1)[u][v] dw ⊗ v + (−1)[w] w ⊗ dv = $τ (v ⊗ w) . We can now give an explicit isomorphism of complexes from (V ⊗ W )[s + t] to V [s] ⊗ W [t] by simply composing the following isomorphisms: 2s [t]

(V ⊗ W )[s][t] −−−→ (V [s] ⊗ W )[t] −→ 

τ [t]

−−−→ (W ⊗ V [s])[t] −→ 

2t

−−−→ (W [t] ⊗ V [s]) −→ 

τ

−−−→ (V [s] ⊗ W [t]) . 

An elementary computation shows that the resulting isomorphism associates: v ⊗ w → (−1)t ([v]+s) v[s] ⊗ w[t] , which is not obvious a priori. 6. Exercise (fn. (10 ), p. 18) (p. 18). Let f : M → N be proper. In the locally compact space N , a subset A ⊆ N is closed if and only, for all K ⊆ N compact, the set A ∩ K is compact. As a consequence, the map f : M → N is closed if and only if, for all Y ⊆ M closed, and all K ⊆ N compact, the set f (Y ) ∩ K = f (Y ∩ f −1 (K)) is compact, which is always the case when f is proper, since then f −1 (K), Y ∩ f −1 (K)) and f (Y ∩ f −1 (K)) are compact. Conversely, suppose f : M → N closed with compact fibers. Let K ⊆ N be compact. Given a family {Yi }i∈I of nonempty closed subsets of f −1 (K), . which we assume to be stable by finite intersections, we necessary have L := i f (Yi ) = ∅, since otherwise, K being compact, some finite sub-intersection must be empty, −1 which is not the case. But then, for the fiber f (x) meets every Yi in a . x ∈ L,−1 nonempty set. As a consequence, i Yi ∩ f (x) = ∅, since the fiber is compact. . Therefore i Yi = ∅. We have thus proved that the set f −1 (K) is compact for every compact subspace K ⊆ N , i.e. that f is a proper map. Notice that we did not use the fact that the spaces are locally compact, but only that they are Hausdorff. 7. Exercise 2.4.1.2 (p. 25). To show that the Poincaré adjunctions are not surjective, let ϕ : U → W ⊆ Rm (m := dM ) be a chart of M and let (fn )n∈N be any sequence of nonzero functions in W with two by two disjoint supports and uniformly converging to the zero function.

E Hints and Solutions to Exercises

333

Given 0 ≤ i ≤ n, consider the family {βi,n }n∈N ⊆ ic (M) where  βi,n := ϕ ∗ fn dx1 ∧ · · · ∧ dxi . By construction, for all α ∈ m−i (M) (resp. cm−i (M)) we have lim  α, βi,n M = 0 .

(E.4)

n→+∞

On the other hand, since the family of i-forms {βi,n }n∈N is linearly free, it can be extended to a basis of ic (M) (resp. i (M)), so that there exist linear forms  : ic (M) → R such that

(∀n ∈ N) (βi,n ) = 1 ,

(resp.  : i (M) → R). Such linear forms do not verify the condition (E.4), hence they are not of the form  α, − M with α ∈ m−i (M) (resp. cm−i (M)). 8. Exercise 2.4.2.1 (p. 30). (1) An orientation of M defines, on each connected component C ∈ 0 (M), an orientation [C] and a nonzero class in HcdM (C), viz. the fundamental class ζ[C] . Conversely, let ω ∈ HcdM (C)  {0}. If [C] is an orientation of C of fundamental class ζ[C] , we will have ω = λ(ω, [C]) ζ[C] for a unique scalar λ(ω, [C]) ∈ R  {0}. Changing [C] by the opposite orientation −[C], changes the sign of λ(ω, [C]). As a consequence, there is a unique orientation of C, which we denote [C]ω , such that λ(ω, [C]ω ) > 0. The correspondences [C] → ζ[C]

and

ω → [C]ω ,

are then clearly inverse of each other. (2) When |0 (M)| < +∞, we have H 0 (M) =

%

H 0 (C)

and

C∈0 (M)

and 1M =



C∈0 (M) 1C ,



DM (1M ) =

%

HcdM (M) =

HcdM (C) ,

C∈0 (M)

with 1C ∈ H 0 (C). Therefore, 

DC (1C ) =

C∈0 (M)

C

C∈0 (M)



↔

ζC ∈ HcdM (M) .

C∈0 (M)

(3) When |0 (M)| < ℵ0 , we rather have H 0 (M) =

 C∈0 (M)

H 0 (C)

and

HcdM (M) =

% C∈0 (M)

HcdM (C) ,

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E Hints and Solutions to Exercises

in which case 1M is the infinite family {1C | C ∈ 0 (M)}, and the element DM (1M ) =

       C ∈ 0 (M) ↔ ζC ∈ HcdM (C)  C ∈ 0 (M) C

can no more be represented by a single cohomology class in HcdM (M). 9. Exercise 2.5.2.4–(1) (p. 33). Fix a locally finite good cover V := {Vi }i∈N of N (fn. (15 ), p. 28) by trivializing For each m ∈ N, let Wm :=  open subsets for f .−1 {V0 , . . . , Vm } and let Wm := Wm . The manifold f Wm is of finite type. Indeed, thanks to Leray, one knows that H (f −1 Wm ) is the abutment of a spectral sequence p,q of second page IE 2 = H p (Wm ; Hq ), where Hp is the local system Hq IR f∗ R M q of germs H (F ), where F denotes the fiber of f . To show that dim(H (f −1 Wm )) < +∞, it will then suffice to show that dim(IE 2 ) < +∞, and since q is bounded, only that dim(H (Wm ; Hq )) < +∞, for each q. For that, one uses the fact that Wm is a good cover of Wm , and that, therefore, one can compute H (Wm ; Hq ) using the q 1 ˇ ˇ Cech complex C(W m ; H ), which is easily seen to be finite dimensional. 10. Exercise 2.5.2.4–(2) (p. 33). If M is orientable, then M = M˜ and dim(Hc (M)) = dim(H (M)) after Poincaré duality. If M is not orientable, the orientation manifold M˜ is connected and it is the total space of a 2-fold covering π : M˜ → M with Aut(π ) = Z/2Z. In particular, one has ˜ M = M/τ  as manifolds. The generator τ of Aut(π ) is an orientation-reversing involutive diffeomorphism τ : M → M. Its action by pullback, also denoted by ‘ τ ’, gives rise to the canonical decompositions 

˜ = H (M)+ ⊕ H (M)− H (M)

(E.5)

˜ = Hc (M)+ ⊕ Hc (M)− Hc (M)

where (−)+ denotes the vector subspaces of τ -invariants, i.e. τ (x) = x, while (−)− denotes the space of τ -anti-invariants, i.e. τ (x) = −x. The key fact is that subspaces with same invariance  ∈ {+, −} are orthogonal under the Poincaré pairing ·, ·M˜ . ˜  , and β ∈ Hc (M) ˜  , then Indeed, if α ∈ H (M) 

α, β

       = τ α, τβ = τ α ∧ τβ = τ (α ∧ β) = − α ∧ β = − α, β M˜ , M˜ M˜



M˜

M˜

M˜

since τ reverses orientation, which immediately implies that 

1 See

α, β



M˜

= 0.

Godement [46], sec. 5.3–4, Theorem 5.4.1 and its Corollary pages 212–213.

E Hints and Solutions to Exercises

335

As a consequence, the Poincaré duality map DM˜ (Theorem 2.4.1.3–(2.29)) establishes an isomorphism ˜ − −−→ (Hc (M) ˜ + )∨ , DM˜ : H (M) 

which, introduced in the decomposition (E.5), gives rise to a canonical isomorphism of vector spaces ˜ + )∨ . ˜  H (M) ˜ + ⊕ (Hc (M) H (M)

(E.6)

˜ Aut(π ) and since π is proper, the pullback On the other hand, since M ∼ M/ ˜ + and Hc (M) ↔ Hc (M) ˜ + , so that the map π ∗ identifies H (M) ↔ H (M) decomposition (E.6), can be best seen as the decomposition ˜  H (M) ⊕ Hc (M)∨ H (M) whence the claim. 11. Exercise 2.6.1.1–(1) (p. 38). We have    α ∧ d β = (−1)[α]+1 dα∧β ID M d β (α) = M



M



= (−1)[α]+1 ID M (β) d α = (−1)[α]+1 (−1)dM −[β]+1 (DID M (β))(α) = (−DID M (β))(α) . Hence, ID M : ((M)[dM ], d) → (c (M)∨ , −D) is a morphism of complexes. 12. Exercise 2.6.1.1–(2) (p. 38). The differential in (c (M)[dM ], d) is d, while in ((M), d)[dM ] is (−1)dM d, which immediately shows that ι is not a morphism of complexes, contrary to . For ID M and ID M  , we know already that they are morphisms of complexes. For Ξ , we have  Ξ (−Dλ) = Ξ (−1)[λ] λ ◦ d = (−1)[λ] (−1)[λ]+1+dM λ ◦ d ◦ ι  D(Ξ (λ)) = D (−1)[λ]+dM λ ◦ ι = (−1)[λ]+dM (−1)[λ]+1 λ ◦ ι ◦ d and Ξ is also a morphism of complexes. To check the commutativity of the diagram, let β, β  ∈ c (M) be homogeneous. Then, ID M ((ω))(ω ) = (−1)[ω]dM

 M

ω ∧ ω = (−1)[ω]

 M

ω ∧ ω

= (−1)[ω] ID  (ω)(ω ) = Ξ (ID  (ω))(ω ) .

336

E Hints and Solutions to Exercises

13. Exercise 2.6.2.3 (p. 42). Simple application of Proposition 2.5.1.1 on adjoint operators.

Ch. 3. Relative Poincaré Duality 14. Exercise 3.1.1.1 (p. 50). Since the projection map π is open and since E  is connected, the space B  := π(E  ) is an open connected submanifold of B, hence of some well-defined dimension dB  . On the other hand, from general topology, for all b ∈ B  , the connected components of the fibers Fb are contained in the connected components of E, hence the fact that Fb := Fb ∩E  is union of connected components of Fb . The triviality of π above small open balls IB b centered at b ∈ B  then warrants that Fb is of dimension dE  − dB  , and, furthermore, that the fibers Fx are diffeomorphic for all x ∈ IB b , hence for all x ∈ B  , since B  is connected. Notice that we have also proved that B  is a closed subspace of B. The map π  : E  → B  is therefore a locally trivial fibration of manifolds onto a connected component of B. 15. Exercise 3.1.4.1 (p. 52). (1) Let π ∈ MorC (X, W ) and let X := X ×(π,idW ) W . The map Ξ (Y ) : Mor (Y, X) η

η

Mor (Y, X ) , (η, π ◦ η)

X

W

Y π◦η

π

W

is a natural bijection for all Y ∈ C. The morphism (idX , π ) : X → X corresponds to idX : X → X through the isomorphism Ξ (X), and it has as left inverse the morphism ψ := Ξ (X )−1 (idX ) : X → X since the diagram ψ∗

Mor(ψ,X)

Mor (X, X)

Mor (X , X)

Ξ (X)

Ξ (X)

Ξ (X ) Mor(ψ,X )

Mor (X, X )

ψ

idX

(idX , π)

Mor (X , X )

Ξ (X )

ψ∗

idX

is commutative. A symmetric argument shows that ψ is a right inverse to (idX , π ), hence that (idX , π ) is an isomorphism. (2) If the notation (X1 ×W W ) ×Z X2 denotes (X1 ×W W ) ×(ν,π2 ) X2 , we set π1 := ν ◦ (idX , π ), in which case we have a commutative diagram X

(idX ,π)

X ×W W

π1

Z

ν

Z

E Hints and Solutions to Exercises

337

with horizontal isomorphisms, which implies that the induced morphism (X1 ×W W ) ×(ν,π2 ) X2 → X1 ×(π1 ,π2 ) X2 is an isomorphism. 16. Exercise 3.1.4.3 (p. 54). We give the answer for H := {e} and leave the general case to the reader. We thus consider the diagram X f

Y

νX

⊕ νY

X/G fG

Y/G.

Assume the diagram Cartesian. Given x ∈ X, such that f (x) = f (k · x), the two elements x and k · x must coincide after Proposition 3.1.4.2–(1) since they verify simultaneously f (x) = f (k · x) and νX (x) = νX (k · x). Hence, StabG (x) ⊆ StabG (f (x)). The opposite inclusion being obvious, the map f : G · x → G · f (x)

(E.7)

is bijective. Conversely, assume the maps (E.7) bijective for all x ∈ X. Then  w : X → Y ×(νY ,fG ) (X/G) ,

w(x) := (f (x), G · x) ,

is surjective since, given y ∈ Y and G · x ∈ X/G such that y ∈ f (G · x), there always exist k ∈ G verifying y = f (k · x), in which case w(k · x) = (y, G · x). But such k is unique because (E.7) is bijective. The map w is therefore a continuous bijection. It is also an open map. Indeed since f is continuous, an open subspace U ⊆ X is the union of open subspaces of the form f −1 (V ) where V ⊆ Y is open. It therefore suffices to show that   w f −1 (V ) = V ×(νY ,fG ) fG−1 νY (V )

(E.8)

is open in Y ×(νY ,fG ) (X/G). But this is clear since νY is an open map, in which case the right-hand side of (E.8) is the induced fiber product of the two open subspaces −1  V ⊆ Y and f νY (V ) ⊆ X/G. 17. Counterexample to a fiber product of manifolds 3.1.6 (p. 56). Given two maps of manifolds pi : Mi → N, assume that their fiber product exists in the category of manifolds, denote it by (M1 ×N M2 )diff . Denote by (M1 ×N M2 )top the fiber product in the category of topological spaces. By the universal property of fiber product in Top, we have a natural continuous map ξ : (M1 ×N M2 )diff → (M1 ×N M2 )top ⊆ M1 × M2 .

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E Hints and Solutions to Exercises

The universal property applied to 0-dimensional manifolds, for which continuous and differentiable maps are the same, implies immediately that ξ is a bijection. Now, suppose that for (x1 , x2 ) ∈ (M1 ×N M2 )top the map p1 is étale on x1 , i.e. that there exists an open neighborhood Ux1 # x1 such that V1 := p1 (Ux1 ) is open in N and such that p : U1 → V1 is a diffeomorphism. Then, the subset   W (x1 , x2 ) := U1 × p2−1 (V1 ) ∩ M1 ×N M2 is an open neighborhood of (x1 , x2 ) ∈ (M1 ×N M2 )top and is clearly the graph of the differentiable map x2 → p1−1 (p2 (x2 )) for x2 ∈ p2−1 (V1 ) ⊆ M2 . Hence, W (x1 , x2 ) is a locally closed differentiable submanifold of M1 × M2 of dimension dim(M2 ). Since W (x1 , x2 ) is also open in (M1 ×N M2 )diff , this gives a constructive proof of the existence of this manifold at a neighborhood of (x1 , x2 ). We thereby see that to find a counterexample to a fiber product of manifolds, we need a case were, for (x1 , x2 ) ∈ M1 ×N M2 , neither xi is étale for pi . For example pi : R → R, pi (t) = t 2 , and xi = 0. In that case (R ×R R)top = {(t, t)} ∪ {(t, −t)} ⊆ R × R . Since the paths γ± : t → (t, ±t) ∈ (R×R R)top are differentiable in R×R, they must lift to (R ×R R)diff , if this manifold exist. But this manifold should be a curve after the previous paragraph, and the sets {γ± (R)} must therefore coincide in (R×R R)diff , but this is impossible since they are different in R × R. We have thus proved that the fiber product of pi : R → R, pi (t) = t 2 , does not exist in the category of manifolds. 18. Exercise 3.1.7.4(1) (p. 62). We can assume, after Exercise 3.1.1.1, that E, B and F are equidimensional, respectively of dimensions m, n and d. – If B is orientable and ωB ∈ n (B) is nowhere vanishing, then π ∗ ωB ∧ ωπ is nowhere vanishing. Indeed, for x ∈ Fb , fix a family {ξi , . . . , ξn } of vector fields on some trivializing open neighborhood U of b such that {ξ1 (b), . . . , ξn (b)} spans Tb B, and fix a family {ζ1 , . . . , ζd } of vector fields on Fb such that {ζi (x), . . . , ζn (x)} spans Tx Fb . Extend the vector fields to the whole π −1 (U ) using some trivializing diffeomorphism ϕ : π −1 (U ) → U × Fb . Then, we have (π ∗ ωB ∧ ωπ )(ξ1 , . . . , ξn , ζ1 , . . . , ζd )(x) = ωB (ξ1 , . . . , ξn )(b) ωπ

Fb (ζ1 , . . . , ζd )(x)

= 0 .

The differential form π ∗ ωB ∧ ωπ is therefore nowhere vanishing and of highest degree m, hence E is orientable. – Assume E orientable and choose ωE ∈ m (E) nowhere vanishing. Fix an open cover U := {Ui }i∈I of B by trivializing subspaces for π , where each Ui is diffeomorphic to Rn , hence is orientable. For each i ∈ I, we can then choose a nowhere vanishing form ωU ∈ n (U ) such that π ∗ ωU ∧ ωπ (which

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339

is nowhere vanishing after the previous paragraph) and ωE define the same orientation on π −1 (Ui ). Then,

for any partition of unity {φi }i∈I subordinate to U, the differential form i φi ωUi ∈ n (B) is nowhere vanishing since, by construction, for every b ∈ B, the differential forms ωUi such that b ∈ Ui , define the same orientation on a neighborhood of b. Hence B is orientable. 19. Exercise 3.1.7.4(1) (p. 62). The Möbius Strip is naturally fibered over its central circle S1 with fibers diffeomorphic to R. More generally, the normal bundle to any non orientable submanifold N of any Euclidean space Rm . 20. Exercise 3.1.9.3-(1) (p. 64). In the commutative diagram X

g

X

⊕

π

B

π

B

g

we have, for all P ⊆ X and all K ⊆ B  ,  g −1 (P ) ∩ π −1 (K) = g −1 P ∩ π −1 (g(K)) . Consequently, if g and p : P → Z are proper maps, then the right-hand term in the last equality is always compact, which proves that g −1 (P ) is π  -proper. Counterexamples to the converse are easily found applying Proposition 3.1.9.2-(1), for example any direct product Y ×K

p2

K

⊕

p1

Y

{•}

with K compact and Y noncompact. 21. Exercise 3.1.9.3-(2) (p. 64). To see that p2 is open, it suffices to show that for all (x1 , x2 ) ∈ X1 ×B X2 and for all pair of open neighborhoods xi ∈ Vxi ⊆ Xi , the set Wx2 of elements x2 ∈ Vx2 verifying (Vx1 × {x2 }) ∩ (X1 ×B X2 ) = ∅, is a neighborhood of x2 . In the present case, where π1 is open, the set π1 (Vx1 ) is open in B and the set Wx2 := π2−1 (π1 (Vx1 )) ∩ Vx2 fulfills the requirements. As counterexample for the ‘closed’ statement, take πi := R → {•}. The maps πi are closed and R ×{•} R = R × R, but it is well-known that the projection maps pi : R2 → R are not closed. The hyperbola H := {x1 x2 = 1} is closed in R2 while its projections pi (H ) := R  {0} are not closed in R. 22. Exercise 3.1.9.3-(3) (p. 64). In a locally compact space Y , a subset A ⊆ Y is closed if and only, for all K ⊆ Y compact, the set A ∩ K is compact. As a consequence, a map f : W → Y is closed if and only if, for all A ⊆ W closed, and all K ⊆ Y compact, the set f (A) ∩ K = f (A ∩ f −1 (K)) is compact, but this

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is always the case if f is proper map. We can thus conclude that a proper map is universally closed, since, by Proposition 3.1.9.2-(1), if π1 : X → B is proper, then for all π2 : X2 → B the map p2 : X1 ×B X2 → X2 is proper, hence closed. Conversely, let π1 : X → B be universally closed and let K ⊆ B be compact. Then, for π2 : K ⊆ B the inclusion map, the map p2 : X1 ×B K → K is universally closed, and since K → {•} is universally closed also, their composition, i.e. the constant map X1 ×B K → {•}, is universally closed. But a constant map on a locally closed space W → {•} is universally closed (if and) only if W is compact. Indeed, & the Alexandroff compactification of W . The map p2 : W ×{•} W &→W & denote by W & is a closed map and the image of the diagonal $W ⊆ W × W , which is closed since the spaces are Hausdorff, is also closed, hence compact. But this image is homeomorphic to $W which, in turn, is homeomorphic to W . We have thus proved that X1 ×B K is compact. We can now conclude that π1 is proper. Indeed, for K  ⊆ B compact, the subspace K := K  ∩ π1 (X) is also compact since π1 (X) is closed. But then, π1−1 (K  ) = π1−1 (K) = X1 ×B K, which we already showed it is compact. 23. Exercise 3.1.9.3-(4) (p. 65). Applying Proposition 3.1.9.2-(2), we get the factorization ϕ = p2 ◦ ξ through the fiber product: π −1 (U )

F

ϕ

π

π −1 (U ) π

{• }

U

U

ξ

U ×F

p2

F

p1

U

{•}

where for ξ , a priori a continuous bijection, to be a homeomorphism it is necessary an sufficient that it be proper, i.e. that ϕ −1 preserves properness after Proposition 3.1.9.2-(2a). In the category of manifolds, to see that ξ is a diffeomorphism it suffices to show that it is locally invertible, i.e. étale, which is clearly equivalent to ask that the restrictions of ϕ to fibers are all étales, since the first coordinate π already provides the full horizontal tangent spaces. 24. Exercise 3.1.10.1 (p. 65). We have |φ β| ⊆ π −1 (|φ|) ∩ |β|, so that if β ∈ c (E), then φ β ∈ c (E). Conversely, by Urysohn’s lemma, given K ⊆ B compact, there exists φ ∈ 0c (B) such that K ⊆ |φ|. Whence π −1 (K) ∩ |β| is compact since closed and contained in |φ β|. 25. Construction of a Thom form 3.2.3 (p. 81). Given an oriented fiber bundle of manifolds (E, B, π, M) with connected fiber M of dimension dM , we show that there always exists a differential form ζπ ∈ dcvM (E) such that  M

π ∗ (α) ∧ ζπ = α ,

∀α ∈ (B) .

Let U := {Ui }i∈I be a trivializing cover of B, and fix some oriented family of trivializations {Φi : π −1 (Ui ) → Ui × M}i∈I . We have the Cartesian diagram of

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341

oriented fiber bundles of fiber M Φ

π −1 (U )

U ×M

p2

M

p1

π

U

U

and, after Proposition 3.2.1.3, a commutative diagram cv (U

× M)[dM ]

Φ∗

cv (π

−1 (U

))[dM ]

M

M

)

).

Hence, if we define ζπ,i := Φi∗ (p2∗ (ζM )) ∈ dcvM (π −1 (Ui )) , where ζM ∈ dc M (M) is a differential form representing ζM , we get the equality  M

π ∗ (αi ) ∧ ζπ,i =

 M

π ∗ (αi ) ∧ p2∗ (ζM ) = αi ,

∀αi ∈ (Ui ).

(E.9)

Now, taking a partition of unity {φi }i∈I subordinate do U , we define ζπ :=

 i

π ∗ (φi ) ζπ,i ∈ dcvM (E) .

By (E.9), we then have, for all α ∈ (B),  M

π ∗ (α) ∧ ζπ =

 M

=

π ∗ (α) ∧

  i M

 i

π ∗ (φi ) ζπ,i

π ∗ (φi α) ∧ ζπ,i =

 i

φi α = α .

26. Exercise 3.7.1-(1) (p. 104). The sequence is a complex since, the support |β| of β ∈ c (U ), being compact disjoint to F , there exists an open neighborhood M (β) = 0. W ⊇ F disjoint to |β|, hence RW M ) ⊆  (U ) are clear. The injectivity of j! and the fact that ker(RF c M is onto, let now W ⊇ F be relatively compact, and let {φ , φ } To see that RF 1 2 be a partition of unity relative to the open cover {W, M  F }. Notice that |φ1 | is compact since closed in W which is compact. Then, for any open subspace V ⊇ F

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V M and any α ∈ (V ), we have φ1 α ∈ c (M) and RW ∩V (φ1 α) = RW ∩V (α). Hence, M. the surjectivity of RF

27. Exercise 3.7.1-(1) (p. 104). The Tubular Neighborhood Theorem states that every submanifold N in M has an open neighborhood V diffeomorphic to the normal bundle of N in M (Sect. 3.2.3.1). Since the statement is also true for any open neighborhood V ⊇ F in place of M, the collection FT of tubular neighborhoods of F is cofinal in F , in which case the natural map lim (V ) → lim F (V ) −→ T −→ F is an isomorphism. We can now easily conclude since all the restrictions RVV are quasi-isomorphic and that a filtrant inductive limit of quasi-isomorphisms is a quasi-isomorphism. M is the restriction 28. Exercise 3.7.1-(1) (p. 104). Since the composition RFF ◦ RF ∗ morphism i : c (M) → (F ), we get the short exact sequence of complexes: i∗

j!

0 → c (U ) −−→ c (M) −−→ (F ) → 0 inducing the long exact sequence 3.85. 29. Exercise 3.7.1-(1) (p. 105). By the general definition of the connecting morphism corresponding to the short exact sequence i∗

j!

0 → c (U ) −−→ c (M) −−→ (F ) → 0 , we have to lift ω ∈ (F ) to c (M). For this, we multiply π ∗ ω ∈ (B2 ) by a Urysohn function φ : M → R ≥ 0 of compact support equal to 1 on a neighborhood of F . The differential form φ π ∗ ω has now compact support and verifies i ∗ (φ π ∗ ω) = ω. When ω the snake diagram 0

c (U )

j!

i∗

c (M)

0

∗

φπ ω

ω

d ∗

dφ ∧ π ω

d ∗

dφ ∧ π ω

0

gives the connecting morphism c : H (F ) −→ Hc (U )[1] ,

ω → dφ ∧ π ∗ ω .

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343

Looking closely, we see that in fact dφ ∧ π ∗ ω ∈ c (B2  F ), where we have a Cartesian diagram of fibrations B2

R×S

F

p

p2

S

S

so that we can apply the study of the zero section of a vector bundle in Sect. 3.6.5. The Gysin map σ! : H (S ) → Hc (B2  F ) is therefore an isomorphism with inverse the morphism p! of integration along fibers (cf . 3.6.2). Hence,   dφ ∧ π ∗ ω = σ! p! (dφ ∧ π ∗ ω) = σ! p! (dφ ∧ p∗ (π ∗ ω))   = σ! p! (dφ) ∧ π ∗ ω) = σ! π ∗ ω) after the projection formula (2.6.2.1-(2)). To finish, we have still to pushforward dφ ∧ π ∗ ω from c (B2  F ) to U , which is obviously done through the extension by zero morphism j! . 30. Exercise 3.7.1-(2) (p. 105). This is almost tautological. By duality, we have the long exact sequence j! ∨

i ∗∨

c∨

· · · → Hc (F )∨ −−→ Hc (M)∨ −−→ Hc (U )∨ −−→ Hc (F )∨ → · · · that we can link through Poincaré adjunctions, H (F )[dF ] ID F

Hc (F )∨

i∗

(I) i ∗∨

H (M)[dM ] ID M

Hc (M)∨

j∗

(II) j! ∨

H (U )[dM ]

δ [1]

ID F [1]

ID U

Hc (U )∨

H (F )[dF ][1]

c∨ 1

Hc (F )∨ [1]

Where i∗ is the Gysin morphism in cohomology corresponding to the closed embedding i : N ⊆ M. The subdiagrams (I) and (II) are commutative thanks to the exchanges of Poincaré adjoints j! ↔ j ∗ and i ∗ ↔ i∗ (see 2.5.1.1-(4)). The connecting morphism δ is then the left Poincaré adjoint to c.

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E Hints and Solutions to Exercises

31. Exercise 3.7.1-(2) (p. 106). The connecting morphism δ is the left Poincaré adjoint to c which was decomposed in (1) as H (F )

π∗

Hc (S )

σ! [−dS ]

Hc (B2

F )[1]

j ! [1]

Hc (U )[1]

c

By the theorem of adjoint morphisms 2.5.1.1-(4), we see that δ is obtained by successively applying • the adjoint to the zero extension j! , i.e. the restriction j∗ • the adjoint to Thom isomorphism σ! , i.e. the restriction i ∗ , • the adjoint of the pullback π ∗ , i.e. the integration along fibers π! . Hence, the announced formula H (U ) # α → δ(α) =

 S

α

S

∈ H (F ) .

32. Exercise 3.7.2-(1) (p. 106). Both result by adjunction. The first:  M

Gr(f )∗ (δ∗ (1)) =

 M

Gr(f )∗ (δ∗ (1)) ∪ 1 =

 M×M

δ∗ (1) ∪ Gr(f )∗ (1) .

The second:  M×M

δ∗ (1) ∪ Gr(f )∗ (1) = (−1)dM .dM = (−1)dM



δ M



Gr(f )∗ (1) ∪ δ∗ (1) Gr(f )∗ (1) .

M×M

 ∗

33. Exercise 3.7.2-(2) (p. 106). If f has no fixed points, then the map Gr(f ) factors through the open subspace (M × M)  $M : M

Gr (f )

(M × M)

j M

M ×M

Gr(f )

where Gr (f ) denotes the restriction of Gr(f ) and j is the open inclusion. But then Gr(f )∗ = Gr(f )! = j! ◦ Gr (f )! , where j! is the extension by zero, in which case δ ∗ ◦ j! = 0 and δ ∗ ◦ Gr(f )∗ = Gr(f )! = (δ ∗ ◦ j! ) ◦ Gr (f )! = 0 .

E Hints and Solutions to Exercises

345

Hence, f = 0, after the last equality in 3.89. 34. Exercise 3.7.2-(3) (p. 106). Since δ∗ (1) ∈ H dM (M × M) and that by Künneth’s theorem we have H (M × M) = H (M) ⊗ H (M), the decomposition of δ∗ (1) in the basis {ei ⊗ ej } looks like: δ∗ (1) =

 [ei ]+[ej ]=dM

xi,j ei ⊗ ej .

The coefficients xa,b are given by the formula xa,b = (−1)[eb ][ea ]

 M×M

(ea ⊗ eb ) ∪ δ∗ (1) ,

which gives, by adjunction, 

xa,b = (−1)[eb ][ea ]

M

= (−1)[eb ][ea ]

δ ∗ (ea ⊗ eb )



ea ∪ eb = { (−1)[ea ] if a = b,0 if a = b.

M

Hence, the announced formula: δ∗ (1) =

 i∈I

(−1)deg(ei ) ei ⊗ ei .

Therefore,  M

δ∗ (1)

$



=



(−1)k

 ei ∪ ei =

[ei ]=k

0≤k≤dM



 (−1)k dim H k (M) .

0≤k≤dM

35. Exercise 3.7.2-(4) (p. 106). After 3.7.2-(3), we have (f ) :=

 M

Gr(f )∗ δ∗ (1) =

=



(−1)k

0≤k≤dM

 0≤k≤dM



f ∗ (ei ) ∪ ei



[ei ]=k

 (−1)k Tr f ∗ : H k (M) → H k (M) .

The fixed points criterion results then from 3.7.2-(2). 36. Exercise 3.7.2.1 (p. 107). 1) By the Lefschetz Fixed point theorem 3.7.2-(3), we have  M

Eu($M ) = χM .

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E Hints and Solutions to Exercises

2) j : M → M is an open embedding of fiber bundles of base M and the orientations of the fibers of π induce an orientation of the fibers of π  . (2) The fibers of π  are clearly homeomorphic to RdM  {0}. (2) The sequence in question contains the exact subsequence: c

j!

ρ

dM dM −→ H dM −1 (M) − → → Hcv (M ) −→ Hcv (M ) −→ H dM (M) −→



0

R # Φπ  −→ Eu($M ) dM where π : M → M is a vector bundle, so that δ! : H 0 (B) → Hcv (M ) is the Thom isomorphism (3.6.5.1). The restriction map ρ is therefore necessarily injective, since ρ(δ! (1)) = Eu($M ) = 0, after (1). As a consequence, j! = 0 and the connecting morphism c is surjective. (2) After (2), it suffices to take any manifold M such that χM = 0 and H dM −1 (M) = 0. For example, an even dimensional sphere M := S2m .

Ch. 4. Equivariant Background 37. Exercise 4.2.2.1 (p. 120). (3)⇒(1)⇒(2) is obvious. (2)⇒(1). Let A be the set whose members are the sets A of simple submodules S ⊆ V , such that the sums



A := S∈A S are direct sums. Notice that A is nonempty as it contains the singletons {S}. We endow A with the partial order of set inclusion. We claim that (A, ⊆) is an inductive poset. Indeed,  let C := {Aa }a∈(A,≤) be a totally ordered subset in (A, ⊆). Express the union a∈A Aa as the family of its elements {Sb }b∈B . If we have b∈B sb = 0 with almost all sb ∈ Sb equal to zero, then the same vanishing expression would be verified in some Aa , for a ∈ (A, ≤) big enough, in which case sb = 0 for all b ∈ B. As a consequence, the set  a∈A Aa is a member of A and is also an upper bound for C. We can therefore apply the Zorn Lemma, and state that there exists a maximal element A ∈ (A, ⊆). We claim every simple submodule

that the sum A contains

S ⊆ V . Indeed, if S ⊆ A then the sum S + A is direct, and A  {S} is a member of (A, ⊆) strictly greater than A, which is not possible. We have thus proved that if V is a sum of simple submodules, then there exists a

set A of simple submodules of V , such that  the sum A is a direct sum containing every simple submodule of V . Hence V = A, and V is semisimple. (1,2)⇒(3,4). If V is a sum of simple modules, then the same holds for any quotient ν : V → Q, and Q is semisimple after (2⇒1). Moreover, in proving the semisimplicity of Q, we consider the poset (A(ν), ⊆), whose members are the sets A of simple

submodules S ⊆ V such that ν(S) ⊆ Q is simple, and such that the sum S∈A ν(S) is a direct sum in Q. This condition automatically implies



that A := S∈A S is a direct sum in V , and, therefore, that the restriction ν : A → ν( A) is an isomorphism. In particular, if A is maximal in (A(ν), ⊆), then A is a semisimple complement to ker(ν), which proves (4). (4)⇒(1). By induction on dim V . If dim V ≤ 1 semisimplicity is obvious. Otherwise, the module V is either simple, or it has a proper submodule 0  W  V

E Hints and Solutions to Exercises

347

with some complement W  . But then, since both, W and W  , verify (4) and are of strictly smaller dimensions than V , we can conclude that they are semisimple, hence, that V is semisimple. 38. Exercise 4.2.5.2 (p. 122). Denote by ν = M → M/N the canonical projection. Given M  ⊆ M such that N ⊆ M  and that M  /N is finite dimensional, let ϕ : M  /N → M lift the inclusion M  /N ⊆ M/N. The induced morphism ν : ϕ(M  /N) → M  /N is then isomorphism, and we have ϕ(M  /N ) ⊕ N = M  . Conversely, let V be finite dimensional and let ϕ : V → M/N be given. Since N is of finite codimension in M  := ν −1 (ϕ(V )), it admits a complementary g-submodule H ⊆ M  by hypothesis. But then the restriction of ν induces an isomorphism −1 νH : H → ϕ(V ), and νH ◦ ϕ : V → M lifts ϕ. 39. Basic Elements in Cartan-Weil Complexes (Sect. 4.3.2) (p. 125). To show that the map  (g) ⊗ S(g ∨ ) ⊗ C = S(g ∨ ) ⊗ C Ξ  : (g) ⊗ S(g ∨ ) ⊗ C hor → +  (g) ⊗ S(g ∨ ) ⊗ C is a θ -equivariant isomorphism we can disregard the S(g ∨ ) factor (since interior products are S(g ∨ )-linear), which reduces us to simply showing ⎧ (i) ((g ∨ ) ⊗ C)hor ∩ (+ (g ∨ ) ⊗ C) = 0 ; ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

(ii) ((g ∨ ) ⊗ C)hor + (+ (g ∨ ) ⊗ C) = (g ∨ ) ⊗ C .

(iii) ((g ∨ ) ⊗ C)hor is θ -stable.

(i) For d ∈ Z, defining Cd :=

i≤d

C i , one gets an increasing -filtration

 ((g ∨ ) ⊗ C) := · · · ⊆ ((g ∨ ) ⊗ C)d ⊆ ((g ∨ ) ⊗ C)d+1 ⊆ · · · , by graded subspaces which are stable by interior products ι(X), for all X ∈ g. The action of ι(X) on Gr ((g ∨ ) ⊗ C) ) is then given by the map ι(X) ⊗ 1 : (g ∨ ) ⊗ C → (g ∨ ) ⊗ C , so that  Gr ((g ∨ ) ⊗ C) )hor = (g ∨ )hor ⊗ C . But, elementary reasons give the equalities (g ∨ )hor =

, X∈g

ker(ι(X) : (g ∨ ) → (g ∨ )) =

, X∈g

(X⊥ ) = ({0}) ,

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E Hints and Solutions to Exercises

where X⊥ := {λ ∈ g ∨ | λ(X) = 0}, which lead us to conclude that (+ (g ∨ ) ⊗ C)hor = 0 , since the -filtration is regular. (ii) We need only check the inclusion 1 ⊗ C ⊆ ((g ∨ ) ⊗ C)hor + (+ (g ∨ ) ⊗ C) , which is obvious after the equality   

 ei ⊗ ι(ei )ω + ei ⊗ ι(ei )ω , 1⊗ω = 1⊗ω− i

i

where {ei } is a basis of g, of dual basis {ei }, and where the first term in the r.h.s. is clearly horizontal, as it is an obvious zero of the interior products ι(ej ). (iii) If  ∈ ((g ∨ ) ⊗ C)hor , then, for all X, X ∈ g, we have  ι(X )θ (X)( ) = ι(X )θ (X) − θ (X)ι(X ) ( ) = ι([X , X])( ) = 0 , 1

where (=1 ) is condition 4.13-(iii) for g-dgm’s (in Sect. 4.2.3). We have thus proved that θ (X)( ) ∈ ((g ∨ ) ⊗ C)hor , for all X ∈ g, as announced. 40. Exercise 4.3.4.2 (p. 128). The statement results by proving that, given two Gmodules N ⊆ M, we have N |M as G-modules, if and only if we have N |M as G0 -modules. Recall that the subgroup G0 is an open normal subgroup of G, and the quotient W := G/G0 is a finite group since G is compact. Recall also that on the category of R[G]-modules, we have natural identifications of bifunctors:  HomG (−, −) = HomR (−, −)G = HomR (−, −)G0 W = HomG0 (−, −)W . Furthermore, given a G-module V and a G0 -module V  , we have canonical isomorphisms of functors from Mod(G) to Vec(R): 

(i)

HomG (V , −)  HomG0 (V , −)W ,

(ii) HomG (R[G] ⊗G0 V  , −)  HomG0 (V  , −) .

(E.10)

– Assume that N |M as G0 -modules. Let V be a finite dimensional G-module. Since V is also a G0 -module, the map HomG0 (V , M) → HomG0 (V , M/N ) is surjective, but then HomG0 (V , M)W → HomG0 (V , M/N )W is surjective too, since the functor W -invariants (−)W is exact on the category Mod(R[W ]). Whence we have N |M as G-modules, by isomorphism (i)-(E.10).

E Hints and Solutions to Exercises

349

– Assume that N|M as G-modules. If V  is a finite dimensional G0 -module, then V := k[G] ⊗G0 V  is a finite dimensional G-module (since W is finite), the map HomG (V , M) → HomG (V , M/N) is surjective, and we can conclude that N |M as G-modules, by isomorphism (ii)-(E.10). 41. Exercise 4.7.2.1 (p. 150). For every space X, the set Ω d (X; k) consists of all the set-theoretic maps f : Xd+1 → k, and for Z ⊆ X, the restriction morphism Ω d (X;k) → Ω d (Z; k) is simply the restriction of maps from Xd to Z d . If X = ↑n∈N Zn is an increasing cover, any d-tuple in Xd , belongs to Znd for  d d some n ∈ N, so that X = ↑n∈N Z n . The bijectivity of (i,ii) in (4.52) then immediately follows. For (iii), let M = ↑k∈N Kk be an increasing cover of M by compact subspaces. The subspaces Kn,G (n) ⊆ MG (n) are compact and the d (M(n); k). The family {1 } indicator function 1n of Kn,G (n)d belongs to Ωcv n n∈N d → {1}, but is a projective family whose limit is the constant map 1∞ : MG d (M ; k) if and only if M is compact. 1∞ ∈ Ωcv G 42. Exercise 4.10.1.3 (p. 174). (1,2) These are straightforward applications of Theorem  g 4.3.3.1-(2). The filtration is given by the subcomplexes Km = S ≥m (g ∨ ) ⊗ c(α) ˆ for all m ∈ N. The rest of the argument is the same as in the proof of 4.3.3.1-(2). (3) The hint refers to the fact that “the spectral sequence of the mapping cone of a morphism of filtered complexes is the mapping cone of the induced morphism on the corresponding spectral sequences ”. The exercise is then immediate. 43. Proof of Proposition 5.1.2.2-(3) (p. 178). Let A := S(g ∨ )g = R[X1 , . . . , Xr ] and let W := G/G0 . Following Hilbert’s Syzygy Theorem, for every AW -graded module M, the module A ⊗AW M admits a resolution of A-graded modules 0 → Lr → · · · → L1 → L0 → A ⊗AW M → 0 ,

(E.11)

where the Li ’s are free A-modules. The submodules (Li )W are then free AW modules and, applying the exact functor (−)W to (E.11), we conclude that  dim-projAW (A ⊗AW M)W ≤ r . On the other hand, (A ⊗AW M)W  M since the action of W on M is trivial. Hence, dim-projAW (M) ≤ r for all M ∈ Mod(AW ), and dimh (AW ) ≤ dimh (A).

Ch. 6. Equivariant Euler Classes 44. Exercise 6.5.2.3 (p. 229). Use the projection formula for Gysin morphisms. 45. Exercise 6.5.4.3 (p. 233). A maximal torus T ⊆ SO(3) is the group of rotations around a one dimensional vector subspace L ⊆ R3 . Hence, although dim(R3 )G = 0, we have dim(R3 )T = 1, and then EuG (0, R3 ) = EuT (0, R3 ) = 0, after 6.5.4.2–(3).

350

E Hints and Solutions to Exercises

Ch. 7. Localization 46. Exercise 7.2.2 (p. 236). Let M be an HG -gm. Show that the canonical map Show that M → QG ⊗HG M is injective if and only if M is  torsion-free.  if M is also of finite type, then the natural map Hom•HG M, HG → Hom•HG M, QG induces an isomorphism   QG ⊗HG Hom•HG M, HG  Hom•QG QG ⊗HG M, QG . Apply 7.2.1. 47. Exercise 7.4.1.1–(3) (p. 237). Let λ : V → QG be HG -linear. If V is torsion and v ∈ V , we have P .v = 0 for some P ∈ HG  {0}. But then P .λ(v) = λ(P .v) = 0, in which case λ(v) = 0, because P is invertible in QG . Since this is true for all v ∈ V , we conclude that λ = 0. Conversely, if v ∈ V is not torsion, we have HG  HG · v ⊆ V , and we claim that the linear map λ : HG · v → HG , P · v → P , can be extended to the whole of V . Indeed, applying the localization functor QG ⊗HG (−) to the inclusion ι : HG · v V , we get a commutative diagram ι

HG · v

V jV

jv α

QG = QG ⊗HG HG · v

(E.12)

QG ⊗HG V

ρ

where jV (u) := 1 ⊗ u, jv (P · v) = P ⊗ v, and α is induced by ι. The morphism α is injective, since the localization functor is exact (QG is a flat HG -module). But, as injection between QG -vector spaces, α admits a QG -linear retraction ρ, i.e. ρ ◦ α = id. Then, the HG -linear map ρ ◦ ιV : V → QG is nonzero because it extends jv . We have therefore proved that if V in not a torsion module, then HomHG (V , QG ) = 0 48. Exercise 7.4.1.2–1 (p. 237). For P ∈ HG , let W (P ) := HG /(HG · P ) and take  V := P ∈HG W (P ). 49. On the General Slice Theorem 7.5.1 (p. 238). The G-equivariant maps p

G ×Gx S(x)

G/Gx

c

{•}

where p([g, m]) := [g] and c is the constant map, give rise to maps of Borel constructions p

IEG

G

(G

Gx

S(x))

c

IEG

G

G/Gx

IEG

G •

E Hints and Solutions to Exercises

351

where IEG×G (G×Gx S(x)) = V (x)G = S(x)Gx . Here the space G/Gx is compact, which implies that for a subspace P ⊆ V (x)G being (c ◦ p)-proper is equivalent to being p-proper. Consequently, Ωcv (V (x)G ; k) = Ωcv (S(x)Gx ; k) and the equality HG,c (Vx ) = HGx ,c (S(x)) is proved. 50. Exercise 7.5.2.2 (p. 239). Use the slice theorem. If x ∈ M T , show that the slice S(x) is a strict submanifold of M stable under Gx and that OG (G · S(x)) = OGx (S(x)), then conclude by induction on dim(M). Otherwise, if x ∈ M T , linearize the action as in Proposition 6.5.4.2 and conclude showing that there is a one-to-one correspondence between isotropy groups in the T -space Tx M and subsets of the set of nonzero weights of the linear representation of T on Tx M.

Ch. A. Appendix: Basics on Derived Categories 51. Exercise A.1.4.2 (p. 270). The equivalence (1)⇔(2) is tautological. The equivalence (2)⇔(3) comes from the fact that, since F is additive, it respects the mapping cone. We can then apply 3-(3b) and conclude. 52. Exercise A.1.5.1 (p. 270). Given α ∈ MorC(Ab) (A, B) and h : A[1] → B, let α  = α + h d + d h. The map L(h) : (c(α), ˆ $) → (c(α ˆ  ), $ ) ,

L(h)(b, a) := (b − h(a), a) .

(E.13)

verifies   L(h) $(b, a) = L(h) db + α(a), −da) = (db + α(a) + h(da), −da)   $ L(h)(b, a) = $ b − h(a), a = (db − dh(a) + α(a) + hd(a) + dh(a), −da) = (db + α(a) + h(da), −da)

and (E.13) is a morphism of complexes. For the same reasons, the map L(−h) : (c(α ˆ  ), $ ) → (c(α), ˆ $) is a morphism of complexes, and we can conclude since we have the obvious equalities L(−h) ◦ L(h) = idc(α) ˆ

and

L(h) ◦ L(−h) = idc(α ˆ ) .

The fact that mapping cones are canonically defined in the homotopy category K(Ab) is then a straightforward consequence of the fact that L(h) is uniquely determined by the homotopy h. 53. Exercise A.1.5.3-(1) (p. 272). After Proposition A.1.5.2, applying the functor p ι α ˆ −→A[1] we obtain the long MorK (C, −) to the mapping cone A −→ B −→ c(α) exact sequence p∗

α∗

ι∗

p∗

ˆ −→ MorK (C, A) −→ MorK (C, B) −→ MorK (C, c(α)) ˆ −→ MorK (C, c(α)[−1])

352

E Hints and Solutions to Exercises

where α∗ is injective since α is assumed to be a monomorphism. This immediately implies that p∗ = 0, which, in the particular case where C = c(α), ˆ gives us 0 = p∗ (idc(α) ˆ B) → MorK (c(α), ˆ c(α)) ˆ is ˆ ) = p, and the fact that ι∗ : MorK (c(α), ˆ B) such that ι ◦ σ = idc(α) surjective, hence the existence of σ ∈ MorK (c(α), ˆ , which is the announced section of ι. We have ρ

c(α) ˆ

ι

B

c(α) ˆ ,

id

which allows to conclude that i ∗ : MorK (c(α), ˆ C) → MorK (B, C) is injective, hence, that ι is an epimorphism in K (Ab). By a symmetric argument, replacing α is mono, by ι is epi, and MorK (C, −) by MorK (−, C), one shows that there exists ρ : B → A such that idA = ρ ◦ α, which is the announced retraction of α. When the category K(Ab) is abelian we can further look at the kernel of ρ and the cokernels of ι, and show by standard arguments that the sequence α

0

A

ι

B ρ

σ

c(α) ˆ

0,

is a split sequence. The abelian category K(Ab) is therefore split. Now, if α : X → Y is a morphism in Ab, the morphism α[0] : X[0] → Y [0] is split in K(Ab), which means that we have direct decompositions X[0] = K ⊕ L and Y [0] = L ⊕ M in K(Ab), through which α[0] reads as (ι ◦ p)(k, l) = (l, 0). All things preserved by the cohomology functor X[0] K ⊕L

α[0]

ι◦p

Y [0] L⊕M

X

α

Y [0]

ι◦p

h(L) ⊕ h(M)

h

h(K) ⊕ h(L)

the morphism α is split in Ab, and this ends the proof that Ab is split. 54. Exercise A.1.5.3-2 (p. 272). If Ab is a split category every complex C ∈ C(Ab) is homotopy-equivalent to its cohomology (cf . answer of Exercise 2.1.6.1, p. 329) and the fully faithful functor AbZ → K(Ab) which associates with a graded object the corresponding complex with 0 differential, is then essentially surjective, hence is an equivalence of categories. 55. Exercise A.1.5.7-(1) (p. 273). The map  : c(α) ˆ → C is a morphism of complexes since ($(x, y)) = (dx + α(y), −dy) = β(dx) = d((x, y)) . If β(x) ∈ C is a cocycle, we have dx = y for some cocycle y ∈ A. Hence (x, −y) is a cocycle in c(α) ˆ and (x, −y) = β(x). But then, in cohomology, we get   c h()(x, −y) = y = −h(p) (x, −y) ,

E Hints and Solutions to Exercises

353

where c : h(C) → h(A[1]) is the connecting morphism. We therefore have a commutative diagram h(A)

h(α)

h(B)

h(ι)

h(p)

c(α) ˆ

h(α)[1]

h(A)[1]

h(B)[1]

h

h(A)

h(α)

h(B)

h(β)

−c

h(C)

h(A)[1]

h(α)[1]

h(B)[1]

where we apply the Five Lemma to conclude that h() is an isomorphism.  ×2 In C(Mod(Z)) we can take E := 0 → Z[0] −→ Z[0]  → (Z/2Z)[0] → 0 . ˆ = 0. Then c(×2) ˆ = Z⊕Z is a non-torsion module and HomgrZ (Z/2Z)[0], c(×2) 56. Exercise A.1.5.7-(2) (p. 273). In a triangle T := (A[0], B[0], C[0], p[0], q[0], γ ), any morphism γ : C[0] → A[1] vanishes since the complexes are concentrated in different degrees. Hence γ = 0. But then, if T is an exact triangle, the arguments in Exercise A.1.5.3-(1) show that the exact sequence E is slit in Ab. Conversely, if p ι →A⊕C − → C → 0), and denoting E is split, it is isomorphic to E  := (0 → A − by ζ : C[−1] → A the zero morphism, we have c(ζ ˆ ) = A ⊕ C and the triangle (A[0], (A ⊕ C)[0], C[0], ι[0], p[0], ζ ) is exact since up to a rotation it coincides with the mapping cone of ζ , i.e. with (C[−1], A[0], c(ζ ˆ ), ζ, ι[0], p[1]). 57. Exercise A.1.6.1-(1) (p. 276). 1) The complexes C(m) are all exact hence acyclic. The morphism of complexes f := (0, 0, 0) : C(m) → C(n) is therefore a quasi-isomorphism. Given a morphism of complexes f := (f0 , f1 , f2 ) : C(m) → C(m), consider the following notations 0

f0

0

Z

(m) (m)

h1

f1

Z

νm

Z/(m)

h2 νm

f2

Z/(m)

0 0

For m > 0 and f2 = 0, the morphism f cannot be homotopic to the identity, since then idZ/(m) = νm ◦h2 , which is impossible because h2 = 0. As a consequence, a morphism of complexes g : C(m) → C(0) does not accept any homotopic inverse. Indeed, if g  : C(0) → C(m) is such, then f := g  ◦ g would be homotopic to the identity while f2 = g2 ◦ g2 = 0, because g2 = 0. We have thus proved that C0 is not homotopic to any other C(m). For m > 0 and n > 0, write m = m gcd(m, n) and n = n gcd(m, n). Then, in a morphism of complexes g : C(m) → C(n), the map g1 is the multiplication by a multiple of n , and the same for a homotopic inverse g  : C(n) → C(m). As a consequence, if f := g  ◦ g, the map f1 is the multiplication by a multiple of n m , but then, as f ∼ id, we would have 1 − λn m ∈ im(h1 ) ⊆ (m), which implies that m = 1. A symmetric argument shows that n = 1. Hence m = n.

354

E Hints and Solutions to Exercises

58. Exercise A.1.6.1-(2) (p. 276). Given a monomorphism ι : X → Y in Ab, let Y → → K := coker(ι) denote its cokernel. The short exact sequence: / X /

0

ι

ν

/ Y

// K

/ 0

is an acyclic complex and if it is homotopic to 0, any homotopy X

0 h0

0

ι

Y

ι

Y

ν

0

K h3

h2

h1

X

ν

K

0

uses a section h2 : K → Y of ν : Y → K, i.e. ν ◦ h2 = idK . In that case, the morphism X⊕K →Y ,

(x, k) → x + h2 (k) ,

is an isomorphism with inverse Y →X⊕K,

 y → y − h2 ◦ ν(y), ν(y) .

Now, any morphism α : X → Y in Ab factors through its image im(α), so that we get two short exact sequences: 0 → ker(α) → X → im(α) → 0

and

0 → im(α) → Y → coker(α) → 0 ,

and, after what we have just seen, two direct decompositions X ∼ ker(α) ⊕ im(α)

and

Y ∼ im(α) ⊕ coker(α) ,

through which α clearly reads as (a, b) → (b, 0). Hence α is a split morphism. Conversely, if Ab is a split category, any acyclic complex of Ab is isomorphic to a complex of the form: di−1 di+1 di+2  di C := · · · −−→ Ai−1 ⊕ Ai −−→ Ai ⊕ Ai+1 −−→ Ai+1 ⊕ Ai+2 −−→ · · · ,

where di (a, b) = (b, 0). But then, the morphisms hi+1 : Ai ⊕ Ai+1 → Ai−1 ⊕ Ai ,

hi (x, y) = (0, y) ,

constitute an obvious homotopy idC ∼ 0. 59. Proof of Proposition A.1.6.3 (p. 277). We have to check the conditions (S-1) and (S-2) of multiplicative families of morphisms (Sect. A.1.6.1).

E Hints and Solutions to Exercises

355

(S-1) Given s ∈ S, we have to complete in K(Ab) a diagram D := to a commutative diagram

•

t

X

Y

t

Z

s

s

X Y t

s

Z

.

We begin by completing D, in C(Ab), to the commutative diagram ι1 ◦s

X D Y

t

c(t) ˆ

ι3 (3)

s

Z

t

c(ι ˆ 1 ◦ s)

I

(4) u ι1 (1)

ι2

c(t) ˆ

(2)

X[1]

c(ι ˆ 1)

s[1]

Z[1]

p3

(E.14)

ˆ 1 ◦ s) (Sect. A.1.3). The by successively constructing the cones c(t), ˆ c(ι ˆ 1 ) and c(ι natural morphism u closing the diagram is then a quasi-isomorphism. By the definition A.1.3-(A.4), one has c(ι ˆ 1 ) = Z ⊕ Y [1] ⊕ Z[1] with differential  D(z, y  , z ) = d z + t (y  ) + z , −d y  , −d z . Hence, a commutative diagram of complexes in C(Ab) t

c(ι ˆ 1 ◦ s) c(ι ˆ 1 ) = Z⊕Y [1] ⊕ Z[1] (5) u

X[1]

I

(4) u

s[1] p3

Z[1]

p2

Y [1]

t [1]

Z[1]

where u (y  ) = (0, −y  , t (y  )). Now, the key point here is that although it is easy to see that u is a quasi-isomorphism (exercise), it cannot be composed with u, in C(Ab), since it goes in the wrong direction. There is nothing to do about this since simple counterexamples show that S is not a multiplicative collection of morphisms in C(Ab) (cf . p. 277). Nonetheless, u has a homotopic inverse, viz. the projection p2 : c(ι ˆ 1 ) → Y [1], p2 (z, y  , z ) := y  , and while the morphisms t[1] ◦ p2 and p3 do not coincide in C(Ab), they do coincide in K(Ab) since they are homotopic (exercise). We can therefore set s  := p2 ◦ u in K(Ab) to complete the diagram D as needed. The statement (S-1) for shapes

s

• t •

•

follow by the same ideas.

(S-2) We limit ourselves to showing which steps to follow, leaving the details as exercises. We have to show that in K(Ab) if s ◦ u = 0 for some s ∈ S, then there is s  ∈ S such that u ◦ s  = 0.

356

E Hints and Solutions to Exercises X u

Y s

We complete the diagram following diagram in C(Ab)

Z

adding cones, in order to construct the

c(t)[−1] ˆ q

X

u 0

Y

ι

c(u) ˆ := Y ⊕ X[1]

s :=p2 ◦q p2

X[1]

u[1]

Y [1]

s t

Z The morphism t : Y ⊕ X[1] → Z is defined as t (y, x  ) := y + h(x  ) , where h : X[1] → Z is a homotopy to zero for the morphism s ◦ u : X → Z. The other arrows are those of the corresponding mapping cones. • Show that t is a well-defined morphism of complexes. • Show that t ◦ ι is homotopic to s. • Show that s  := p2 ◦ q is a quasi-isomorphism such that u ◦ s  = 0. The converse statement (u ◦ s = 0) ⇒ (s  ◦ u = 0), follows by the same ideas. 60. Counterexample in A.1.6.3 (p. 277). Let X = 0 and assume there is some commutative diagram

Y s ι X

0 c(id ˆ X)

, where s a quasi-isomorphism. Then Y = 0

and s = 0. On the other hand, since c(id ˆ X ) = X ⊕ X[1] and ι(x) = (x, 0), we have ι ◦ s = 0, contrary to the vanishing of the composition Y → 0 → c(id ˆ X ). In K(Ab) the diagram

X idX X

0 ι

is commutative, i.e. the morphism ι :

c(id ˆ X)

X → c(id ˆ X ) := X ⊕ X[1], ι(x) := (x, 0), is homotopic to zero. Indeed, if we set h : X → c(id ˆ X )[−1] := X[−1] ⊕ X ,

h(x) = (0, x) ,

we get the diagram: X h

X[−1] ⊕ X

ι

X ⊕ X[1]

where $(x, x  ) := (dX x + x  , −dX x  ). Whence,

dX h

X[1]

E Hints and Solutions to Exercises

(h ◦ dX + $ ◦ h)(x) =

357

(0, dX x) + $(0, x) =

(0, dX x) + (x, −dX x) = ι(x) ,

and ι = 0 in K(Ab) as expected. 61. Exercise A.1.6.6 (p. 278). Given a path representing a morphism α in D(Ab) s1

t1

s

t

(C, d) ←−− • −−→ • ←−− • −−→ · · · ,

(E.15)

the exchange property (S-2) for multiplicative families (Sect. A.1.6.1) tells us that there exist s2 ∈ S en t2 , such that t1 ◦ s1−1 = s2−1 ◦ t2 . The path (E.15) is then equivalent by A.1.6.4-(A.19) to the path t2

s2

s

t

(C, d) −−→ • ←−− • ←−− • −−→ · · · where we can compose s and s2 in a single quasi-isomorphism s2 ◦ s, hence decreasing the number of steps in the path representing α in D(Ab). An obvious induction then finishes the proof. 62. Exercise A.2.3.6 (p. 285). 1) For X ∈ Ob(Ab), to say that  : X[0] → I := (0 → I0 → I1 → I2 → · · · ) is a quasi-isomorphism is equivalent to say that the complex 0

0 → X −→ I0 → I1 → I2 → · · · is acyclic. Therefore, if F is left exact, the sequence F (0 )

0 → F (X) −−−→ F (I0 ) → F (I1 ) → F (I2 ) → · · · is left exact, or, in other terms, the induced morphism X : F (X) → H 0 (F (I)) is an isomorphism. As a consequence, if in addition I is a complex of injective objects, then F (I) = IR F (X[0]) and we get the announced natural isomorphism  : F (X) → IR 0 F (X[0]) . The same arguments apply, reversing arrows, for right exact functors. 2) No other changes that the source of the functor IR F which is D− (Ab), and the source of the functor ILF which is D+ (Ab). 63. Exercise B.3.2 (p. 304). Let s ∈ Γ (S; F). After Proposition B.3.1-(2), we know that for each x ∈ S, there exists an open subspace Vx ⊆ X, neighborhood of x, and a global section sVx ∈ Γ (X; F), such  that (sVx )t = st , for all t ∈ S ∩ Vx . In that case, and since the open subspace x∈S Vx is paracompact, there exists a partition

of unity {φx }x∈S subordinate to {Vx }x∈S . Then s˜ := x∈S φx sVx is a well-defined global section such that s˜x = sx for all x ∈ S.

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Glossary

Chapter 2.

Nonequivariant Background

k

an arbitrary field

9

R, C

the fields of real and complex numbers

9

Vec(k)

the category of k-vector spaces and k-linear maps

9

V∨

alias for Homk (V , k), dual of the vector space V

9

Homgrdk (−, −)

space of graded homomorphisms of degree d

10

graded space of graded homomorphisms

10

GV(k)

category of graded k-vector spaces

10

Vec(k)Z

alias for GV(k)

Homgr∗k (−, −)

GV+ (k), GV− (k), GVb (k), . . . , subcategories of bounded graded vector spaces Endgr∗k (−)

10 10

algebra of graded endomorphisms of a graded space

11

g

Lie algebra of a Lie group G

11

S(g ∨ )g

algebra of g-invariant polynomials functions over g

11

G

alias for S(g ∨ )g

11

DGV(k)

the category of complexes of k-vector spaces

13

C(Vec(k))

alias for DGV(k)

13

h(V , d)

cohomology of V ∈ DGV(k)

13

h

: DGV(k)  GV(k), the cohomology functor

13

[−] Hom•k (−, −)

shift functors

13

functor on DGV(k)

14

functors on DGV(k)

14

(−)∨ (V , d)∨

alias for Hom•k (−, k[0]), duality functor

16

:= (V ∨ , D), dual complex

17

Man

category of manifolds and differentiable maps

18

(− ⊗k −)•

Manpr

category of manifolds and differentiable proper maps

18

(−), c (−),

: Man  DGV(R) de Rham differential forms functors

18

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Arabia, Equivariant Poincaré Duality on G-Manifolds, Lecture Notes in Mathematics 2288, https://doi.org/10.1007/978-3-030-70440-7

363

364

Glossary

H (−), Hc (−) G- Man

: Man  ModN (R) de Rham cohomology functors

18

category of G-manifolds

19

G- Manpr

category of G-manifolds and proper maps

19

[M]

orientation of M

20

dc M (M) → R, integration operation

21

∧

exterior product of differential forms

23

·, ·M

Poincaré pairing

23

∪

cup-product of cohomology classes

23

ID M

left Poincaré adjunction for complexes

23

DM

left Poincaré adjunction in cohomology

24

ζ[M]

30

ID M

the fundamental class of the oriented manifold (M, [M])   : c (M)[dM ], d) −→ (M)∨ , −D , right Poincaré adjunction in complexes

 DM

right Poincaré adjunction in cohomology

38

f!

: Hc (M)[dM ] → Hc (N )[dN ], Gysin morphism

39

(−)!

: Manor  GV(R), Gysin functor

40

f∗

: H (M)[dM ] → H (N )[dN ], Gysin morphism

45

(−)∗

: Manor pr  GV(R), Gysin functor for proper maps

46

[M]

37

Chapter 3.

Poincaré Duality Relative to a Base Space

X1 ×(π1 ,π2 ) X2

fiber product of two maps πi : Xi → (π1 , π2 )

51



inside a diagram, it indicates a Cartesian sub-diagram

53

h−1 (−) : TopB  TopB  , pullback or base change functor

54

cv (E)

complex of compactly supported differential forms

65

π!  E ⊆ π∗  E

sheaves on B of  B -differential graded modules

68

H(π! ) := H(π!  E ) graded sheaf cohomology of π!  E

68

H(π∗ ) := H(π∗  E ) graded sheaf cohomology of π∗  E

68

F  G ∈ Sh(X × Y ) external tensor product of F ∈ Sh(X) and G ∈ Sh(Y )

70

ID U,V relative left Poincaré adjunction for complexes :  (E)[d ] → (B) , integration along the fiber cv M M

75

relative left Poincaré adjunction for complexes

77

·, ·B,M Poincaré pairing relative to B M : π!  E [dM ] →  B , sheafification of integration along fibers  ID B,M π∗  E → Hom • B π!  E ,  B , sheafification of the left Poincaré

77

ID B,M

73

78 83

adjunction (−)∗

: Fibor B,pr  D(DGM((B), d)), Relative Gysin functor for proper maps

96

(−)!

: Fibor B  D(DGM((B), d)), Relative Gysin functor for arbitrary maps

96

j!

: c (U ) → c (M), extension by zero for j : U ⊆ M open

100

Gr(f )

: M → M × M, the graph map x → (f (x), x)

106

L(f )

∈ H dM (M) , the Lefschetz class

106

Glossary

365

(f )

the Lefschetz number

Chapter 4.

Equivariant Background

106

dg

differential of the Cartan complex

G (M)

the Cartan complex of the G-manifold M

113

XG

alias for IEG ×G X, the Borel Construction of a G-space X

114

KG (X)

equivariant K-theory of G-space X

115

Vg

subspace of g-invariant elements of a g-module V

120

112

g-dgm/dga

alias for g-differential graded module/algebra

120

C := (C, d, θ, ι)

a g-differential graded module/algebra

120

DGM(g)

category of differential graded g-modules over R

121

DGM(g)

category of differential graded g-modules over R

121

(−)|(−)

g-split inclusion of g-modules

122

S(g ∨ )

the ring of real polynomial functions on g

124

Cg

the Cartan Complex associated with a g-dgm C

126

dg

126

Hg (C)

the Cartan Differential  alias for h Cg , dg , g-equivariant cohomology of C

Y(−)

fundamental vector field on a G-manifold associated with Y ∈ g

129

Endgr−1 GV(k) ((M)),

126

ι

:g→

L(Y)

Lie derivative with respect to the vector field Y

130

θ

: g → EndgrGV(k) ((M)), Lie derivative representation

130 130

interior product representation

130

((M), d, θ, ι)

g-dga of differential forms

(c (M), d, θ, ι)

g-dga of differential forms compactly supported

130

(G (−), dG )

Cartan complex functor

130

(G,c (−), dG )

Cartan complex functors for compact supports

130

HG (−), HG,c (−)

G-equivariant cohomology functors

130

quotient of IE × X under the diagonal action of G

144

IE ×G X IEG

universal fiber bundle for G

146

IEG(n)

n-th approximation of universal fiber bundle for G

146

IBG

:= IEG/G, classifying space for G

146

IBG(n)

n-th approximation of classifying space for G

146

O(r)

group of orthogonal (r × r)-matrices

147

(−)G

(IEG × M)/G, Borel construction functor

148

HG,N (M)

local equivariant (de Rham) cohomology

171

(G,N (M), dG )

local Cartan complex of M relative to N

172

Chapter 5.

Equivariant Poincaré Duality

A

graded k-algebra

175

Homgr∗A (−, −)

graded space of graded homomorphisms of A-gm’s

175

GM(A)

category of A-graded modules

175

DGM(G )

category of G -differential graded modules

179

366

Glossary

Hom•G (−, −)

(− ⊗G −)• Hom•G (−, G )

: DGM(G ) × DGM(G )  DGM(G )

180

: DGM(G ) × DGM(G )  DGM(G )

180

: DGM(G )  DGM(G ), duality functor

180

: DGM(G )  GM(G ), forgetful functor

180

Tot⊕ , Tot

: C(GM(G ))  DGM(G )

183

Hom•G (−, −)

: C(GM(G )) × C(GM(G ))  C(GM(G ))

184

(− ⊗G −)• IR F, ILF

: C(GM(G )) × C(GM(G ))  C(GM(G ))

186

: GM(G )  K(Ab), right and left derived functors

187

C ," (Ab)

alias for C (C" (Ab))

188

IR Hom•G (−, −)

: D(GM(G )) × D+ (GM(G ))  D(GM(G ))

191

(−) ⊗ILG

: D(GM(G

o

(−)

Exti,∗ G (−, −) Tor∗G ,i (−, −)

)) × D− (GM(

G ))

 D(GM(G ))

: D(GM(G )) × D+ (GM(G ))  GM∗ (G ) : D(GM(G

)) × D+ (GM(

G ))

∗

 GM (G )

•

IR HomG (−, G ) : D(GM(G ))  D(GM(G )) , derived duality functor IR Hom•G (−, G ) : D(DGM(G ))  D(DGM(G )) , derived duality functor

191 191 191 192 192

Exti,∗ G (−, G )

: D(DGM(G ))  GM∗ (G )

192

·, ·M,G

: G (M) × G,c (M) → G , equivariant Poincaré pairing

201

·, ·M,G

: HG (M) × HG,c (M) → HG , idem in cohomology  : G (M) → Hom•G G,c (M), G , equivariant Poincaré left adjunction  : G,c (M) → Hom•G G (M), G , equivariant Poincaré right adjunction  : HG (M)[dM ] → Hom•HG HG,c (M), HG , equivariant Poincaré duality morphism is cohomology

ID G,M ID G,M D G,M

201 201 201 204

Chapter 6.

Equivariant Gysin Morphism

f!

: HG,c (M)[dM ] → HG,c (N )[dN ], equivariant Gysin morphism

212

(−)!

G-Manor  GM(HG ), equivariant Gysin functor

213

f∗

: HG (M)[dM ] → HG (N )[dN ], equivariant Gysin morphism

214

(−)∗ ResG H

G-Manor pr

Chapter 6.

Equivariant Euler Classes

Φ(N, M)

i∗ (1) ∈ H dM −dN (M), equivariant Thom class of (N ⊆ M)

229

Eu(N, M)

i ∗ i∗ (1) = Φ(N, M), equivariant Euler class of (N ⊆ M)

229

Chapter 7.

Localization

QG ⊗G (−) D G,M

: GM(G )  Vec(QG ), localization functor 235  • : Q ⊗ HG (M)[dM ] → HomQG Q ⊗ HG,c (M), QG , localized 236 equivariant Poincaré duality

 GM(HG ), equivariant Gysin functor

group restriction morphism

214 217

Glossary

367

D G,M

 : Q ⊗ HG (M)[dM ] → Hom•QG Q ⊗ HG,c (M), QG , localized 236 equivariant Poincaré duality

f!

: Q ⊗ HG,c (M) → Q ⊗ HG,c (N ), localized Gysin morphism

236

f∗

: Q ⊗ HG (M) → Q ⊗ HG (N ), localized Gysin morphism

236

Chapter 8.

Changing the Coefficients Field

k

field of arbitrary characteristic

245

dualizing complex on the space (−) category of sheaves of k-vector spaces on X

246

•

ω ( ) (k) −

Sh(X) := Sh(X; k)

246

C(X) := C(X; k)

category of complexes of sheaves on X

246

K(X) := K(X; k)

homotopy category of C(X)

246

D(X) := D(X; k)

derived category of C(X)

246

IB, IBG

classifying space of the compact Lie group G

247

Sh(X ),

f∗ , f!

: Sh(X) 

IR f∗ , IR f!

: D+ (X)  D+ (X ), derived direct images

247

kX

constant sheaf on X with fiber k

247

Ω (X; k)

sheaf of Alexander-Spanier cochains with coefficients in k

248



paracompactifying family of supports

248

ΩG

:= Ω (IB), sheafification of G

249

Ω G (X)

:= π∗ Ω (XG ), sheafification of ΩG (M)

249

Ω G,c (X)

:= π∗ Ω (XG ), sheafification of G,c (M)

249

DGM(Ω G )

category of sheaves of differential graded modules over Ω G

249

direct images

IR Hom •Ω G (−, Ω G ) duality functor in DGM(Ω G )

247

250

ID G,M

left Poincaré adjunction in D DGM(Ω G )

256

D G,M

left Poincaré adjunction in D(DGM(ΩG ))

256

ID G,M

right Poincaré adjunction in D DGM(Ω G )

256

f!

HG,c (M; k)[dM ] → HG,c (N ; k)[dN ], Gysin morphism over k

262

f∗

HG,c (M; k)[dM ] → HG,c (N ; k)[dN ], Gysin morphism over k

263

Appendix A.

Basics on Derived Categories

Ab

abelian category

C(Ab)

category of complexes on Ab

266 266

h : C(A)  AbZ

cohomology functor

266

[m]

: C(Ab) → C(Ab), shift functor

266

c(α) ˆ

cone of a morphism of complexes α

268

HotC(Ab) (−, −)

the group of homotopy classes of complex morphisms

269

K(Ab)

homotopy category of C(Ab)

270

D(Ab)

derived category of C(Ab)

275

multiplicative collection of morphisms in a category C

276

localization of the category C by S

276

: D+ (Ab)  D+ (Ab ), right derived functor of F

283

S Q : C  C[S IR F

−1

]

368

Glossary

ILF

: D− (Ab)  D− (Ab ), left derived functor of F

283

HotDGM(G ) (−, −)

the group of homotopy classes of G -dgm’s

288

D(DGM(G ))

:= K(DGM(G ))[S−1 ], derived category of DGM(G )

289

Appendix B.

Sheaves of Differential Graded Algebras

A ⊗ (−)

A-localization presheaf functor

A ⊗ (−) Sx ⊆ OX (X)

307

A-localization functor

307

set of functions equal to 1 in a neighborhood of x

308



(paracompactifying) family of supports

311

Γ (Y ; −)

functor of global sections with supports in 

312

0

functor on GM(A(X))

313

(A, d) ⊗ (−)

(A, d)-localization presheaf functor

315

(A, d)-localization functor

316

localization of an (A, d)-dgm by S ⊆ A.

317

(A, d) ⊗ (−)

S −1 (M, d)

Index

Underlined page numbers refers to definitions. A acyclic complex, 123, 266, 275 dgm, 318 object, 285 resolution, 285, 312 sheaf, 312 additive functor, 14, 267 adjoint, 4 map, 17 pair, 30, 30, 99 adjointness equivariant Gysin morphisms, 218, 226 adjunction, 1 left nonequivariant, 24 left relative, 73, 77 left,right equivariant, 201 map, 9 right nonequivariant, 37 A-graded algebra, 175 module, A-gm, 175 Alexander-Spanier cochains, 152, 154, 248, 301 algebraic connection, 110 André, M., 111 annihilator of an element, 237 of module, 237 anticommutative, 11, 45, 176, 179, 302, 331 antiderivation, 129 ascending chain property, 33 Atiyah, M.F., 117

augmentation, 188, 193, 206 average, 141, 157, 163

B base change, 55 base space, 50, 51 basic element, 110, 124, 323 Berline, 117 bicomplex, 188 bidual embedding, 17, 29, 37 bilinear map, 9 Borel, A., 114, 117 construction, 93, 94, 114, 148, 149 Bott, R., 117 bounded, below, above, . . . , 10 bounded cohomologically, 254 Bruhat decompostion, 258 Brylinski, J.-L., vi, 1, 119

C cancellable, 271 Cartan, H., 11, 28 complex, 113, 126, 133 differential, 126 isomorphism, 111, 125, 133 model, 113 Cartan-Weil morphisms, 111, 133 Cartesian diagram, 53, 103 category of A-graded modules, 175 of complexes, 182, 266

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Arabia, Equivariant Poincaré Duality on G-Manifolds, Lecture Notes in Mathematics 2288, https://doi.org/10.1007/978-3-030-70440-7

369

370 derived, 277 of dg-vector spaces, 13 of g-dg-algebras, 121 of g-dg-modules, 121 of G-manifolds, 19 of g-modules, 120 of graded vector spaces, 10 of H -graded modules, 176 homotopy, 270, 318 of manifolds, 18 of G -dg-modules, 179 split, 13, 181 Chern–Weil homomorphism, 109, 232 Chevalley, C., 111 isomorphism, 232 restriction theorem, 232 Theorem, 178 classifying space, 148 closed embedding, 104, 226, 227, 241 closed map, 64 cohomologically bounded, 254 cohomology, 13, 266 cokernel, 271 column -filtration, 189 compact vertical supports, 65 completely reducible module, 120 complex of an abelian category, 266 bounded (below, above), 279 double, 188 of objects, 182 simple or total, 189 of vector spaces, 12, 13 concentrated complex, 266 connection algebraic, 110 infinitesimal, 110 projective system of, 160 universal, 110 cup product, 248

D de Rham cohomology, 18 (compactly supported), 19 complex, 18 (compactly supported), 18 derivation, 120, 130 derived category, 47, 84, 277 duality functor, 192 functor (left and right), 187 functor (left), 283

Index functor (right), 283 determinant bundle, 20 diagonal action, 144, 148 embedding, 106 differential, 12, 179, 266 graded module, 179 graded space, 12 direct image, 245, 247 direct image with proper supports, 245, 247 double (cochain) complex of the first quadrant, 189 double complex, 188 dual of a complex, 17 vector space, 9 duality functor, 2, 16, 192 dualizing complex, 246, 254 Duistermaat-Heckman, 117

E embedding bidual, 17, 29, 37 closed, 104, 226, 227, 241 diagonal, 106 open, 226, 227 enough injective objects, 281 projective objects, 282 epimorphism, 271 equivariant cohomology, 49, 130 (compactly supported), 130 with supports in M, 5 cohomology of a g-complex, 126 cohomology with supports, 171 differential form, 130 differential form with supports, 172 Gysin functor for general maps, 213 for proper maps, 214 Gysin morphism for general maps, 212 for proper map, 214 integration, 198, 198 left, right adjunction, 201 map, 19 projection formula for proper maps, 214 Euler characteristic, 106, 107 class, 107, 211, 229 extension by zero, 39, 42, 100, 104, 344

Index extension functor, 192 external tensor product, 70 Exti,•G (−, −), 191 F faithful functor, 11 fiber bundle, 50 principal, 140 fiber product, 51 fiber space, 50 filtrant cover, 42 filtration column, 189 regular, 127, 189, 204, 205 row, 189 finite de Rham type, 33, 204, 210, 213, 236, 256 finite orbit type, 239 first quadrant bicomplex, 189 fixed point theorem, 239 flat module, 310 forgetful functor, 180 formality, 91 of dg-algebras, 91, 293 of dg-moldules, 295 of manifolds, 91 free G -graded module, 176, 193 free presentation, 305 Fubini, 23, 221 functor additive, 14, 279 adjointness of, 294 (A, d)-localization, 316 A-localization, 307 duality, 192 exact (left, right), 14, 267 faithful, full, 11, 310 forgetfull, 180 global section, 305 i’th extension, 192 pullback, 55 shift, 14 fundamental class, 30, 81, 106 fundamental vector field, 129 G g-complex, 120 g-dg-algebra, 121 g-dg-module, 120 g-invariant, 120 G-manifold, 19 g-module, 119

371 g-module morphism, 119 Godement resolution, 71, 80, 152 good cover, 28, 33, 34, 137 graded algebra over a field, 10 over a graded ring, 175 evenly, 11, 176 homomorphism, 175 morphism of degree d, 10 object, 266 positively, 11, 176 vector space, 10 bounded (below, above), 10 graded commutative, 11, 176 graph map, 106 Grothendieck, A., 51, 275, 282 group (orthogonal), 147 g-split complex, 122, 128 Gysin exact sequence, 105 Gysin functor, 3, 39, 40 for proper maps, 46 Gysin morphism, 3, 39 for arbitrary maps, 32, 40 for general maps, 47 for proper maps, 33 (relative) for arbitrary maps, 96 (relative) for proper maps, 96

H harmonic polynomials, 206 hereditary algebra, 295 Hilbert’s Syzygy Theorem, 178 homogeneous component, 303 homotopic inverse, 355 morphisms in C(Ab), 269 to zero in C(Ab), 268 to zero in DGM(G ), 288 homotopy category, 84, 270, 318 quotient, 62, 93, 114, 148, 149 homotopy-equivalent complexes, 269 horizontal element, 124 Houzel, C., 28 Hsiang, W., 117

I incomplete basis theorem, 330 indicator function, 304, 313, 349 infinitesimal connection, 110

372 injective envelope, 178 module, 328 object, 281 G -graded module, 176 resolution, 281 integral, 22 integration along fibers, 75, 343 interior product, 109, 117, 120, 129 isomorphism modulo torsion, 240 isotropy group, 238

J Jacobian, 19 Josua, 119

K K-injective, 293, 318 Koszul differential, 133 sign rule, 15, 179, 331 K-projective, 295 Künneth, 167, 319, 345

L Lefschetz class, number (of a map), 106 (of an equivariant map), 227 Lefschetz fixed point formula, 106 left adjoint, 31 derived functor, 283 Leibniz rule, 124, 317 Leray cover, 28, 334 Leray spectral sequence, 334 Lie algebra, 119 derivative, 109, 117, 130 group, 34 local cohomology, 157 localization of categories, 275, 276 functor, 235 localization formula, 243 localized Cartan complex, 235 equivariant cohomology, 235 locally trivial fibration, 49

Index M manifold, 18 map closed, 64 equivariant, 19 of manifolds, 18 proper, 18, 64 universally closed, 64 mapping cone, 267 Mayer-Vietoris, 26, 33, 36, 239–241 mild space, 8, 302 Mittag-Leffler condition, 155 Möbius Strip, 339 module completely reducible, 120 reducible, 120 semisimple, 120 simple, 120 monomorphism, 271 morphism of additive functors, 280 of A-graded modules, 175 augmentation, 188, 193, 206 of complexes, 13, 266 of fiber bundles, 50 of g-complexes, 121 of g-dg-algebras, 121 of g-dg-modules, 121 of graded spaces, 10 of G -dg-modules, 179 split, 13, 276 of triangles, 272 multiplicative system, 116, 178, 276, 276, 277, 327

N n-acyclic space, 146 n-connected space, 146 nondegenerate pairing, 9, 23, 25, 202, 208 nontorsion element, 237 module, 237 normal space, 18

O G -dg-module, 179 G -duality, 180 open embedding, 226, 227 orbit type, 239 orientable G-manifold, 197 orientation of a fiber bundle, 59, 62

Index of a manifold, 19 oriented atlas of a manifold, 19 family of trivializations, 59, 75 orthogonal group, 147

P pairing nondegenerate, 9 perfect, 9, 25, 33, 208, 236 paracompactifying family, 312 partition of unity, 19, 22, 27, 34, 137, 144, 306, 339, 341, 357 perfect complex, 254 π -proper, 62 Poincaré adjoint pair, 30, 30 adjunction, 252, 253 duality map, 25 duality theorem, 1, 25, 203, 209, 256 left, right adjoint, 31 pairing, 1 (equivariant), 201 (nonequivariant), 23, 25, 33 Poincaré duality morphism (G-equivariant de Rham), 2, 204 (G-equivariant over k), 256 (T -equivariant de Rham), 210 Poincaré left adjunction nonequivariant, 24 nonequivariant relative, 73, 77 Poincaré right adjunction nonequivariant, 31, 37 projection formula, 40, 213, 343 for proper maps, 45 projection map, 50 projective object, 281 G -graded module, 176, 193, 195 resolution, 282 projective system of connections, 160 proper base change, 100 proper map, 18, 64 properness, 63 proper support, 65 pullback bundle, 56 functor, 55 morphism, 41, 42, 44, 66 pushforward, 39, 43, 100 pushforward action of G, 135

373 Q quadrant (first,third), 189 quasi -injection, 13, 37 -isomorphism, 13 -surjection, 13 quasi-coherence, 305 Quillen, D., 117

R reducible module, 120 reflexive module, 208 regular filtration, 127, 189, 204, 205 regular representation, 231 relative to a base space, 51 right adjoint, 31 adjunction, 37 derived functor, 283 row "-filtration, 189

S Schubert cells, 258 semisimple module, 120 sheaf acyclic, 312 soft, 304 shift functor, 14 for complexes, 266 simple complex, 80, 84, 189 module, 120 object, 181 simultaneous resolution, 79 slice, 140, 238 centered at a point, 141 theorem, 238 space above a base space, 50 mild, 302 (topological), 246 Spivak, M., 82 split category, 13, 181, 354 complex, 330 morphism, 13, 276, 354 Stiefel manifolds, 147

374 support, 311 (compact vertical), 65 in , 312 proper, 65 symmetrization operator, 135 Syzygy, 178 T Theorem of Chevalley, 178 of equivariant Gysin morphism for general maps, 212 for proper maps, 214 of fixed points, 239 of Hilbert (syzygy), 178 of Hurewicz, 146 of Kostant, 206 of nonequivariant Gysin morphism for arbitrary maps, 39 for proper maps, 45 of Peter-Weyl, 147 of Poincaré duality G-equivariant (de Rham), 203 G-equivariant (over k), 256 nonequivariant, 25 of Sard, 36 T -equivariant (de Rham), 209 third quadrant bicomplex, 189 Thom class, 81, 102 class of a pair (N, M), 229 isomorphism, 81, 102, 344 topological space, 246 Tor•G (−, −), 191 torsion element, module, 237 torsion-free module, 208, 237 total complex, 80, 84, 189 total degree, 128, 182, 186, 198, 223 total space, 50 triangle, 268, 272

Index exact, distinguished, 273, 274 morphism of, 272 trivial module, 120 representation, 120 trivializing cover, 49 truncated complex, 255 tube, 82 tubular neighborhood, 82, 228 theorem, 342

U universal connection, 110 universal fiber bundle, 146 universally closed map, 64 universal property of fiber products, 52, 57, 337, 338 of injective objects, 281 of localized categories, 276, 278, 280 of projective objects, 282 Urysohn, 340

V vector bundle, 81 Vergne, M., 117

W weakly contractible, 146, 258 weight, 232 Weil, A., 28 algebra, 110 morphism, 111 Weyl group, 94, 178, 231, 257 Witten, E., 117

Z zero section, 101, 223

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