Lagrangian shadows and triangulated categories 9782856299401

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Paul Biran Octav Cornea Egor Shelukhin

LAGRANGIAN SHADOWS AND TRIANGULATED CATEGORIES

ASTÉRISQUE 426

Société Mathématique de France 2021

Astérisque est un périodique de la Société mathématique de France Numéro 426, 2021

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ISSN 0303-1179 (print) 2492-5926 (electronic) ISBN 978-2-85629-940-1 DOI 10.24033/ast.1148 Directeur de la publication : Fabien DURAND

ASTÉRISQUE 426

LAGRANGIAN SHADOWS AND TRIANGULATED CATEGORIES

Paul Biran Octav Cornea Egor Shelukhin

Société Mathématique de France 2021

Paul Biran Department of Mathematics, ETH-Zürich, Rämistrasse 101, 8092 Zürich, Switzerland. E-mail : [email protected] Octav Cornea Department of Mathematics and Statistics, University of Montreal, C.P. 6128 Succ. Centre-Ville Montreal, QC H3C 3J7, Canada. E-mail : [email protected] Egor Shelukhin Department of Mathematics and Statistics, University of Montreal, C.P. 6128 Succ. Centre-Ville Montreal, QC H3C 3J7, Canada. E-mail : [email protected]

2010 Mathematics Subject Classification. — 53D40 (Primary) 53D37, 18E30 (Secondary). Key words and phrases. — Lagrangian submanifold, Lagrangian cobordism, derived category, weakly filtered category, Fukaya category. Mots clefs. — Sous-variété lagrangienne, cobordisme lagrangien, catégorie dérivée, catégorie faiblement filtrée, catégorie de Fukaya.

The second author was supported by an individual NSERC Discovery grant. The third author was supported by an individual NSERC Discovery grant, and by the FRQNT start up grant. Received 07/01/2019, revised 29/04/2020, accepted 31/08/2020.

LAGRANGIAN SHADOWS AND TRIANGULATED CATEGORIES Paul Biran, Octav Cornea, Egor Shelukhin

Abstract. — We introduce new metrics on spaces of Lagrangian submanifolds, not necessarily in a fixed Hamiltonian isotopy class. Our metrics arise from measurements involving Lagrangian cobordisms. We also show that splitting Lagrangians through cobordism has an energy cost and, from this cost being smaller than certain explicit bounds, we deduce some forms of rigidity of Lagrangian intersections. We also fit these constructions in the more general algebraic setting of triangulated categories, independent of Lagrangian cobordism. As a main technical tool, we develop aspects of the theory of (weakly) filtered 𝐴∞ -categories. Résumé (Ombres des sous-variétés Lagrangiennes et catégories triangulées) Nous introduisons de nouvelles métriques sur des espaces des sous-variétés Lagrangiennes dont la classe d’isotopie Hamiltonienne n’est pas nécessairement fixée. Ces métriques proviennent de certaines quantités associées aux cobordismes Lagrangiens. Nous montrons également que la décomposition d’un Lagrangien à travers un cobordisme a un coût énergétique non-nul et, à partir d’une borne explicite de ce coût, nous déduisons des formes de rigidité des intersections Lagrangiennes. Nous donnons un sens à certaines de ces constructions dans le cadre algébrique plus général des catégories triangulées, indépendamment du cobordisme Lagrangien. Comme outil technique central, nous développons certains aspects de la théorie des catégories 𝐴∞ (faiblement) filtrées.

© Astérisque 426, SMF 2021

CONTENTS

1. Introduction and main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1. Decomposition by Lagrangian cobordism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2. Weighted fragmentation pseudo-metrics on triangulated categories . . . . 10 1.3. Outline of the proof of Theorem B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2. Weakly filtered 𝐴∞ -theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Weakly filtered 𝐴∞ -categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Typical classes of examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Weakly filtered 𝐴∞ -functors and modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Weakly filtered mapping cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. The 𝜆-map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. Structure theorem for weakly filtered iterated cones . . . . . . . . . . . . . . . . . . . . 2.7. Invariants and measurements for filtered chain complexes . . . . . . . . . . . . .

15 16 17 19 24 31 37 44

3. Floer theory and Fukaya categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Families of Fukaya categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Weakly filtered structure on Fukaya categories . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Extending the theory to Lagrangian cobordisms . . . . . . . . . . . . . . . . . . . . . . . 3.5. The monotone case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. Inclusion functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7. Weakly filtered iterated cones coming from cobordisms . . . . . . . . . . . . . . . .

51 55 56 56 62 65 67 70

4. Quasi-exact and quasi-monotone cobordisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Quasi-exact cobordisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Extending the results from Section 3.7 to quasi-exact cobordisms . . . . . . . 4.3. Quasi-monotone cobordisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Extending the results from Section 3.7 to quasi-monotone cobordisms . .

77 77 79 81 82

6

CONTENTS

5. Proof of the main geometric statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.1. Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.2. Proof of Theorem 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.3. The quasi-exact and quasi-monotone cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6. Metrics on spaces of Lagrangians and examples . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.1. Setting up the right class of cobordisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.2. Shadow metrics on spaces of Lagrangian submanifolds . . . . . . . . . . . . . . . . 106 6.3. Some examples and calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.4. Algebraic metrics on L𝑎𝑔 ∗ (𝑀) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

ASTÉRISQUE 426

CHAPTER 1 INTRODUCTION AND MAIN RESULTS

One of the main objectives of this paper is to introduce new metrics and related measurements on certain classes of Lagrangian submanifolds of a given symplectic manifold. The (pseudo) metrics that we look for are supposed to have three features: (i) Have significant symplectic content, in particular, be coherent with respect to Hofer’s norm. (ii) Be non-degenerate. (iii) Be finite for a class of Lagrangians as large as possible. Symplectic topology is characterized by an interplay of flexible and rigid phenomena, flexibility originating in the Gromov ℎ-principle and rigidity being reflected through properties of 𝐽-holomorphic curves. This tension flexibility – rigidity renders non-trivial the definition of metrics with the three properties above: without restricting in an appropriate manner the class of Lagrangians considered, flexibility leads to pseudo-metrics that are degenerate. On the other hand, having finite distances between Lagrangians with different isotopy (and even homotopy) types is non-obvious. Our measurements arise from the perspective of Lagrangian cobordism. The simplest non-trivial setting in which our metrics exist is the case when (𝑀, 𝜔 = d𝜆) is a Liouville manifold. Denote by L𝑎𝑔 ex (𝑀) the collection of exact Lagrangian submanifolds in 𝑀 which are compact without boundary. Given a Lagrangian cobordism 𝑉 ⊂ ℝ2 × 𝑀 (see Section 1.1 for the definition), denote by S(𝑉) the area of the projection of 𝑉 to ℝ2 together with all the bounded regions bounded by this projection. We call this measurement the shadow of 𝑉. More precisely: S(𝑉) = Area(ℝ2 \ U),

(1.1)

where U ⊂ ℝ2 \ 𝜋(𝑉) is the union of all the unbounded connected components of ℝ2 \ 𝜋(𝑉). Here 𝜋 : ℝ2 × 𝑀 → ℝ2 is the projection. Fix a family of exact Lagrangians F ⊂ L𝑎𝑔 ex (𝑀). For every 𝐿, 𝐿0 ∈ L𝑎𝑔 ex (𝑀) define: (1.2) 𝑑 F(𝐿, 𝐿0) := inf S(𝑉) ; 𝑉 : 𝐿



𝑉

(𝐹1 , . . . , 𝐹𝑖−1 , 𝐿0 , 𝐹𝑖 , . . . , 𝐹 𝑘 ), 𝑘 ≥ 0, 𝐹𝑖 ∈ F ,



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CHAPTER 1. INTRODUCTION AND MAIN RESULTS

where the infimum is taken over all (possibly disconnected) exact Lagrangian cobordisms 𝑉 having 𝐿 as its single positive end and whose negative ends consists of 𝐿0 possibly together with other Lagrangians, all taken from the family F. We use the convention that inf ∅ = ∞, so that 𝑑 F(𝐿, 𝐿0) = ∞ if there is no exact cobordism 𝑉 as in (1.2). It is easy to see that 𝑑 F is a pseudo-metric (possibly with infinite values). However, 𝑑 F is generally degenerate (yet not identically zero). Fix a second family of Lagrangians F0 ⊂ L𝑎𝑔 ex (𝑀) and define 0 0 b 𝑑 F,F := max{𝑑 F, 𝑑 F }.

(1.3) One of our main results is:

0 ∩ ( 𝐾0 ∈F0 𝐾 0 ) is totally disconnected, then b 𝑑 F,F is nondegenerate, hence a metric, (possibly with infinite values) on L𝑎𝑔 ex (𝑀).

Theorem A. — If (

Ð

𝐾∈F 𝐾 )

Ð

0

We call b 𝑑 F,F the shadow metric associated to the pair of families F, F0. For example, one can take Fto be a finite family of Lagrangians and for the family F0 one can take a small and generic Hamiltonian perturbation of each of the elements in F. Then Ð Ð ( 𝐾∈F 𝐾 ) ∩ ( 𝐾0 ∈F0 𝐾 0 ) is discrete and Theorem A applies. The shadow metrics bear a simple relation to the well known Lagrangian Hofer metric [Che00] on the space of Lagrangian submanifolds in a given Hamiltonian isotopy class. Indeed, it is not hard to see that if 𝐿0 is Hamiltonian isotopic to 𝐿 then 0 b 𝑑 F,F (𝐿, 𝐿0) ≤ 𝑑 Hofer (𝐿, 𝐿0)

and, in particular, shadow metrics satisfy property (i) from the beginning of the introduction. This is so because any Hamiltonian isotopy {𝜙𝑡 (𝐿)} between two exact 𝐿0 (called Lagrangians 𝐿 and 𝐿0 gives rise to an exact Lagrangian cobordism 𝑉 : 𝐿 the Lagrangian suspension of the isotopy) with S(𝑉) = length Hofer {𝜙𝑡 (𝐿)}. When F = ∅ the pseudo-metric 𝑑 ∅ is already non-degenerate and coincides with the metric introduced in [CS19] which infimizes the shadow of cobordisms having 0 only 𝐿 and 𝐿0 as ends (these are called simple cobordism). Of course b 𝑑 F,F ≤ 𝑑∅ . The use of multiple ended cobordisms and not of only simple ones in the definition 0 of metrics such as b 𝑑 F,F is a crucial novelty brought forth in this paper. Three aspects of this construction are worth underlining at this point. Firstly, in the exact setting, it is conjectured that any simple cobordism is a Lagrangian suspension (progress on this question appears in [Sua17]). Therefore, 𝑑∅ , at least conjecturally, coincides with the Lagrangian Hofer distance and, in particular, 𝑑∅ (𝐿, 𝐿0) is expected to be infinite as soon as 𝐿 and 𝐿0 are not Hamiltonian isotopic. However, for nonempty families F, F0 0 the associated distances b 𝑑 F,F (𝐿, 𝐿0) are often finite for pairs of Lagrangians 𝐿, 𝐿0 that are not even smoothly isotopic or can even have different homotopy types. Secondly 0 and more conceptually, the existence of the metrics b 𝑑 F,F for F, F0 ≠ ∅ is a reflection of the fact that the Lagrangian submanifolds in our setting can be organized in an 𝐴∞ -category which in turn, by a further algebraic process, gives rise to a triangulated category – the derived Fukaya category. As we will explain in detail below (see already 0 Section 1.2), the metrics b 𝑑 F,F reflect the triangulated structure of this category in

ASTÉRISQUE 426

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1.1. DECOMPOSITION BY LAGRANGIAN COBORDISM

the sense that they can be understood as providing the infimum of an “energy” cost required for certain decompositions by iterated exact triangles in this category. Finally, the last point to mention is that, as a technical reflection of the second aspect 0 mentioned just above, proving the non-degeneracy of the metrics b 𝑑 F,F requires, among other steps, a considerable development of 𝐴∞ -algebraic machinery in the filtered setting and this setup could potentially be of use elsewhere. In Chapter 6 we will study further aspects of shadow metrics. In particular we will see that analogues of the shadow metric exist also for other classes of Lagrangian submanifolds, such as weakly exact Lagrangians and monotone ones and variants of Theorem A continue to hold in these settings.

1.1. Decomposition by Lagrangian cobordism A Lagrangian cobordism [Arn80] (see [BC13] for the formalism in use here) is a Lagrangian submanifold 𝑉 ⊂ ℝ2 × 𝑀 with the property that there exists a compact interval [𝑎 − , 𝑎+ ] ⊂ ℝ such that 𝑉 \ [𝑎− , 𝑎+ ] × ℝ × 𝑀 =




𝑘 Þ

ℓ− × {𝑖} × 𝐿 𝑖

𝑘0  Ý Þ

𝑖=1



ℓ+ × {𝑗} × 𝐿0𝑗 ,

𝑗=1

𝐿0𝑗 ’s

where ℓ− = (−∞, 𝑎− ), ℓ+ = (𝑎+ , ∞) and the 𝐿 𝑖 ’s and are Lagrangian submanifolds of 𝑀. The 𝐿 𝑖 ’s are called the negative ends of 𝑉 and the 𝐿0𝑗 ’s the positive ends. We write: 𝑉 : (𝐿01 , . . . , 𝐿0𝑘0 )

(𝐿1 , . . . , 𝐿 𝑘 ).

We allow any of 𝑘 0 or 𝑘 to be 0 in which case the positive or negative end of the cobordism is void. Fix a collection L of Lagrangian submanifolds of 𝑀. Given a Lagrangian submanifold 𝐿 ⊂ 𝑀 we are interested in the “splitting” (or decomposition) of 𝐿 into Lagrangian submanifolds picked from the collection L. The type of splitting that we focus on is through Lagrangian cobordisms 𝑉 with a single positive end equal to 𝐿 and multiple negative ends, 𝑉 : 𝐿 (𝐿1 , . . . , 𝐿 𝑘 ). This perspective on cobordism is natural not least because, as is known from previous work [BC13], [BC14] and under appropriate constraints on L, such cobordisms induce genuine (iterated cone) decompositions of 𝐿 with factors the negative ends 𝐿 𝑖 in the derived Fukaya category of 𝑀. As already mentioned at the beginning of the introduction, the central point of view for this paper is to regard the shadow S(𝑉) of a cobordism 𝑉 as an energy cost for the splitting corresponding to 𝑉. We address two natural questions from this perspective: 1) Assuming 𝐿 and 𝐿1 , . . . , 𝐿 𝑘 fixed, find a lower bound for the minimal energy cost required to split 𝐿 in the factors 𝐿 𝑖 (see Theorem B)? 2) Is there some form of Lagrangian intersections rigidity that is specific to low energy splittings (see Theorem C)?

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For the following results we restrict to the class of Lagrangian submanifolds 𝐿 ⊂ 𝑀 that are closed and weakly exact (i.e. 𝜔 𝜋2 (𝑀,𝐿) = 0). Similarly, cobordisms 𝑉 are assumed to be weakly exact. The next theorem shows that the shadow of cobordisms 𝑉 : 𝐿 (𝐿1 , . . . , 𝐿 𝑘 ) between fixed 𝐿 and (𝐿1 , . . . , 𝐿 𝑘 ) cannot become arbitrarily small unless these Lagrangians are placed in a very particular position.

Theorem B. — Let 𝐿, 𝐿1 , . . . , 𝐿 𝑘 ⊂ 𝑀 be weakly exact Lagrangian submanifolds. Assume

that 𝐿 is not contained in 𝐿1 ∪ · · · ∪ 𝐿 𝑘 . Then there exists 𝛿 = 𝛿(𝐿; 𝑆) > 0 which depends only on 𝐿 and 𝑆 := 𝐿1 ∪ · · · ∪ 𝐿 𝑘 , such that for every weakly exact Lagrangian cobordism (𝐿1 , . . . , 𝐿 𝑘 ) we have 𝑉:𝐿 S(𝑉) ≥ 12 𝛿.

(1.4)

The proof is given Chapter 5. A non-technical outline of the proof is presented in next Section 1.3. The next theorem establishes relations between 𝐿 and 𝐿1 , . . . , 𝐿 𝑘 in case they are related by a Lagrangian cobordism with small shadow.

Theorem C. — Let 𝐿, 𝐿1 , . . . , 𝐿 𝑘 ⊂ 𝑀 be weakly exact Lagrangians and 𝑆 as in Theorem B. Let 𝑁 ⊂ 𝑀 be another weakly exact Lagrangian and assume that the Lagrangians 𝑁 , 𝐿, 𝐿1 , . . . , 𝐿 𝑘 are in general position. There exists 𝛿0 = 𝛿0(𝑁 , 𝑆) > 0 that depends on 𝑁 and 𝑆 (but not on 𝐿) such that for every weakly exact Lagrangian cobordism 𝑉:𝐿 with S(𝑉)
0 can be taken arbitrarily small. From the special form of the differential of M(𝐿) which involves the higher order 𝐴∞ -operations 𝜇𝑑 , we conclude that there is a pseudo-holomorphic polygon 𝑢 in 𝑀 with boundary on 𝐿 and on some of the 𝐿 𝑖 ’s that appears in the differential of M(𝐿) and that passes through 𝑃. Moreover, the area of this polygon is not more than 𝜌 + 𝜖0. In essence, this concludes the argument by making 𝜖, 𝜖0 → 0.

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ACKNOWLEDGMENTS

13

Acknowledgments We thank Misha Khanevsky for mentioning to us the argument in Remark 5.1.3. We also thank Alexandre Perrier for raising important points regarding the composition of cobordisms. The second author thanks Luis Diogo for useful comments. The last two authors thank the Institute for Advanced Study and Helmut Hofer for generously hosting them there for a part of this work. The second author also thanks the Forschungsinstitut für Mathematik for support during repeated visits to Zürich. We thank the referee for a very careful reading of the paper.

SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2021

CHAPTER 2 WEAKLY FILTERED 𝐴∞ -THEORY

In this chapter we develop a general framework for weakly filtered 𝐴∞ -categories, with an emphasis on weakly filtered modules over such categories. In our context “weakly filtered” generally means that the morphisms in the category are filtered chain complexes but the higher 𝐴∞ -operations do not necessarily preserve these filtrations. Rather they preserve them up to prescribed errors which we call discrepancies. In the same vein one can consider also weakly filtered 𝐴∞ -functors and modules. Related notions of filtered 𝐴∞ -structures have been considered in the literature (e.g. [FOOO09a], [FOOO09b]), but the existing theory seems to differ from ours in its scope and applications. Below we will cover only the most basic concepts of 𝐴∞ -theory in the weakly filtered setting. In particular we will not go into the topics of derived categories, split closure or generation in the weakly filtered framework. Our main goal is in fact much more modest: to provide an effective description of iterated cones of modules in the weakly filtered setting in terms of weakly filtered twisted complexes. Some readers may find the details of the weakly filtered setting somewhat overwhelming, especially in what concerns keeping track of the discrepancies. If one assumes all the discrepancies to vanish, the theory becomes “genuinely filtered” and is easier to follow. However, the additional difficulty due to the weakly filtered setting is largely superficial. Indeed, significant parts of the theory developed in this chapter do not become easier if one works in the genuinely filtered setting, except in terms of notational convenience. We also remark that, as far as we know, a good part of the theory developed in this chapter, particularly the study of iterated cones, is new even in the genuinely filtered case. The reason for developing the theory in the weakly filtered setting (rather than filtered) has to do with the geometric applications we aim at which have to do with Fukaya categories of symplectic manifolds. For technical reasons, the weakly filtered framework fits better with the standard implementations of these categories.

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CHAPTER 2. WEAKLY FILTERED 𝐴∞ -THEORY

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2.1. Weakly filtered 𝐴∞ -categories In the following we will often deal with sequences  = (𝜖1 , . . . , 𝜖 𝑑 , . . . ) of real numbers that we will refer to as discrepancies. We will use the following abbreviations and conventions: ⊲ For two sequences , 0 we write  ≤ 0 in order to say that 𝜖 𝑑 ≤ 𝜖0𝑑 for all 𝑑. ⊲ For 𝑐 ∈ ℝ we write  + 𝑐 for the sequence (𝜖1 + 𝑐, . . . , 𝜖 𝑑 + 𝑐, . . . ). ⊲ For a finite number of sequences (1) , . . . , (𝑟) we define max{(1) , . . . , (𝑟) } to be (1) (𝑟) the sequence  = (𝜖1 , . . . , 𝜖 𝑑 , . . . ) with 𝜖 𝑑 := max{𝜖 𝑑 , . . . , 𝜖 𝑑 }. Fix a commutative ring 𝑅, which for simplicity we will henceforth assume to be of characteristic 2 (i.e. 2𝑟 = 0 for all 𝑟 ∈ 𝑅). Unless otherwise stated all tensor products will be taken over 𝑅. The 𝐴∞ -theory developed below will be carried out in the ungraded framework. Also, in contrast to standard texts on the subject such as [Sei08], we will work in a homological (rather than cohomological) setting, following the conventions from [BC14]. Let A be an 𝐴∞ -category over 𝑅. To simplify notation, in what follows we will denote the morphisms between two objects 𝑋 , 𝑌 ∈ Ob(A) by 𝐶(𝑋 , 𝑌) := homA(𝑋 , 𝑌). , 𝑑 ≥ 1. We denote the composition maps of A by 𝜇A 𝑑 Let A = (𝜖1A, 𝜖2A, . . . , 𝜖 A , . . . ) be an infinite sequence of non-negative real numbers, 𝑑 with 𝜖1A = 0. We call A a weakly filtered 𝐴∞ -category with discrepancy ≤ A if the following holds: 1) For every 𝑋 , 𝑌 ∈ Ob(A), 𝐶(𝑋 , 𝑌) is endowed with an increasing filtration of 𝑅-modules indexed by the real numbers. We denote by 𝐶 ≤𝛼 (𝑋 , 𝑌) ⊂ 𝐶(𝑋 , 𝑌) the part of the filtration corresponding to 𝛼 ∈ ℝ. By increasing filtration we mean that 0 00 𝐶 ≤𝛼 (𝑋 , 𝑌) ⊂ 𝐶 ≤𝛼 (𝑋 , 𝑌) for every 𝛼0 ≤ 𝛼00. 2) The 𝜇𝑑 -operation preserves the filtration up to an “error” of 𝜖 A . More precisely, 𝑑 for every 𝑋0 , . . . , 𝑋𝑑 ∈ Ob(A) and 𝛼 1 , . . . , 𝛼 𝑑 ∈ ℝ we have A

𝜇𝑑 𝐶 ≤𝛼1 (𝑋0 , 𝑋1 ) ⊗ · · · ⊗ 𝐶 ≤𝛼 𝑑 (𝑋𝑑−1 , 𝑋𝑑 ) ⊂ 𝐶 ≤𝛼1 +···+𝛼 𝑑 +𝜖 𝑑 (𝑋0 , 𝑋𝑑 ).




Note that since 𝜖1A = 0, 𝜇1A preserves the filtration, each 𝐶 ≤𝛼 (𝑋 , 𝑌), 𝛼 ∈ ℝ, is a subcomplex of 𝐶(𝑋 , 𝑌). Note also that the discrepancy is not uniquely defined – in fact we can always increase it if needed. Namely, if 0 = (𝜖01 = 0, 𝜖02 , . . . , 𝜖0𝑑 , . . . ) is another sequence like A but with A ≤ 0 then Ais also weakly filtered with discrepancy ≤ 0. By analogy with symplectic topology we will often refer to the index of the filtration as an action and say that elements of 𝐶 ≤𝛼 (𝑋 , 𝑌) have action ≤ 𝛼.

ASTÉRISQUE 426

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17

2.1.1. Unitality. — Let A be a weakly filtered 𝐴∞ -category and assume that A is homologically unital (h-unital for short). We say that A is h-unital in the weakly filtered sense if there exists 𝑢 A ∈ ℝ≥0 such that for every 𝑋 ∈ Ob(A) we have a cycle 𝑒 𝑋 A in 𝐶 ≤𝑢 (𝑋 , 𝑋) representing the homology unit [𝑒 𝑋 ] ∈ 𝐻 𝐶(𝑋 , 𝑋), 𝜇1A .




We view the choices of 𝑒 𝑋 , 𝑋 ∈ Ob(A) and 𝑢 A as part of the data of a weakly filtered h-unital 𝐴∞ -category. We call 𝑢 A the discrepancy of the units. Occasionally we will have to impose the following additional assumption on A.

Assumption 𝑈 𝑒 . — Let A be a weakly filtered 𝐴∞ -category which is h-unital in the weakly filtered sense. Let 2𝑢 A + 𝜖2A ≤ 𝜁 ∈ ℝ. We say that Asatisfies Assumption 𝑈 𝑒 (𝜁) if for every 𝑋 ∈ Ob(A) and for some 𝑐 ∈ 𝐶 ≤𝜁 (𝑋 , 𝑋) we have 𝜇2A(𝑒 𝑋 , 𝑒 𝑋 ) = 𝑒 𝑋 + 𝜇1A(𝑐). Put in different words, the assumption 𝑈 𝑒 says that [𝑒 𝑋 ] · [𝑒 𝑋 ] = [𝑒 𝑋 ] in 𝐻∗ (𝐶 ≤𝜁 (𝑋 , 𝑋)), where the dot ‘·’ stands for the product induced by 𝜇2A in homology. (The superscript 𝑒 in 𝑈 𝑒 indicates that the assumption deals with the cycles 𝑒 𝑋 representing the units.) Below we will sometimes write A ∈ 𝑈 𝑒 (𝜁) to say that A satisfies Assumption 𝑈 𝑒 (𝜁).

2.2. Typical classes of examples Before we go on with the general algebraic theory of weakly filtered 𝐴∞ -structures, we make a short digression in order to exemplify what types of filtrations will actually occur in our applications. We resume with the general algebraic theory in Section 2.3 below. The weakly filtered 𝐴∞ -categories that will appear in our applications are Fukaya categories associated to symplectic manifolds. They will mostly be of the following types, described in §§ 2.2.1–2.2.4 below. 2.2.1. Filtrations induced by an “action” functional on the generators. — In this class of weakly filtered 𝐴∞ -categories the collection of morphisms 𝐶(𝑋 , 𝑌) between any two objects is assumed to be a free 𝑅-module with Êa distinguished basis 𝐵(𝑋 , 𝑌), i.e. 𝐶(𝑋 , 𝑌) = 𝑅𝑏. 𝑏∈𝐵(𝑋 ,𝑌)

We also have a function A : 𝐵(𝑋 , 𝑌) → ℝ, which (by analogy to symplectic topology) we call the action function, defined for every 𝑋 , 𝑌 ∈ Ob(A), and this function induces the filtration, namely: Ê 𝐶 ≤𝛼 (𝑋 , 𝑌) =

𝑅𝑏.

𝑏∈𝐵(𝑋 ,𝑌), A(𝑏)≤𝛼

We will mostly assume that 𝐶(𝑋 , 𝑌) has finite rank and that 𝑅 is a field.

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2.2.2. Filtration coming from the Novikov ring. — Here we fix a commutative ring 𝐴 and consider the (full) Novikov ring over 𝐴: (2.1)

Λ=

∞ nÕ

o

𝑎 𝑘 𝑇 𝜆 𝑘 ; 𝑎 𝑘 ∈ 𝐴, lim 𝜆 𝑘 = ∞ , 𝑘→∞

𝑘=0

as well as the positive Novikov ring: (2.2)

Λ0 =

∞ nÕ

o

𝑎 𝑘 𝑇 𝜆 𝑘 ; 𝑎 𝑘 ∈ 𝐴, 𝜆 𝑘 ≥ 0, lim 𝜆 𝑘 = ∞ . 𝑘→∞

𝑘=0

The weakly filtered 𝐴∞ -categories A of the type discussed here are defined over Λ, but the weakly filtered structure is only over the ring 𝑅 = Λ0 . As in § 2.2.1 above, we assume

Ê

𝐶(𝑋 , 𝑌) =

Λ𝑏.

𝑏∈𝐵(𝑋 ,𝑌)

The filtration on 𝐶(𝑋 , 𝑌) is then defined by 𝐶 ≤𝛼 (𝑋 , 𝑌) =

Ê

𝑇 −𝛼 Λ0 𝑏.

𝑏∈𝐵(𝑋 ,𝑌)

Note that 𝐶 ≤𝑎 (𝑋 , 𝑌) is not a Λ-module but rather a Λ0 -module. We will mostly assume that 𝐵(𝑋 , 𝑌) are finite (hence 𝐶(𝑋 , 𝑌) have finite rank) and that 𝐴 is a field (in which case Λ is a field too). 2.2.3. Mixed filtration. — In some situations the filtrations on our 𝐴∞ -categories occur as combination of §§ 2.2.1–2.2.2 above. More specifically, we have 𝐶(𝑋 , 𝑌) = Λ𝐵(𝑋 , 𝑌) as in § 2.2.2 and an action functional A : 𝐵(𝑋 , 𝑌) → ℝ as in § 2.2.1. We then extend A to a functional A : 𝐶(𝑋 , 𝑌) = Λ · 𝐵(𝑋 , 𝑌) −→ ℝ ∪ {−∞} by first setting A(0) = −∞. Then for 𝑃(𝑇) ∈ Λ and 𝑏 ∈ 𝐵(𝑋 , 𝑌) we define: A 𝑃(𝑇)𝑏 := −𝜆0 + A(𝑏),




where 𝜆0 ∈ ℝ is the minimal exponent that appears in the formal power series of Í 𝜆 𝑖 with 𝑎 ≠ 0 and 𝜆 > 𝜆 for every 𝑖 ≥ 1. 𝑃(𝑇) ∈ Λ, i.e. 𝑃(𝑇) = 𝑎0𝑇 𝜆0 + ∞ 0 𝑖 0 𝑖=1 𝑎 𝑖 𝑇 Finally, for a general non-trivial element 𝑐 = 𝑃1 (𝑇)𝑏 1 + · · · + 𝑃𝑙 (𝑇)𝑏 𝑙 ∈ 𝐶(𝑋 , 𝑌), define A(𝑐) = max A(𝑃𝑘 (𝑇)𝑏 𝑘 ) ; 1 ≤ 𝑘 ≤ 𝑙 . The filtration on 𝐶(𝑋 , 𝑌) is then induced by A:





𝐶 ≤𝛼 (𝑋 , 𝑌) = 𝑐 ∈ 𝐶(𝑋 , 𝑌) ; A(𝑐) ≤ 𝛼 .



It is easy to see that 𝐶 ≤𝛼 (𝑋 , 𝑌) is a Λ0 -module.

ASTÉRISQUE 426



2.3. WEAKLY FILTERED 𝐴∞ -FUNCTORS AND MODULES

19

2.2.4. Families of weakly filtered 𝐴∞ -categories. — The weakly filtered 𝐴∞ -categories in our applications will naturally occur in families {A𝑠 } 𝑠∈P parametrized by choices of auxiliary structures 𝑠 needed to define the 𝐴∞ -structure. The parameter 𝑠 will typically vary over a subset P ⊂ 𝐸 \ {0} where 𝐸 is a neighborhood of 0 in a Banach (or Fréchet) space. The subset P will usually be residual (in the sense of Baire) so that 0 is in the closure of P. Typically all the members of the family {A𝑠 } 𝑠∈P will be mutually quasi-equivalent (see [Sei08, Section 10] for several approaches to families of 𝐴∞ -categories). Of course, in the weakly filtered setting the quasi-equivalences between different A𝑠 ’s are supposed to bear some compatibility with respect to the weakly filtered structures on the A𝑠 ’s. Apart from the above, in our applications the families {A𝑠 } 𝑠∈P will enjoy the following additional property which will be crucial. The bounds A𝑠 for the discrepancies of the A𝑠 ’s can be chosen such that 𝑠 lim 𝜖 A = 0, 𝑑

𝑠→0

for all 𝑑.

Moreover, the categories A𝑠 will mostly be h-unital with discrepancy of units 𝑢 A𝑠 and satisfy Assumption 𝑈 𝑒 (𝜁 𝑠 ). The latter two quantities will satisfy lim 𝑢 A𝑠 = lim 𝜁 𝑠 = 0. 𝑠→0

𝑠→0

Below we will encounter further notions in the framework of weakly filtered 𝐴∞ categories such as weakly filtered functors and modules. Each of these comes with its own discrepancy sequence . In our applications everything will occur in families and we will usually have lim𝑠→0 𝜖 𝑑 (𝑠) = 0 for each 𝑑. While the algebraic theory below is developed without a priori assumptions on the size of discrepancies, it might be useful to view the discrepancies as quantities that can be made arbitrarily small. 2.2.5. The case of Fukaya categories. — The general description in § 2.2.4 applies to the case of Fukaya categories which will be central in our applications. More specifically, in order to define the 𝐴∞ -structure of Fukaya categories one has to make choices of perturbation data (e.g. choices of almost complex structures as well as Hamiltonian perturbation – see e.g. [Sei08, Sections 8–9]). The space P will consist of those perturbation data that are regular (or admissible). This is normally a second category subset of the space of all perturbations 𝐸. The discrepancies occur as “error” curvature terms (associated to the perturbations) when defining the 𝜇𝑑 -operations. These discrepancies can be made arbitrarily small (for a fixed 𝑑) by choosing smaller and smaller perturbations. The same holds for the discrepancy of the units and the 𝜁 𝑠 ’s. 2.3. Weakly filtered 𝐴∞ -functors and modules Let A, B two weakly filtered 𝐴∞ -categories and F : A → B an 𝐴∞ -functor. Let  F = (𝜖 1F, 𝜖2F, . . . , 𝜖 𝑑F, . . . ) be a sequence of non-negative real numbers. In contrast to A and B we do allow here that 𝜖 1F ≠ 0.

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We say that F is a weakly filtered 𝐴∞ -functor with discrepancy ≤  F if for all 𝑋0 , . . . , 𝑋𝑑 ∈ Ob(A) and 𝛼 1 , . . . , 𝛼 𝑑 ∈ ℝ we have ≤𝛼

(2.3)

≤𝛼 1 +···+𝛼 𝑑 +𝜖 𝑑F

≤𝛼1 F𝑑 𝐶 A (𝑋0 , 𝑋1 ) ⊗ · · · ⊗ 𝐶 A 𝑑 (𝑋𝑑−1 , 𝑋𝑑 ) ⊂ 𝐶 B




(F𝑋0 , F𝑋𝑑 ).

Here we have denoted by 𝐶 A and 𝐶 B the hom’s in A and B respectively and by F𝑑 the higher order terms of the functor F. There is also a notion of weakly filtered natural transformations between weakly filtered functors but we will not go into this now as our main focus will be on a special case – weakly filtered modules and weakly filtered morphisms between them. 2.3.1. Weakly filtered modules. — Let Abe a weakly filtered 𝐴∞ -category with discrepancy A. Let M be an A-module with composition maps 𝜇M , 𝑑 ≥ 1. Let 𝑑 M = (𝜖1M, 𝜖 2M, . . . , 𝜖 M 𝑑 ,...) be an infinite sequence of non-negative real numbers with 𝜖 1M = 0. We say that M is weakly filtered with discrepancy ≤ M the following holds: 1) For every 𝑋 ∈ Ob(A), M(𝑋) is endowed with an increasing filtration M≤𝛼 (𝑋) indexed by 𝛼 ∈ ℝ. 2) The 𝜇M -operation respects the filtration up to an “error” of 𝜖 M . Namely, for all 𝑑 𝑑 𝑋0 , . . . , 𝑋𝑑−1 ∈ Ob(A) and 𝑎1 , . . . , 𝑎 𝑑 ∈ ℝ we have M

≤𝛼 1 (𝑋0 , 𝑋1 ) ⊗ · · · ⊗ 𝐶 ≤𝛼 𝑑−1 (𝑋𝑑−2 , 𝑋𝑑−1 ) ⊗ M≤𝛼 𝑑 (𝑋𝑑−1 ) ⊂ M≤𝛼1 +···+𝛼 𝑑 +𝜖 𝑑 (𝑋0 ). 𝜇M 𝑑 𝐶




Again, since 𝜖 1M = 0, every (M≤𝛼 (𝑋), 𝜇1M) is a sub-complex of (M(𝑋), 𝜇1M). 2.3.2. Remark. — It is easy to see that weakly filtered A-modules are the same as opp weakly filtered functors F : A → Ch f (having some discrepancy). Here Ch f is opp the dg-category of filtered chain complexes (of 𝑅-modules) and Ch f stands for opp its opposite category. (Note that Ch f and Ch f are in fact filtered dg-categories, i.e. they have discrepancies 0.) The correspondence between weakly filtered functors and weakly filtered modules is the same as in the “unfiltered” case [Sei08, Section (1j)]. opp Note that if F : A → Ch f has discrepancy ≤  F then the weakly filtered module M F corresponding to it has discrepancy ≤ M with 𝜖 M = 𝜖 𝑑−1 for every 𝑑 ≥ 2. 𝑑 Next we define morphisms between weakly filtered A-modules. Let M0 , M1 be two weakly filtered A-modules, both with discrepancy ≤ 𝑚 . Let 𝑓 : M0 → M1 be a pre-module homomorphism. We write 𝑓 = ( 𝑓1 , . . . , 𝑓𝑑 , . . . ) where the 𝑓𝑑 -component is an 𝑅-linear map 𝑓𝑑 : 𝐶(𝑋0 , 𝑋1 ) ⊗ · · · ⊗ 𝐶(𝑋𝑑−2 , 𝑋𝑑−1 ) ⊗ M0 (𝑋𝑑−1 ) −→ M1 (𝑋0 ). 𝑓

𝑓

Let 𝛼 ∈ ℝ and  𝑓 = (𝜖 1 , . . . , 𝜖 𝑑 , . . . ) be a vector of non-negative real numbers. In 𝑓 contrast to A and 𝑚 we do allow that 𝜖 1 ≠ 0.

ASTÉRISQUE 426

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21

We say that 𝑓 shifts action by ≤ 𝜌 and has discrepancy ≤  𝑓 if for every 𝑑, the map 𝑓𝑑 𝑓 shifts action by not more than 𝜌 + 𝜖 𝑑 , namely: 𝑓

≤𝛼 𝑑

𝑓𝑑 𝐶 ≤𝛼1 (𝑋0 , 𝑋1 ) ⊗ · · · ⊗ 𝐶 ≤𝛼 𝑑−1 (𝑋𝑑−2 , 𝑋𝑑−1 ) ⊗ M0




𝑎 1 +···+𝑎 𝑑 +𝜌+𝜖 𝑑

(𝑋𝑑−1 ) ⊂ M1

(𝑋0 ).

We will generally refer to such 𝑓 ’s as weakly filtered pre-module homomorphisms. As before, if 𝜌 ≤ 𝜌0 and  𝑓 ≤  then 𝑓 also shifts filtration by ≤ 𝜌0 and has discrepancy ≤ . We will now define a filtration on the totality of pre-module homomorphisms. Denote ⊲ hom(M0 , M1 ) the pre-module homomorphisms M0 → M1 and ℎ

⊲ hom (M0 , M1 ) ⊂ hom(M0 , M1 ) the weakly filtered pre-module homomorphisms of discrepancy ≤  ℎ (and arbitrary action shift). The filtration will depend on an additional “discrepancy” parameter  ℎ = (𝜖 1ℎ , 𝜖 2ℎ , . . . , 𝜖 𝑑ℎ , . . . ) which is a sequence of non-negative real numbers (the superscript ℎ stands for “homomorphisms”). Again, we do not assume here that 𝜖1ℎ is 0. Our filtration is indexed by ℝ and is defined as follows. The part of the filtration corresponding to 𝜌 ∈ ℝ is denoted by ℎ

hom≤𝜌; (M0 , M1 ) and consists of all pre-module homomorphisms 𝑓 : M0 → M1 which shift action by not more than 𝜌 and have discrepancy ≤  ℎ . Clearly this yields an increasing filtration ℎ on hom (M0 , M1 ). Note however that, when viewed as a filtration on hom(M0 , M1 ), this filtration might in general not be exhaustive since not every pre-module homomorphism must be weakly filtered. Recall that A-modules (and pre-module homomorphisms between them) form a dg-category modA with differential 𝜇mod and composition 𝜇mod (see [Sei08, Sec1 2 tion (1j)] for the definitions). We now analyze these operations in the weakly filtered framework. For the operation 𝜇mod one encounters the following problem. For general choices 1 of A, 𝑚 and  ℎ and two weakly filtered modules M0 , M1 with discrepancy ≤ 𝑚 the ℎ differential 𝜇mod does not preserve hom≤𝜌; (M0 , M1 ). Nevertheless it is possible to 1 correct this problem by restricting the choice of  ℎ as follows:

Assumption E. — A sequence ε = (𝜀1 , . . . , 𝜀𝑑 , . . . ) is said to satisfy Assumption E if for every 𝑑 ≥ 1 we have A 𝜀𝑑 ≥ max{𝜖 𝑚 𝑖 + 𝜀 𝑗 , 𝜖 𝑖 + 𝜀 𝑗 ; 𝑖 + 𝑗 = 𝑑 + 1}. Sometimes we will need to emphasize the dependence of Assumption E on the choices of A and 𝑚 in which case we will refer to it as Assumption E(𝑚 , A). Alternatively we will sometimes write ε ∈ E(𝑚 , A).

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An inspection of definition of 𝜇mod (see e.g. [Sei08, Section (1j)]) shows that if 1 ℎ  ℎ satisfies Assumption E then hom≤𝜌; (M0 , M1 ) is preserved by 𝜇mod hence is a 1 chain complex. The following will be useful later on:

Lemma 2.1. — For every A and 𝑚 there exists ε that satisfies Assumption E(𝑚 , A). Moreover, there exists a sequence of real numbers {𝐴 𝑑 } 𝑑∈ℕ which is universal in the sense that it does not depend on A or 𝑚 and has the following property: for every sequence δ = (𝛿1 , . . . , 𝛿 𝑑 , . . . ) of non-negative real numbers there exists an ε that satisfies Assumption E(𝑚 , A) and such that for all 𝑑: 𝛿 𝑑 ≤ 𝜀𝑑 ≤ 𝐴 𝑑

(2.4)

𝑑 Õ

𝑚 (𝜖 A 𝑗 + 𝜖 𝑗 + 𝛿 𝑗 ).

𝑗=1

Proof of Lemma 2.1. — One can easily construct 𝜀𝑑 and 𝐴 𝑑 inductively: start with 𝜀1 := 𝛿1 then set 𝜀2 := max{𝜖2𝑚 + 𝜀1 , 𝜖 2A + 𝜀1 , 𝛿2 } and so on. (Note that 𝜖 1A = 𝜖 1𝑚 = 0 so that the inequality in Assumption E is obviously satisfied for 𝑖 = 1, 𝑗 = 𝑑.)  2.3.3. Remarks 1) If ε ∈ E(𝑚 , A) then the same holds for ε e := ε + c, where c = (𝑐, . . . , 𝑐, . . . ) is a constant sequence. ℎ 2) When dealing with hom≤𝜌; we can always arrange that 𝜖 1ℎ = 0 by applying the following procedure. Suppose that  ℎ ∈ E(𝑚 , A). Put e 𝜌 := 𝜌 + 𝜖 1ℎ , e 𝜖 𝑑ℎ := 𝜖 𝑑ℎ − 𝜖 1ℎ . Note that e 𝜖 1ℎ = 0, e 𝜖 𝑑ℎ ≥ 0 and that e  ℎ still satisfies Assumption E. It is easy to see that ℎ

ℎ

𝜌;e  hom≤e (M0 , M1 ) = hom≤𝜌; (M0 , M1 ).

We now turn to the 𝜇mod operation. Let M0 , M1 , M2 be weakly filtered A-modules 2 with discrepancy ≤ 𝑚 . Let 𝑓 : M0 → M1 , 𝑔 : M1 → M2 be two weakly filtered 𝑓 𝑓 𝑔 𝑔 pre-module homomorphisms with 𝑓 ∈ hom≤𝜌 ; (M0 , M1 ), 𝑔 ∈ hom≤𝜌 ; (M1 , M2 ). Set 𝜑 := 𝜇mod ( 𝑓 , 𝑔) : M0 → M2 . A simple calculation shows that 𝜑 is weakly filtered 2 and that 𝜑 ∈ hom≤𝜌 defined as:

𝑓 +𝜌 𝑔 ; 𝑓 ∗ 𝑔

(M0 , M2 ), where the sequence of discrepancies  𝑓 ∗  𝑔 is 𝑓

𝑔

( 𝑓 ∗  𝑔 )𝑑 = max 𝜖 𝑖 + 𝜖 𝑗 ; 𝑖 + 𝑗 = 𝑑 + 1 .



(2.5)



Moreover, a simple calculation shows that if  𝑓 ,  𝑔 ∈ E(𝑚 , A) then the same holds for  𝑓 ∗  𝑔 . A few words are in order about the structure of the totality of weakly filtered A-modules. Ideally one would like to view the weakly filtered modules (say with discrepancy ≤ 𝑚 , and with morphisms of discrepancy ≤  ℎ ) as a sub-category of modA and define a weakly filtered structure on it. As seen above, Assumption E ℎ assures that the hom≤𝜌; (M0 , M1 )’s are closed under 𝜇mod . However without further 1 restrictions on  ℎ , the operation 𝜇mod does not map 2 0

ℎ

00 ; ℎ

hom≤𝜌 ; (M0 , M1 ) ⊗ hom≤𝜌

ASTÉRISQUE 426

(M1 , M2 )

0

to

00 ; ℎ

hom≤𝜌 +𝜌

(M0 , M2 ).

2.3. WEAKLY FILTERED 𝐴∞ -FUNCTORS AND MODULES

23

Thus for general  ℎ ∈ E(𝑚 , A) we still do not get a dg-category. We refer the reader to the expanded version of this paper [BCS] for possible solutions to this issue, as well as to further discussion on categorical aspects of weakly filtered modules such as the Yoneda embedding and triangulated structure in the weakly filtered framework. For the applications in this paper, we do not need the weakly filtered modules to form a dg-category, and therefore will generally not restrict  ℎ beyond Assumption E. We stress that Assumption E will continue to play an important role since it assures that ℎ the hom≤𝜌; ’s are preserved by 𝜇mod . Thus we will mostly continue to assume it. 1 2.3.4. Action-shifts. — Let M be a weakly filtered module over a weakly filtered 𝐴∞ category A. Let 𝜈0 ∈ ℝ. Define a new weakly filtered A-module 𝑆 𝜈0 M to be the same module as M only that its filtration is shifted by 𝜈0 , namely: (𝑆 𝜈0 M)≤𝛼 (𝑁) := M≤𝛼+𝜈0 (𝑁) for all 𝑁 ∈ Ob(A), 𝛼 ∈ ℝ . Clearly 𝑆 𝜈0 M has the same discrepancy as M. We call 𝑆 𝜈0 M the action-shift of M by 𝜈0 . In what follows we will use the same notation 𝑆 𝜈0 also for action shifts of other filtered objects such as filtered chain complexes or more generally filtered 𝑅-modules. 2.3.5. Homologically unital A-modules. — We have already discussed h-unital 𝐴∞ categories in the weakly filtered sense on page 17. In what follows we will sometimes need an analogous, yet somewhat stronger, notion for modules.

Assumption 𝑈𝑚 . — Let A be a weakly filtered 𝐴∞ -category with discrepancy ≤ A. Assume that Ais h-unital in the weakly filtered sense as defined in Section 2.1, i.e. we A have 𝑢 A ≥ 0 and choices of cycles 𝑒 𝑋 ∈ 𝐶 ≤𝑢 (𝑋 , 𝑋) for every 𝑋 ∈ Ob(A) representing the units in homology. Let M be a weakly filtered A-module with discrepancy ≤ 𝑚 , and let 𝑢 A + 𝜖2M ≤ 𝜅 ∈ ℝ. We say that M satisfies Assumption 𝑈𝑚 (𝜅) (or M ∈ 𝑈𝑚 (𝜅) for short) if for every 𝑋 in Ob(A) and every 𝛼 ∈ ℝ the map (2.6)

M≤𝛼 (𝑋) −→ M≤𝛼+𝜅 (𝑋),

𝑏 ↦−→ 𝜇2M(𝑒 𝑋 , 𝑏)

induces in homology the same map as the one induced by M≤𝛼 (𝑋) ↩→ M≤𝛼+𝜅 (𝑋). Note that in particular, M is an h-unital module. Sometimes the module M will be a Yoneda module 𝒴 associated to an object 𝑌 ∈ Ob(A). In that case we will sometimes write 𝑌 ∈ 𝑈𝑚 (𝜅) instead of 𝒴 ∈ 𝑈𝑚 (𝜅). Note that in this case the map in (2.6) becomes 𝐶 ≤𝛼 (𝑌, 𝑋) −→ 𝐶 ≤𝛼+𝜅 (𝑌, 𝑋),

𝑏 ↦−→ 𝜇2A(𝑒 𝑋 , 𝑏).

There is also a homotopical version of 𝑈𝑚 (see [BCS, Section 2.3.4], where it is called 𝑈 𝑠 ) but we will not need it here.

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24

2.3.6. Pulling back weakly filtered modules. — Let A, B be two weakly filtered 𝐴∞ categories and F : A → B a weakly filtered 𝐴∞ -functor with discrepancy ≤  F. Let Mbe a weakly filtered B-module with discrepancy ≤ M. Consider the A-module F∗ M which is obtained by pulling back M via F. We filter F∗ M by setting (F∗ M)≤𝛼 (𝑁) = M≤𝛼 (F𝑁). The following can be easily proved. ∗

Lemma 2.2. — The module F∗ M is weakly filtered with discrepancy ≤  F M, where

for all 𝑑 ≥ 2 ∗

𝜖 𝑑F M = max 𝜖 𝑠F1 + · · · + 𝜖 𝑠F𝑘 + 𝜖 M 𝑘+1 ; 1 ≤ 𝑘 ≤ 𝑑 − 1, 𝑠 1 + · · · + 𝑠 𝑘 = 𝑑 − 1 .





In particular, if the higher order terms of F vanish, i.e. F𝑠 = 0 for all 𝑠 ≥ 2, then ∗

𝜖 𝑑F M = (𝑑 − 1)𝜖 1F + 𝜖 M 𝑑 ,

for all 𝑑.

Let M0 , M1 be two weakly filtered B-modules and 𝑓 : M0 → M1 a weakly filtered module homomorphism that shifts action by ≤ 𝜌 and has discrepancy ≤  𝑓 . Pulling back we obtain a homomorphism of A-modules F∗ 𝑓 : F∗ M0 → F∗ M1 . The following can be easily verified.

Lemma 2.3. — The module homomorphism F∗ 𝑓 is weakly filtered with action shift ≤ 𝜌 F∗ 𝑓

and discrepancy ≤  F 𝑓 , where 𝜖1 ∗

F∗ 𝑓

𝜖𝑑

𝑓

= 𝜖 1 and

𝑓

= max 𝜖 𝑠F1 + · · · + 𝜖 𝑠F𝑘 + 𝜖 𝑘+1 ; 1 ≤ 𝑘 ≤ 𝑑 − 1, 𝑠 1 + · · · + 𝑠 𝑘 = 𝑑 − 1 ,





for all 𝑑 ≥ 2.

In particular, if the higher order terms of F vanish, i.e. F𝑠 = 0 for all 𝑠 ≥ 2, then F∗ 𝑓

𝜖𝑑

𝑓

= (𝑑 − 1)𝜖 1F + 𝜖 𝑑 ,

for all 𝑑.

2.4. Weakly filtered mapping cones Let M0 , M1 be two weakly filtered A-modules with discrepancies ≤ M0 and ≤ M1 respectively. Let 𝑓 : M0 → M1 be a module homomorphism, i.e. 𝑓 is a pre-module homomorphism which is a cycle: 𝜇mod ( 𝑓 ) = 0. Assume that 𝑓 shifts action by ≤ 𝜌 and 1 𝑓 𝑓 has discrepancy ≤  , or in other words 𝑓 ∈ hom≤𝜌; (M0 , M1 ). We generally do not assume that  𝑓 satisfies Assumption E(𝑚 , A) unless explicitly specified. Consider the mapping cone C := Cone( 𝑓 ) viewed as an A-module and endowed with its standard 𝐴∞ -composition maps 𝜇𝑑C. We endow C with a weakly filtered structure as follows. For 𝑋 ∈ Ob(A) and 𝛼 ∈ ℝ, put 𝑓

(2.7)

≤𝛼−𝜌−𝜖1

C≤𝛼 (𝑋) := M0

(𝑋) ⊕ M1≤𝛼 (𝑋).

Define (see page 16 for the precise meaning of this notation) 𝑓

 C := max M0 , M1 ,  𝑓 − 𝜖1 .



ASTÉRISQUE 426

25

2.4. WEAKLY FILTERED MAPPING CONES

Then C is weakly filtered with discrepancy ≤  C. This follows from (2.7) and the fact that 0 𝜇𝑑C 𝑎 1 , . . . , 𝑎 𝑑−1 , (𝑏 0 , 𝑏1 ) = 𝜇M (𝑎1 , . . . , 𝑎 𝑑−1 , 𝑏0 ), 𝑑




1 𝑓𝑑 (𝑎1 , . . . , 𝑎 𝑑−1 , 𝑏0 ) + 𝜇M (𝑎1 , . . . , 𝑎 𝑑−1 , 𝑏1 ) . 𝑑




2.4.1. Remark. — If we assume in addition that M0 , M1 ≤ 𝑚

 𝑓 ∈ E(𝑚 , A),

and

𝑓

𝑓

then we have  𝑓 − 𝜖1 ≥ 𝑚 , hence  C =  𝑓 − 𝜖1 . It is important to note that the filtration we have defined on Cone( 𝑓 ) in (2.7) 𝑓 strictly depends on the choices of 𝜌 and 𝜖 1 . Therefore, whenever these dependencies are relevant we will denote the weakly filtered cone of 𝑓 by ( 𝑓 ;𝜌, 𝑓 )

Cone( 𝑓 ; 𝜌,  𝑓 ) or by

(2.8)

C𝑜𝑛𝑒(M0 −−−−−−→ M1 ).

2.4.2. Remark. — We opted to define the filtration on the cone as in (2.7) so that the inclusion M1 → C becomes a strictly filtered map. We now discuss several elementary properties of weakly filtered mapping cones that will be useful later on. We begin with the effect of action-shifts (see § 2.3.4) on mapping cones. The following follows immediately from the definitions.

Lemma 2.4. — Let 𝑓 : M0 → M1 be a weakly filtered module homomorphism between two

weakly filtered A-modules. Assume that 𝑓 shifts action by ≤ 𝜌 and has discrepancy ≤  𝑓 . Let 𝜈0 ∈ ℝ. Then we have the following equality of weakly filtered A-modules: ( 𝑓 ;𝜌, 𝑓 )

( 𝑓 ;𝜌, 𝑓 )

𝑆 𝜈0 C𝑜𝑛𝑒(M0 −−−−−−→ M1 ) = C𝑜𝑛𝑒 𝑆 𝜈0 M0 −−−−−−→ 𝑆 𝜈0 M1




( 𝑓 ;𝜌, 𝑓 −𝜈0 )

= C𝑜𝑛𝑒 M0 −−−−−−−−−→ 𝑆 𝜈0 M1 ( 𝑓 ;𝜌−𝜈0 , 𝑓 )







= C𝑜𝑛𝑒 M0 −−−−−−−−−→ 𝑆 𝜈0 M1 .




Next, we analyze (a special case of) cones over a composition of module homomorphisms, from the weakly filtered perspective. Let 𝑓 : M0 → M1 be as at the beginning of the present chapter. Let M10 be another weakly filtered A-module with 0 discrepancy ≤ M and let 𝜉 : M1 → M10 be a weakly filtered module homomorphism 𝜉

with 𝜉 ∈ hom≤𝑠; (M1 , M10 ). Denote the composition of 𝑓 and 𝜉 by 𝑓 0 = 𝜉 ◦ 𝑓 := 𝜇mod ( 𝑓 , 𝜉) : M0 −→ M10 . 2 𝑓0

We have 𝑓 0 ∈ hom𝜌+𝑠; (M0 , M10 ), where 𝑓0

𝑓

𝜖 𝑑 = ( 𝑓 ∗ 𝜉 )𝑑 =: max 𝜖 𝑖 + 𝜖 𝜉𝑗 ; 𝑖 + 𝑗 = 𝑑 + 1 .





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Lemma 2.5. — There exists a weakly filtered module homomorphism 𝜓 : C𝑜𝑛𝑒( 𝑓 ; 𝜌,  𝑓 ) −→ C𝑜𝑛𝑒( 𝑓 0; 𝜌 + 𝑠,  𝑓 ) 0

that shifts action by ≤ 𝑠 and has discrepancy ≤ 𝜉 . The homomorphism 𝜓 fits into the following (chain level) commutative diagram of A-modules: 𝑓

M0 (2.9)

/ M1

𝑓0

/ 𝑇M0

𝜓

𝜉

id

 M0

/ C𝑜𝑛𝑒( 𝑓 )

id

 / C𝑜𝑛𝑒( 𝑓 0)

 / M0 1

 / 𝑇M0

where the horizontal unlabeled maps are the standard inclusion and projection maps (with zero higher order terms), and 𝑇M0 stands for the shift of M0 with respect to grading. Moreover, if 𝜉 is a quasi-isomorphism then so is 𝜓. As indicated earlier, in this paper we work in the ungraded setting, hence the equality 𝑇M0 = M0 . Nevertheless we have written 𝑇M0 in (2.9) as a suggestion for how the statement should look like in the graded case. Proof of Lemma 2.5. — Simply define 𝜓1 (𝑏0 , 𝑏1 ) = 𝑏0 , 𝜉1 (𝑏 1 ) and for 𝑑 ≥ 2:




𝜓 𝑑 𝑎1 , . . . , 𝑎 𝑑−1 , (𝑏0 , 𝑏1 ) := 0, 𝜉𝑑 (𝑎1 , . . . , 𝑎 𝑑−1 , 𝑏1 ) .







All the statements asserted by the lemma can be verified by direct calculation.



Next we discuss how the weakly filtered mapping cone changes if we alter the cycle 𝑓 by a boundary. Assume now that M0 and M1 have both discrepancies ≤ 𝑚 . ℎ Fix a sequence  ℎ that satisfies Assumption E(𝑚 , A). Let 𝑓 ∈ hom≤𝜌; (M0 , M1 ) be ℎ a module homomorphism and 𝑓 0 = 𝑓 + 𝜇mod (𝜃) for some 𝜃 ∈ hom≤𝜌; (M0 , M1 ). 1 Consider the two weakly filtered mapping cones Cone( 𝑓 ; 𝜌,  ℎ ) and Cone( 𝑓 0; 𝜌,  ℎ ).

Lemma 2.6. — There exists a module homomorphism 𝜗 : Cone( 𝑓 ; 𝜌,  ℎ ) −→ Cone( 𝑓 0; 𝜌,  ℎ ) with the following properties: (i) 𝜗 is a quasi-isomorphism. (ii) 𝜗 does not shift action (i.e. it shifts the action filtration by ≤ 0) and has discrepancy ≤ 𝜗 :=  ℎ − 𝜖1ℎ . In particular (since 𝜖 1𝜗 = 0) the chain map 𝜗1 : Cone( 𝑓 ; 𝜌,  ℎ )(𝑋) −→ Cone( 𝑓 0; 𝜌,  ℎ )(𝑋) preserves the action filtration for every 𝑋 ∈ Ob(A). Proof. — Define 𝜗1 (𝑏 0 , 𝑏1 ) := (−1)|𝑏0 |−1 𝑏 0 , (−1)|𝑏1 | 𝑏1 + 𝜃1 (𝑏 0 ) and for 𝑑 ≥ 2 define:




𝜗 𝑑 𝑎 1 , . . . , 𝑎 𝑑−1 , (𝑏 0 , 𝑏1 ) = 0, 𝜃𝑑 (𝑎1 , . . . , 𝑎 𝑑−1 , 𝑏0 ) .




Cf. [Sei08, Formula 3.7, p. 35].

ASTÉRISQUE 426




27

2.4. WEAKLY FILTERED MAPPING CONES

Note that in this paper we work with a base ring 𝑅 of characteristic 2, hence the signs in the preceding formula for 𝜗1 can actually be ignored. Nevertheless we included them, just as an indication for a possible extension to more general rings.  The next lemma shows that weakly filtered cones are preserved under pulling back by weakly filtered functors.

Lemma 2.7. — Let: A, B be two weakly filtered 𝐴∞ -categories and F : A → B a weakly filtered 𝐴∞ -functor with discrepancy ≤  F; ⊲

M0 , M1 weakly filtered B-modules and 𝑓 : M0 → M1 a weakly filtered module homomorphism which shifts action by ≤ 𝜌 and has discrepancy ≤  𝑓 . ⊲

Then we have the following equality of weakly filtered A-modules: ∗

( 𝑓 ;𝜌, 𝑓 )

∗

(F∗ 𝑓 ;𝜌, F 𝑓 )

∗

F C𝑜𝑛𝑒(M0 −−−−−−→ M1 ) = C𝑜𝑛𝑒 F M0 −−−−−−−−−−→ F∗ M1 ,







where  F 𝑓 is given in Lemma 2.3. ∗

The proof is straightforward, hence omitted. We now return briefly to unitality of modules, more specifically to Assumption 𝑈𝑚 . The following lemma shows that this assumption is preserved under certain quasiisomorphisms of weakly filtered modules.

Lemma 2.8. — Let: 0

00

⊲ M0, M00 be weakly filtered A-modules with discrepancies ≤ M and ≤ M 0. 0 𝜙 ⊲ 𝜙0 : M0 → M00 be a weakly filtered module homomorphism, 𝜙0 ∈ hom𝜌 ; (M0 , M00). ⊲ Let 𝜙00 be a collection of chain maps 𝜙00𝑋 : M00(𝑋) → M0(𝑋), defined for all 𝑋 in Ob(A), and assume for all 𝑋 ∈ Ob(A) and 𝛼 ∈ ℝ, 00 +𝜖00

𝜙00𝑋 M00≤𝛼 (𝑋) ⊂ M0≤𝛼+𝜌




for some fixed

𝜌00 , 𝜖00

∈ ℝ. (For example, if 00 𝜌00 ;𝜙

𝜙00

M00

:

(𝑋)

→ M0 is a weakly filtered module 𝜙00

homomorphism with 𝜙00 ∈ hom (M00 , M0), where 𝜖1 ≤ 𝜖00, then the assumptions 00 on 𝜙 are clearly satisfied.) ⊲ Let 𝜈, 𝜅00 ∈ ℝ and assume further that 𝛼) For every 𝑋 ∈ Ob(A) and every 𝛼 ∈ ℝ the composition of chain maps 𝜙00𝑋 ◦𝜙01

00 +𝜖 𝜙 1

0

M0≤𝛼 (𝑋) −−−−−−→ M0≤𝛼+𝜌 +𝜌

0

+𝜖00

0

inc

00 +𝜖 𝜙 1

(𝑋) −−−→ M0≤𝛼+𝜌 +𝜌

0

+𝜖00 +𝜈

(𝑋)

induces in homology the same map as the one induced by the inclusion 0

00 +𝜖 𝜙 1

M0≤𝛼 (𝑋) −→ M0≤𝛼+𝜌 +𝜌

0

+𝜖00 +𝜈

(𝑋).

𝛽) ∈ 𝑈𝑚 Then M0 belongs to 𝑈𝑚 (𝜅0), where M00

(2.10)

(𝜅00).

𝜙0

0

𝜙0

𝜙0

𝜅0 = 𝜌0 + 𝜌00 + 𝜖00 + max 𝜖 1 + 𝑢 A + 𝜖2M + 𝜈, 𝜖 1 + 𝜅00 , 𝜖 2 + 𝑢 A .





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28

Proof. — Fix 𝑋 ∈ Ob(A), 𝛼 ∈ ℝ and let 𝑏 ∈ M0≤𝛼 (𝑋) be a cycle. Since 𝜙0 is a module homomorphism (i.e. 𝜇mod (𝜙0) = 0) we have 1 0

00

00

𝜙01 𝜇2M (𝑒 𝑋 , 𝑏) = 𝜇2M 𝑒 𝑋 , 𝜙01 (𝑏) ± 𝜇1M 𝜙02 (𝑒 𝑋 , 𝑏).




Applying 𝜙00𝑋 to both sides of this identity we obtain 0

00

0

𝜙00𝑋 𝜙01 𝜇2M (𝑒 𝑋 , 𝑏) = 𝜙00𝑋 𝜇2M 𝑒 𝑋 , 𝜙01 (𝑏) ± 𝜇1M 𝜙00𝑋 𝜙02 (𝑒 𝑋 , 𝑏).

(2.11) 0

Since 𝜇2M (𝑒 𝑋 , 𝑏) ∈ M0≤𝛼+𝑢

A+𝜖 M0 2




(𝑋) our assumption on 𝜙00𝑋 ◦ 𝜙01 implies that there

0 𝜙0 0≤𝛼+𝑢 A+𝜖2M +𝜌0 +𝜌00 +𝜖1 +𝜖00 +𝜈

exists 𝑥 ∈ M

(𝑋) such that

M0

0

0

𝜙00𝑋 𝜙01 𝜇2 (𝑒 𝑋 , 𝑏) = 𝜇2M (𝑒 𝑋 , 𝑏) − 𝜇1M (𝑥).

(2.12)

𝜙0

0

Since M00 ∈ 𝑈𝑚 (𝜅00) there exists an element 𝑦 ∈ M00≤𝛼+𝜌 +𝜖1 M00

𝜇2

+𝜅00

(𝑋) such that

M00

𝑒 𝑋 , 𝜙01 (𝑏) = 𝜙01 (𝑏) + 𝜇1 (𝑦).




Substituting the last identity together with (2.12) into (2.11) yields: 0

0

0

0

𝜇2M (𝑒 𝑋 , 𝑏) = 𝜇1M (𝑥) + 𝜙00𝑋 𝜙01 (𝑏) + 𝜇1M 𝜙00𝑋 (𝑦) + 𝜇1M 𝜙00𝑋 𝜙02 (𝑒 𝑋 , 𝑏) .

(2.13)







Using our assumption on 𝜙00𝑋 ◦ 𝜙01 , we can write the second term of (2.13) as 0

0

00 +𝜖 𝜙 1

𝜙00𝑋 𝜙01 (𝑏) = 𝑏 + 𝜇1M (𝑧) for some 𝑧 ∈ M0≤𝛼+𝜌 +𝜌

0

+𝜖00 +𝜈

(𝑋). Substituting this in (2.13)

we obtain 0

(2.14)

0

0

0

0

𝜇2M (𝑒 𝑋 , 𝑏) = 𝑏 + 𝜇1M (𝑥) + 𝜇1M (𝑧) + 𝜇1M 𝜙00𝑋 (𝑦) + 𝜇1M 𝜙00𝑋 𝜙02 (𝑒 𝑋 , 𝑏) ,







where 𝑥 ∈ M0≤𝛼+𝑢

0 A+𝜖 M0 +𝜌0 +𝜌00 +𝜖 𝜙 +𝜖00 +𝜈 2 1

𝜙 0≤𝛼+𝜌0 +𝜌00 +𝜖1

0

𝑧∈M

The estimate (2.10) for

𝜅0

+𝜖00 +𝜈

(𝑋),

(𝑋),

0

00 +𝜖 𝜙 1

𝜙00𝑋 (𝑦) ∈ M00≤𝛼+𝜌 +𝜌

0

+𝜖00 +𝜅00

𝜙 0≤𝛼+𝑢 A+𝜌0 +𝜌00 +𝜖00 +𝜖 2

𝜙00𝑋 𝜙02 (𝑒 𝑋 , 𝑏) ∈ M

(𝑋),

0

.



readily follows.

It is known that h-unitality is preserved under mapping cones [Sei08, Section 3e]. The following Lemma is a weakly filtered analogue, concerning Assumption 𝑈𝑚 .

Lemma 2.9. — Assume that A satisfies Assumption 𝑈 𝑒 (𝜁) (see page 17). Let: M0 , M1 be weakly filtered A-modules with discrepancies ≤ M0 and ≤ M1 respectively and assume that M0 ∈ 𝑈𝑚 (𝜅 M0 ) and M1 ∈ 𝑈𝑚 (𝜅 M1 ). ⊲

𝑓

𝑓 ∈ hom≤𝜌; (M0 , M1 ) be a module homomorphism. Then the weakly filtered module C𝑜𝑛𝑒( 𝑓 ; 𝜌,  𝑓 ) satisfies Assumption 𝑈𝑚 (𝜅), where ⊲

(2.15)

𝜅 = max 2𝜅 M0 , 2𝜅 M1 , 2𝑢 A + 𝜖3C, 2𝑢 A + 2𝜖 2C, 𝜁 + 𝜖2C ,





𝑓

and  C := max{M0 , M1 ,  𝑓 − 𝜖 1 }. (Recall that  C is the standard bound on the discrepancy of C = C𝑜𝑛𝑒( 𝑓 ; 𝜌,  𝑓 ) – see page 24.) To show this lemma we will make use of the following proposition that is of independent interest.

ASTÉRISQUE 426

29

2.4. WEAKLY FILTERED MAPPING CONES

Proposition 2.10. — Assume that A ∈ 𝑈 𝑒 (𝜁). Let M be a weakly filtered A-module with discrepancy ≤ M and let 𝑋 ∈ Ob(A). Then the chain maps 𝑣 : M(𝑋) −→ M(𝑋),

𝑣(𝑏) := 𝜇2M(𝑒 𝑋 , 𝑏)

and 𝑣 ◦ 𝑣 : M(𝑋) → via a chain homotopy that shifts action by  M(𝑋) are chain homotopic not more than max 2𝑢 A + 𝜖 3M, 𝜁 + 𝜖2M . Proof. — The 𝐴∞ -identities for M (+ the fact that 𝑒 𝑋 is a cycle) imply that for every 𝑏 ∈ M(𝑋) we have (2.16)

𝜇1M𝜇3M(𝑒 𝑋 , 𝑒 𝑋 , 𝑏) − 𝜇2M 𝑒 𝑋 , 𝜇2M(𝑒 𝑋 , 𝑏)




+ 𝜇3M 𝑒 𝑋 , 𝑒 𝑋 , 𝜇1M(𝑏) + 𝜇2M 𝜇2A(𝑒 𝑋 , 𝑒 𝑋 ), 𝑏 = 0.







Since A ∈ 𝑈 𝑒 (𝜁) we have 𝜇2A(𝑒 𝑋 , 𝑒 𝑋 ) = 𝑒 𝑋 +𝜇1A(𝑐), for some 𝑐 ∈ 𝐶 ≤𝜁 (𝑋 , 𝑋). Substituting this in (2.16) together with 𝜇2M(𝜇1A(𝑐), 𝑏) + 𝜇2M(𝑐, 𝜇1M(𝑏)) + 𝜇1M𝜇2M(𝑐, 𝑏) = 0 yields: 𝜇2M 𝑒 𝑋 , 𝜇2M(𝑒 𝑋 , 𝑏) − 𝜇2M(𝑒 𝑋 , 𝑏) = 𝜇1M ℎ(𝑏) + ℎ𝜇1M(𝑏),




where ℎ(𝑏) = 𝜇3M(𝑒 𝑋 , 𝑒 𝑋 , 𝑏) − 𝜇2M(𝑐, 𝑏). Clearly the chain homotopy ℎ shifts action by  not more than max 2𝑢 A + 𝜖 3M, 𝜁 + 𝜖2M .  We now return to the proof of the lemma. Proof of Lemma 2.9. — Denote C = C𝑜𝑛𝑒( 𝑓 ; 𝜌,  𝑓 ). Recall that this module has dis𝑓 crepancy ≤  C := max{M0 , M1 ,  𝑓 − 𝜖 1 }. Put 𝛿 := max 𝑢 A + 𝜖2C, 𝜅 M0 , 𝜅 M1 ,



𝜅 := max 2𝛿, 2𝑢 A + 𝜖 3C, 𝜁 + 𝜖2C .







It is easy to see that the latter expression for 𝜅 coincides with (2.15). For an 𝐴∞ -module M and 𝑋 ∈ Ob(A) we will typically denote by 𝑉M : 𝐻∗ M≤𝛼 (𝑋) −→ 𝐻∗ M≤𝛼+𝑟 (𝑋)







the map induced in homology by the chain map 𝑣 M : M≤𝛼 (𝑋) → M≤𝛼+𝑟 (𝑋),

𝑏 ↦−→ 𝜇2M(𝑒 𝑋 , 𝑏).

Here 𝑟 ∈ ℝ is chosen such that 𝑢 A + 𝜖2M ≤ 𝑟 so that 𝑣 M is well defined with the above given target. We will need to consider such maps for different values of 𝑟, and whenever a need to distinguish between them arises we will use additional 0 00 “decorations” such as 𝑉M , 𝑉M , etc. Fix 𝛼 ∈ ℝ, 𝑋 ∈ Ob(A). Since 𝑓

≤𝛼−𝜌−𝜖1

C≤𝛼 (𝑋) = Cone M0

𝑓1

(𝑋) −−→ M1≤𝛼 (𝑋)




we have a long exact sequence in homology: 𝜄

𝜋

𝑓

≤𝛼−𝜌−𝜖1

· · · → 𝐻 𝑘 M1≤𝛼 (𝑋) −−→ 𝐻 𝑘 C≤𝛼 (𝑋) −−−→ 𝐻 𝑘 M0







(𝑋) → · · · ,




where 𝜄 and 𝜋 are the maps in homology induced by the inclusion M1 (𝑋) → C(𝑋) and the projection C(𝑋) → M0 (𝑋) respectively.

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Replacing 𝛼 by 𝛼+𝛿 and by 𝛼+𝜅 we obtain two similar long exact sequences. These three sequences are mapped one to the other via maps induced from the inclusions coming from raising the action level from 𝛼 to 𝛼 + 𝛿 and then to 𝛼 +𝜅. In particular, the degree-𝑘 chunks of these exact sequences gives the following commutative diagram with exact rows: 𝐻 𝑘 M1≤𝛼 (𝑋)

/ 𝐻 𝑘 C≤𝛼 (𝑋)




0 =𝑖 0 𝑉M M 1

0 𝑉C 𝑖 0C

1



(2.17)

𝐻 𝑘 M1≤𝛼+𝛿 (𝑋)




𝜄

00 =𝑖 00 𝑉M M 1

𝑓

/ 𝐻 𝑘 M≤𝛼−𝜌−𝜖1 (𝑋) 0




/ 𝐻 𝑘 C≤𝛼+𝛿 (𝑋)) 00 𝑉C

1



𝐻 𝑘 M1≤𝛼+𝜅 (𝑋)



0

𝑓 ≤𝛼−𝜌−𝜖1 +𝛿

𝜋

/ 𝐻𝑘 M 0

𝑖 00C

/ 𝐻 𝑘 C≤𝛼+𝜅 (𝑋)




0 =𝑖 0 𝑉M M







0

(𝑋)




00 =𝑖 00 𝑉M M 0

𝑓 ≤𝛼−𝜌−𝜖1 +𝜅

/ 𝐻𝑘 M 0







0

(𝑋)




The maps 𝑖 0C, 𝑖 00C are induced by the corresponding inclusions and similarly for 𝑖 0M0 , 𝑖 00M0 , 𝑖 0M1 , 𝑖 00M1 . By assumption (and by the choices of 𝛿 and 𝜅) we have 0 𝑉M = 𝑖 0M𝑗 , 𝑗

00 𝑉M = 𝑖 00M𝑗 , 𝑗

for

𝑗 = 0, 1.

0 Note also that each of the maps 𝑉C and 𝑖 0C makes the above diagram commutative, 00 00 and similarly for 𝑉C and 𝑖 C. Denote by 000 𝑉C : 𝐻 𝑘 C≤𝛼 (𝑋) −→ 𝐻 𝑘 C≤𝛼+𝜅 (𝑋)







the map induced in homology by 𝑣 C : C≤𝛼 (𝑋) → C≤𝛼+𝜅 (𝑋), and by ≤𝛼 ≤𝛼+𝜅 𝑖 000 (𝑋) C : 𝐻 𝑘 C (𝑋) −→ 𝐻 𝑘 C







the map induced by the inclusion. Clearly we have 000 0 00 𝑉C = 𝑖 00C ◦ 𝑉C = 𝑉C ◦ 𝑖 0C. 000 To prove the lemma, we need to show that 𝑉C (𝑥) = 𝑖 000 (𝑥) for all 𝑥 ∈ 𝐻 𝑘 ( C≤𝛼 (𝑋)). C 0 To prove the latter equality, we first note that since both 𝑉C and 𝑖 0C make diagram (2.17) commutative, we have 0 𝑉C (𝑥) − 𝑖 0C(𝑥) ∈ ker 𝜋 = image 𝜄. 0 00 Now write 𝑉C (𝑥) − 𝑖 0C(𝑥) = 𝜄(𝑦) for some 𝑦 ∈ 𝐻 𝑘 (M1≤𝛼+𝛿 (𝑋)). As both 𝑉C and 𝑖 00C make 00 00 diagram (2.17) commutative we also have 𝑉C ◦ 𝜄(𝑦) = 𝑖 C ◦ 𝜄(𝑦). It follows that 00 0 0 𝑉C 𝑉C (𝑥) − 𝑖 0C(𝑥) = 𝑖 00C 𝑉C (𝑥) − 𝑖 0C(𝑥) .







Applying Proposition 2.10 with M = C we obtain 000 00 0 𝑉C (𝑥) − 𝑉C ◦ 𝑖 0C(𝑥) = 𝑖 00C ◦ 𝑉C (𝑥) − 𝑖 000 C (𝑥). 000 00 0 Since 𝑉C = 𝑉C ◦ 𝑖 0C = 𝑖 00C ◦ 𝑉C the lemma follows.

ASTÉRISQUE 426



2.5. THE 𝜆-MAP

31

2.5. The 𝜆-map Let A be an 𝐴∞ -category and M an A-module. Let 𝑌 ∈ Ob(A) and denote by 𝒴 the Yoneda module corresponding to 𝑌. Consider the map: (2.18)

𝜆 : M(𝑌) −→ hom(𝒴, M),

𝑐 ↦−→ 𝜆(𝑐) = 𝜆(𝑐)1 , 𝜆(𝑐)2 , . . . , 𝜆(𝑐)𝑑 , . . . ,




where 𝜆(𝑐)𝑑 (𝑎 1 , . . . , 𝑎 𝑑−1 , 𝑏) = 𝜇M (𝑎 , . . . , 𝑎 𝑑−1 , 𝑏, 𝑐). 𝑑+1 1 This map was defined by Seidel [Sei08, Section (1l)] in the context of the Yoneda embedding of 𝐴∞ -categories. A straightforward calculation shows that it is a chain map. We will refer to it from now on as the 𝜆-map. Seidel [Sei08, Lemma 2.12] proves that, under the additional assumptions that A and M are h-unital, the 𝜆-map is a quasi-isomorphism. Our goal is to establish a weakly filtered analogue of this result. We begin with a technical assumption on a given object 𝑌 ∈ Ob(A).

Assumption 𝑈 𝑅,𝑒 . — Let 𝜅 ≥ 𝜖2A + 𝑢 A be a real number (recall that 𝜖2A and 𝑢 A are the discrepancies associated respectively to the 𝜇2 -operation and units in A, see § 2.1.1). We say that 𝑌 satisfies Assumption 𝑈 𝑅,𝑒 (𝜅) (or 𝑌 ∈ 𝑈 𝑅,𝑒 (𝜅) for short) if for every 𝑋 ∈ Ob(A) the map 𝐶(𝑋 , 𝑌) −→ 𝐶(𝑋 , 𝑌),

𝑏 ↦−→ 𝜇2 (𝑏, 𝑒𝑌 )

is chain homotopic to the identity via a chain homotopy ℎ 𝑋 that shifts action by ≤ 𝜅. The superscript 𝑅, 𝑒 stand for “Right”-multiplication with 𝑒𝑌 . We now define the right setting for the 𝜆-map in the weakly filtered case. Assume that Aand M are both weakly filtered with discrepancies ≤ A and ≤ M respectively. Clearly 𝒴 is also a weakly filtered module with discrepancy ≤ A. Without loss of generality we assume from now on that M ≥ A so that 𝒴 can be regarded also as a weakly filtered module with discrepancy ≤ M. (If needed, we can always increase M and M will continue being weakly filtered with discrepancy less than the increased M.) Let  ℎ be any sequence that satisfies Assumption E and assume in addition that 𝜖 𝑑ℎ ≥ 𝜖 M 𝑑+1

(2.19)

for all 𝑑.

Under these assumptions, the 𝜆-map restricts to maps: ℎ

𝜆 𝛼 : M≤𝛼 (𝑌) −→ hom≤𝛼; (𝒴, M),

(2.20)

defined for all 𝛼 ∈ ℝ. Since  ℎ satisfies Assumption E, the right-hand side of (2.20) is a chain complex with respect to 𝜇mod and the 𝜆-map from (2.20) is a chain map. 1 Let A, M, 𝑌 and 𝒴 be as at the beginning of Section 2.5. Fix also M,  ℎ as above. For every 𝛼 ∈ ℝ set ℎ

H≤𝛼 := hom≤𝛼; (𝒴, M) and for every 𝑘 ≥ 1: ≤𝛼 𝑄 (𝑘) := 𝑡 ∈ H≤𝛼 ; 𝑡1 = · · · = 𝑡 𝑘 = 0 ,





≤𝛼 ≤𝛼 H(𝑘) := H≤𝛼 /𝑄 (𝑘) .

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As explained above, the 𝜆-map restricts to maps 𝜆 𝛼 : M≤𝛼 (𝑌) → H≤𝛼 for every 𝛼 ∈ ℝ and we also have the induced maps: ≤𝛼 𝛼 𝜆(𝑘) : M≤𝛼 (𝑌) −→ H(𝑘) ≤𝛼 defined by composing 𝜆 𝛼 with the quotient map 𝜋(𝑘) : H≤𝛼 → H(𝑘) .

Proposition 2.11. — Suppose that A is h-unital in the weakly filtered sense with discrepancy of units ≤ 𝑢 A. Let 𝜅 ∈ ℝ such that 𝜅 ≥ 𝑢 A + 𝜖2M, 𝑢 A + 𝜖 2A and assume that M ∈ 𝑈𝑚 (𝜅) and 𝑌 ∈ 𝑈 𝑅,𝑒 (𝜅). Let 𝛼 ∈ ℝ. Fix 1 ≤ ℓ ∈ ℤ and put 𝛼0 := 𝛼 + ℓ 𝜅. Consider the commutative diagram in cohomology: 𝐻∗ M≤𝛼 (𝑌) 𝑖𝐻






/ 𝐻∗ H≤𝛼




𝑖𝐻

0

𝐻∗ M≤𝛼 (𝑌)

(2.21)

𝜆∗𝛼





/ 𝐻∗ H≤𝛼0

0 𝜆∗𝛼

𝐻 𝜋(ℓ )

id


0 𝐻∗ M≤𝛼 (𝑌)


/ 𝐻∗ H≤𝛼0 (ℓ )

0 𝜆(ℓ𝛼 )∗

where the 𝑖 𝐻 maps are induced by the inclusions M≤𝛼 (𝑌) → M≤𝛼 (𝑌) and H≤𝛼 → H≤𝛼 0 0 and 𝜋(ℓ𝐻) is induced by the projection 𝜋(ℓ ) : H≤𝛼 → H(ℓ≤𝛼 . Then for every 𝑏 ∈ 𝐻∗ (H≤𝛼 ) there ) 0

0

0

exists 𝑐 ∈ 𝐻∗ (M≤𝛼 (𝑌)) such that 𝜋(ℓ𝐻) ◦ 𝑖 𝐻 (𝑏) = 𝜆(ℓ𝛼 )∗ (𝑐). 0

0

In other words, for every cycle 𝛽 ∈ H≤𝛼 there exists a cycle 𝛾 ∈ M≤𝛼 (𝑌) such that 𝛽 = 𝜆(𝛾) + 𝜇mod (𝜃) + 𝜏, 1

(2.22) 0

0

for some 𝜃 ∈ H≤𝛼 and some cycle 𝜏 = (𝜏1 , 𝜏2 , . . . ) ∈ H≤𝛼 with 𝜏1 = · · · = 𝜏ℓ = 0. Proof. — The proof below follows the general scheme of the proof of Lemma 2.12 from of [Sei08], however the weakly filtered setting entails significant adjustments with respect to [Sei08]. Before we go on, two quick remarks on grading. The first is that in this paper we generally work in an ungraded framework. Nevertheless, the proof below works also in the graded case, hence we have written it in this setting. 1 The second remark is that, in order to keep compatibility with the proof of Lemma 2.12 from of [Sei08], we will work in this proof with cohomological grading (although we generally use homological conventions). We will therefore denote by 𝐻 𝑖 the homology in “cohomological degree” 𝑖. 1. The ungraded case can be viewed as a special case of the graded one, by replacing each chain complex in the statement of Proposition 2.11 by a graded one which equals in all degrees to the original chain complex. Note that, in the graded case, none of the chain complexes in the statement of Proposition 2.11 is assumed to be bounded.

ASTÉRISQUE 426

2.5. THE 𝜆-MAP

33

We begin with some preparations regarding the weakly filtered version of the ≤𝜌 𝜆-map. Let 𝜌 ∈ ℝ, 𝑑 ∈ ℕ . Recall that we have the chain map 𝜆𝜌 : M≤𝜌 (𝑌) → H(𝑑) and consider its mapping cone: 𝜆𝜌

𝜌

≤𝜌


K(𝑑) = Cone M≤𝜌 (𝑌) −−→ H(𝑑) . 𝜌

Define a decreasing filtration 𝐹 𝑟 K(𝑑) , 𝑟 ∈ ℤ≥0 on this chain complex by setting 𝜌 𝐹 𝑟 K(𝑑)

(2.23)

𝜌  K(𝑑)     ≤𝜌 

if 𝑟 = 0, if 𝑟 = 1,

= H(𝑑)

     𝑓 ∈ H≤𝜌 ; 𝑓1 = · · · = 𝑓𝑟−1 = 0  (𝑑)

if 2 ≤ 𝑟. 𝜌

Note that this is a bounded filtration and we actually have 𝐹 𝑟 K(𝑑) = 0 for 𝑟 ≥ 𝑑 + 1. 𝑝,𝑞

Consider now the cohomological spectral sequence {𝐸𝑟 (𝜌), 𝜕𝑟 } 𝑟∈ℤ≥0 associated to the filtration 𝐹 • . Since the filtration is bounded the spectral sequence converges 𝜌 to 𝐻 ∗ (K(𝑑) ). Note also that for 𝜌 ≤ 𝜌0 we have an obvious inclusion of chain com𝜌 𝜌0 plexes 𝑖 : K(𝑑) → K(𝑑) . Moreover, this inclusion preserves the filtrations 𝐹 • on the corresponding chain complexes. Therefore, 𝑖 induces a map of spectral sequences 𝑝,𝑞

𝑝,𝑞

𝑖 𝑟𝐸 : 𝐸𝑟 (𝜌) −→ 𝐸𝑟+1 (𝜌0),

for all 𝑟 ≥ 0 and 𝑝, 𝑞 . 𝑝,𝑞

We now describe more explicitly the first two pages of 𝐸𝑟 (𝜌). A simple calculation gives the following description of the 𝐸0 -page of this spectral sequence. We have 𝑝,• 𝐸0 (𝜌) = 0 for 𝑝 > 𝑑 and for 𝑝 < 0. Next we have 𝐸00,• (𝜌) = M≤𝜌 (𝑌)• , where the superscript • stands here for the (cohomological) grading of the chain complex M≤𝜌 (𝑌). 0,𝑞 0,𝑞+1 The differential 𝜕0 : 𝐸0 (𝜌) → 𝐸0 (𝜌) is simply 𝜇1M. The rest of the columns, 1 ≤ 𝑝 ≤ 𝑑, are 𝑝,•

(2.24)

𝐸0 (𝜌) =

Ö

≤𝜌+𝜖 𝑝ℎ ;•

hom𝑅

𝐶(𝑋0 , 𝑋1 ) ⊗ · · · ⊗ 𝐶(𝑋𝑝−2 , 𝑋𝑝−1 ) ⊗ 𝐶(𝑋𝑝−1 , 𝑌), M(𝑋0 ) ,




where the product is taken for 𝑋0 , . . . , 𝑋𝑝−1 ∈ Ob(A), the superscript ‘• ’ stands again ≤𝜌+𝜖 𝑝ℎ

for (cohomological) grading and hom𝑅 stands for 𝑅-linear homomorphisms that shift action by not more than 𝜌 + 𝜖 𝑝ℎ . (Recall that  ℎ has been fixed at the beginning ≤𝜌

of Section 2.5 and is used in the definitions of H≤𝜌 and H(𝑑) .) For 1 ≤ 𝑝 ≤ 𝑑, the 𝑝,𝑞

𝑝,𝑞+1

differentials 𝜕0 : 𝐸0 (𝜌) → 𝐸0 (𝜌) are induced in a standard way from 𝜇1A and 𝜇1M. 𝑝,• The 𝐸1 -page is consequently the following: 𝐸1 (𝜌) = 0 for all 𝑝 > 𝑑 and for 𝑝 < 0. For 𝑝 = 0 we have 𝐸1 (𝜌) = 𝐻 𝑞 (M≤𝜌 (𝑌)) for all 𝑞. And for 1 ≤ 𝑝 ≤ 𝑑 we have 0,𝑞

𝑝,𝑞

(2.25) 𝐸1 (𝜌) =

Ö

≤𝜌+𝜖 𝑝ℎ

𝐻 𝑞 hom𝑅

𝑋0 ,...,𝑋𝑝−1 ∈Ob(A)

𝐶(𝑋0 , 𝑋1 ) ⊗ · · ·

· · · ⊗ 𝐶(𝑋𝑝−2 , 𝑋𝑝−1 ) ⊗ 𝐶(𝑋𝑝−1 , 𝑌), M(𝑋0 ) . 𝑝,𝑞

We now describe the differentials 𝜕1 : 𝐸1

𝑝+1,𝑞

→ 𝐸1

on the 𝐸1 -page.

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We start with 𝑝 = 0. Let [𝑐] ∈ 𝐸1 = 𝐻 𝑞 (M≤𝜌 (𝑌)), where 𝑐 is a cycle. Then 0,𝑞

𝜕1 [𝑐] ∈ 𝐸1 = 1,𝑞

≤𝜌+𝜖 1ℎ

Ö

𝐻 𝑞 hom𝑅

𝐶(𝑋 , 𝑌), M(𝑋)





𝑋∈Ob(A)

is the cycle represented by the homomorphism 𝐶(𝑋 , 𝑌) −→ M(𝑋),

𝑏 ↦−→ 𝜇2M(𝑏, 𝑐).

It is easy to check that this homomorphism is a cycle and that it shifts action by not more than 𝜌 + 𝜖1ℎ . (The latter hold because 𝜖1ℎ ≥ 𝜖 2M by (2.19).) The formula for 𝜕1 for 1 ≤ 𝑝 ≤ 𝑑 − 1 is the following. Let 𝑓 be an element in the RHS of (2.24) which is a cycle. Then 𝑝+1,𝑞

𝜕1 [ 𝑓 ] = [𝑔] ∈ 𝐸1

,

where 𝑔 is a collection of 𝑅-linear homomorphism 𝑔 : 𝐶(𝑋0 , 𝑋1 ) ⊗ · · · ⊗ 𝐶(𝑋𝑝−1 , 𝑋𝑝 ) ⊗ 𝐶(𝑋𝑝 , 𝑌) → M(𝑋0 ), defined for all objects 𝑋0 , . . . , 𝑋𝑝 ∈ Ob(A) and is given by the formula: (2.26)

𝑔(𝑎 1 , . . . , 𝑎 𝑝 , 𝑏) ↦−→ ±𝜇2M 𝑎1 , 𝑓 (𝑎2 , . . . , 𝑎 𝑝 , 𝑏) ± 𝑓 𝑎1 , . . . , 𝑎 𝑝−1 , 𝜇2A(𝑎 𝑝 , 𝑏)




+

𝑝−2 Õ




± 𝑓 𝑎 1 , . . . , 𝜇2A(𝑎 𝑛+1 , 𝑎 𝑛+2 ), . . . , 𝑏 .




𝑛=0

This follows from a direct calculation. See the proof of Lemma 2.12 in [Sei08] for the precise signs in formula (2.26). Note also that 𝑔 shifts action by ℎ ℎ ≤ max 𝜌 + 𝜖 𝑝−1 + 𝜖 2M, 𝜌 + 𝜖 A 𝑝−1 ≤ 𝜌 + 𝜖 𝑝 ,





where the latter inequality follows from Assumption E(𝑚 , A). 𝜌

𝜌+𝜅

Consider now the inclusion 𝑖 : K(𝑑) → K(𝑑) . As indicated earlier this induces a map of spectral sequences 𝑖 𝐸 : 𝐸(𝜌) → 𝐸(𝜌 + 𝜅), namely 𝑝,𝑞

𝑝,𝑞

𝑖 𝑟𝐸 : 𝐸𝑟 (𝜌) −→ 𝐸𝑟 (𝜌 + 𝜅),

for all 𝑟 ≥ 0.

•,𝑞

•,𝑞

Claim 2.12. — For every 𝑞 the chain map 𝑖1𝐸 : 𝐸1 (𝜌) → 𝐸1 (𝜌 + 𝜅) is chain homotopic to 0 in the degree range 0 ≤ • ≤ 𝑑 − 1. In other words, for every 𝑞 there exist homomorphisms 𝑝,𝑞

𝑝−1,𝑞

𝑆 𝑝,𝑞 : 𝐸1 (𝜌) −→ 𝐸1

(𝜌 + 𝜅),

defined for all 𝑝, such that (2.27)

𝑖1𝐸

𝑝,𝑞

𝐸1 (𝜌)

= 𝜕1 ◦ 𝑆 𝑝,𝑞 + 𝑆 𝑝+1,𝑞 ◦ 𝜕1 ,

for all 0 ≤ 𝑝 ≤ 𝑑 − 1.

We postpone the proof of this claim till later in this section and continue now with the proof of Proposition 2.11. 𝑝,𝑞

𝑝,𝑞

Claim 2.12 implies that 𝑖2𝐸 : 𝐸2 (𝜌) → 𝐸2 (𝜌+𝜅) is the 0 map for every 0 ≤ 𝑝 ≤ 𝑑−1 𝑝,𝑞 𝑝,𝑞 and every 𝑞. It follows that the same holds for the maps 𝑖 𝑟𝐸 : 𝐸𝑟 (𝜌) → 𝐸𝑟 (𝜌 + 𝜅) for every 𝑟 ≥ 2.

ASTÉRISQUE 426

2.5. THE 𝜆-MAP

35

Since both the spectral sequences converge after a finite number of pages (in fact 𝐸 : 𝐸 𝑝,𝑞 (𝜌) → 𝐸 𝑝,𝑞 (𝜌 + 𝜅) is 0 for they collapse at page 𝑟 = 𝑑 + 1) we conclude that 𝑖 ∞ ∞ ∞ 𝜌 𝜌 • ∗ all 0 ≤ 𝑝 ≤ 𝑑 − 1 and all 𝑞. Denote by 𝐹 𝐻 (K(𝑑) ) the filtration on 𝐻 ∗ (K(𝑑) ) induced 𝜌 by 𝐹 • K(𝑑) . Since 𝑝,𝑞

𝜌

𝜌

𝐸∞ (𝜌) = 𝐹 𝑝 𝐻 𝑝+𝑞 (K(𝑑) )/𝐹 𝑝+1 𝐻 𝑝+𝑞 (K(𝑑) ), 𝑝,𝑞

and similarly for 𝐸∞ (𝜌 + 𝜅), we have proved the following auxiliary statement: 𝜌+𝜅

𝜌

Lemma 2.13. — The inclusion 𝑖 : K(𝑑) → K(𝑑) induces in homology the map 𝜌+𝜅

𝜌

𝑖 𝐻 : 𝐻 𝑛 (K(𝑑) ) −→ 𝐻 𝑛 (K(𝑑) ) 𝜌+𝜅

𝜌

which sends 𝐹 𝑝 𝐻 𝑛 (K(𝑑) ) to 𝐹 𝑝+1 𝐻 𝑛 (K(𝑑) ) for every 𝑛 and 0 ≤ 𝑝 ≤ 𝑑 − 1. We are now in position to conclude the proof of Proposition 2.11. Fix 𝛼, ℓ and 𝛼0 as in the statement of the proposition. 𝛼 Choose 𝑑  ℓ and apply what we have proved above to K(𝑑) (i.e. take 𝜌 = 𝛼). 𝛼 𝛼+𝜅 Lemma 2.13, applied with 𝑝 = 0, implies that 𝑖 𝐻 maps 𝐻 𝑛 (K(𝑑) ) to 𝐹 1 𝐻 𝑛 (K(𝑑) ) for all 𝑛. 𝛼+𝜅 𝛼+2𝜅 Apply Lemma 2.13 this time with 𝑝 = 1, 𝜌 = 𝛼 + 𝜅 and K(𝑑) → K(𝑑) . Together 𝛼 𝛼+2𝜅 with the previous conclusion we infer that 2 𝑖 𝐻 maps 𝐻 𝑛 (K(𝑑) ) to 𝐹 2 𝐻 𝑛 (K(𝑑) ) for all 𝑛. Applying the same argument over and over again, ℓ times, we conclude that the 𝛼+ℓ 𝜅 𝛼+ℓ 𝜅 𝛼 𝛼 map 𝑖 𝐻 : 𝐻 𝑛 (K(𝑑) ) → 𝐻 𝑛 (K(𝑑) ) induced by the inclusion K(𝑑) → K(𝑑) maps 𝛼+ℓ 𝜅 𝛼 𝐻 𝑛 (K(𝑑) ) to 𝐹ℓ 𝐻 𝑛 (K(𝑑) ). ≤𝛼0 Let now 𝛽 ∈ H≤𝛼 be a cycle and denote by 𝛽¯ its image in H(𝑑) , where 𝛼0 = 𝛼 + ℓ 𝜅. 0 ¯ ∈ K𝛼 . By what we have proved before we know that [(0, 𝛽)] ¯ Consider the cycle (0, 𝛽) (𝑑)

0

ℎ

𝛼 belongs to 𝐹ℓ 𝐻 ∗ (K(𝑑) ). It follows that there exists 𝜏0 ∈ hom≤𝛼 ; (𝒴, M) such that 0

𝜏10 = · · · = 𝜏ℓ0 = 0 and



¯ = (0, 𝜏0) (0, 𝛽)





0

0



𝛼 in 𝐻 ∗ (K(𝑑) ). 0

ℎ

Therefore, there exist 𝛾 ∈ M≤𝛼 (𝑌) and 𝜃 ∈ hom≤𝛼 ; (𝒴, M) such that ¯ = (0, 𝜏0) + 𝜇M(𝛾), 𝜆 𝛼0 (𝛾) + 𝜇hom (𝜃) (0, 𝛽) 1 1 (𝑑)




𝛼 𝛼 in K(𝑑) . In order to lift the last equation from K(𝑑) to 0

0

0

𝜆𝛼

0

Cone M≤𝛼 (𝑌) −−−→ H≤𝛼

0




we can correct if necessary the terms beyond order 𝑑 by replacing 𝜏0 with a suitable 𝜏 that coincides with 𝜏0 up to order 𝑑 (recall that 𝑑  ℓ ). 2. We have denoted here by the same symbol, 𝑖 𝐻 , the maps induced in homology by the different 𝜌

𝜌+𝜅

𝜌+𝜅

𝜌+2𝜅

inclusions: K(𝑑) → K(𝑑) , K(𝑑) → K(𝑑)

𝜌

𝜌+2𝜅

and K(𝑑) → K(𝑑) . Below we will continue with this notation.

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0

Summing up, we have proved that there exists a cycle 𝛾 ∈ M≤𝛼 (𝑌), and a pre0 ℎ module homomorphism 𝜃 ∈ hom≤𝛼 ; (𝒴, M) such that 𝛽 = 𝜆(𝛾) + 𝜇mod (𝜃) + 𝜏, 1 0

ℎ

where 𝜏 ∈ hom≤𝛼 ; (𝒴, M) is a cycle with 𝜏1 = · · · = 𝜏ℓ = 0, as claimed by the proposition 3. This concludes the proof of Proposition 2.11, modulo the proof of Claim 2.12.  Proof of Claim 2.12. — Fix 𝑞. We define the chain homotopy 𝑆•,𝑞 as follows. Define −1,𝑞 𝑆0,𝑞 = 0 (note that 𝐸1 (𝜌) = 0). Next, to define 𝑆1,𝑞 , let ≤𝜌+𝜖1ℎ

Ö

𝑓 ∈ 𝐸0 (𝜌) = 1,𝑞

hom𝑅

𝐶(𝑋 , 𝑌), M(𝑋)




𝑋∈Ob(A)

be a 𝜕0 -cycle. We define ℎ

𝑆1,𝑞 [ 𝑓 ] := 𝑓 (𝑒𝑌 ) ∈ 𝐸1 (𝜌 + 𝜅) = 𝐻 𝑞 M≤𝜌+𝜖1 +𝜅 (𝑋) .





0,𝑞




Since 𝜅 ≥ 𝑢 A, 𝑓 (𝑒𝑌 ) indeed belongs to 𝐸0 (𝜌 + 𝜅). Moreover, a straightforward calculation shows that 𝑓 (𝑒𝑌 ) is a 𝜕0 -cycle and that its homology class [ 𝑓 (𝑒𝑌 )] depends 1,𝑞 only on the homology class [ 𝑓 ] ∈ 𝐸1 (𝜌). For the range of degrees 2 ≤ 𝑝 ≤ 𝑑 we define 𝑆 𝑝,𝑞 by a similar formula: let 𝑝,𝑞 𝑓 ∈ 𝐸0 (𝜌), i.e. a collection of 𝑅-linear homomorphism as in (2.24). Assume that 𝑓 is 0,𝑞

𝑝−1,𝑞

a 𝜕0 -cycle. Define 𝑆 𝑝,𝑞 [ 𝑓 ] to be the homology class [𝑔] ∈ 𝐸1 𝑔∈

𝑝−1,𝑞 𝐸0 (𝜌

(𝜌 + 𝜅) of the element

+ 𝜅) given by 𝑔(𝑎1 , . . . , 𝑎 𝑝−2 , 𝑏) = 𝑓 (𝑎1 , . . . , 𝑎 𝑝−2 , 𝑏, 𝑒𝑌 ),

for all 𝑎 𝑖 ∈ 𝐶(𝑋𝑖−1 , 𝑋𝑖 ), 𝑖 = 1, . . . , 𝑝 − 2, and 𝑏 ∈ 𝐶(𝑋𝑝−2 , 𝑌). Since 𝜅 ≥ 𝑢 A, we 𝑝−1,𝑞

have 𝑔 ∈ 𝐸0

(𝜌 + 𝜅). A straightforward calculation shows that 𝑔 is a 𝜕0 -cycle and 𝑝−1,𝑞

moreover its homology class, [𝑔] ∈ 𝐸1 (𝜌 + 𝜅) depends only on the homology class [ 𝑓 ] of 𝑓 . This concludes the definition 4 of the maps 𝑆 𝑝,𝑞 . 𝑝,𝑞

We verify now the identity (2.27). We begin with 2 ≤ 𝑝 ≤ 𝑑 − 1. Let 𝑓 ∈ 𝐸0 (𝜌) be a cycle. A straightforward calculation shows that (𝜕1 𝑆 𝑝,𝑞 + 𝑆 𝑝+1,𝑞 𝜕1 )[ 𝑓 ] = [ e 𝑓 ], where e 𝑓 (𝑎1 , . . . , 𝑎 𝑝−1 , 𝑏) = 𝑓 𝑎1 , . . . , 𝑎 𝑝−1 , 𝜇2A(𝑏, 𝑒𝑌 ) . We claim that




𝑝,𝑞

[e 𝑓 ] = [ 𝑓 ] in 𝐸1 (𝜌 + 𝜅). Indeed, since 𝑌 belongs to 𝑈 𝑅,𝑒 (𝜅) there exists a chain homotopy ℎ 𝑋𝑝−1 : 𝐶(𝑋𝑝−1 , 𝑌) → 𝐶(𝑋𝑝−1 , 𝑌) that shifts action by ≤ 𝜅 such that 𝜇2A(𝑏, 𝑒𝑌 ) = 𝑏 + ℎ 𝑋𝑝−1 𝜇1A(𝑏) + 𝜇1Aℎ 𝑋𝑝−1 (𝑏),

for all 𝑏 ∈ 𝐶(𝑋𝑝−1 , 𝑌).

3. The statement concerning diagram (2.21) is a rephrasing of what we have just proved. 4. For 𝑝’s outside of the range 0, . . . , 𝑑 we can define 𝑆 𝑝,𝑞 in an arbitrary way.

ASTÉRISQUE 426

37

2.6. STRUCTURE THEOREM FOR WEAKLY FILTERED ITERATED CONES

𝑝,𝑞−1

Define 𝜓 ∈ 𝐸0

(𝜌 + 𝜅) by 𝜓(𝑎1 , . . . , 𝑎 𝑝−1 , 𝑏) := 𝑓 𝑎1 , . . . , 𝑎 𝑝−1 , ℎ 𝑋𝑝−1 (𝑏) .




𝑝,𝑞

A straightforward calculation shows that e 𝑓 − 𝑓 = 𝜕0 𝜓, hence [ e 𝑓 ] = [ 𝑓 ] in 𝐸1 (𝜌 + 𝜅). This proves (2.27) for 2 ≤ 𝑝 ≤ 𝑑. A similar argument shows that (2.27) holds also for 𝑝 = 1. It remains to verify (2.27) the case 𝑝 = 0. Let 𝑚 ∈ M≤𝜌 (𝑌) be a cycle. We have (𝜕1 𝑆0,𝑞 + 𝑆1,𝑞 𝜕1 )[𝑚] = 𝑆1,𝑞 𝜕1 [𝑚] = 𝜕1 [𝑚] (𝑒𝑌 ) = 𝜇2M(𝑒𝑌 , 𝑚) .








By assumption M ∈ 𝑈𝑚 (𝜅), hence [𝜇2M(𝑒𝑌 , 𝑚)] = [𝑚] in 𝐻 𝑞 (M≤𝜌+𝜅 (𝑌)). This proves (2.27) for 𝑝 = 0 and concludes the proof of Claim 2.12.



2.6. Structure theorem for weakly filtered iterated cones Let A be an h-unital weakly filtered 𝐴∞ -category with discrepancy ≤ A and discrepancy of units 𝑢 A. Let 𝐿0 , . . . , 𝐿𝑟 ∈ Ob(A) and for every 𝑖 denote by L𝑖 the Yoneda module associated to 𝐿 𝑖 , viewed as a weakly filtered module. In this section we analyze iterated cones in the weakly filtered framework. By iterated cones we mean modules of the type 𝜙𝑟

(2.28)

𝜙 𝑟−1

C𝑜𝑛𝑒(L𝑟 −−→ C𝑜𝑛𝑒 L𝑟−1 −−−−→ 𝜙2 𝜙1




C𝑜𝑛𝑒 · · · C𝑜𝑛𝑒 L2 −−→ C𝑜𝑛𝑒 L1 −−→ L0 · · · .

The weakly filtered structure is defined by iterating the construction from Section 2.4. More precisely, we define a sequence of weakly filtered A-modules K0 , . . . , K𝑟 as follows. We start by setting K0 := L0 which is a weakly filtered module with discrepancy ≤ K0 := A. Note that all the modules L𝑖 have discrepancy ≤ A too. Suppose 𝜙1 that 𝜙1 ∈ hom≤𝜌1 ;δ (L1 , K0 ) is a module homomorphism, where 𝜌1 ∈ ℝ and δ 𝜙1 is some sequence. We do not assume that δ 𝜙1 satisfies anything like Assumption E. We define (𝜙1 ;𝜌1 ,δ 𝜙1 )
K1 = C𝑜𝑛𝑒 L1 −−−−−−−−−→ K0 . 𝜙

Since K0 = A, the discrepancy of K1 is ≤ K1 := max{A, δ 𝜙1 − 𝛿 1 1 }. Let 𝑖 ≥ 1 and suppose that we have already defined the weakly filtered modules K0 , . . . , K𝑖 . Let 𝜙 𝑖+1 : L𝑖+1 → K𝑖 be a module homomorphism that shifts action by ≤ 𝜌 𝑖+1 and has discrepancy ≤ δ 𝜙 𝑖+1 . Again, we do not assume that δ 𝜙 𝑖+1 satisfies any assumption of the type E. We define (𝜙 𝑖+1 ;𝜌 𝑖+1 ,δ 𝜙 𝑖+1 )

K𝑖+1 = C𝑜𝑛𝑒 L𝑖+1 −−−−−−−−−−−−−→ K𝑖 .




𝜙

The A-module K𝑖+1 has discrepancy ≤ K𝑖+1 := max{K𝑖 , δ 𝜙 𝑖+1 − 𝛿1 𝑖+1 } because (by induction) K𝑖 ≥ A. The final A-module K𝑟 is precisely the one described by (2.28) and moreover now it also has the structure of a weakly filtered module.

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The following expressions will be used frequently in what follows:

(2.29)

𝜒𝑚,𝑑 :=

𝑚 𝑑+𝑚 Õ Õ

𝜙𝑗

𝛿𝑖 +

𝑑+𝑚 Õ

𝑗=1 𝑖=1

𝜖A 𝑖 ,

𝜉𝑞 := 𝜅 +

𝑖=1

𝑞+3 Õ

𝜖A 𝑖 +

𝑖=1

𝑞 Õ 𝑞+2 Õ

𝜙𝑗

𝛿𝑖 .

𝑗=1 𝑖=1

Theorem 2.14. — Let K𝑖 , 0 ≤ 𝑖 ≤ 𝑟 be as above. Assume that A is h-unital in the weakly filtered sense with discrepancy of units ≤ 𝑢 A. Let 𝜅 ≥ 2𝑢 A + 𝜖 2A be a real number and assume that A and the objects 𝐿 𝑖 satisfy the following two conditions: ⊲ A ∈ 𝑈 𝑒 (𝜅). ⊲ For every 0 ≤ 𝑖 ≤ 𝑟, 𝐿 𝑖 ∈ 𝑈 𝑅,𝑒 (𝜅), and 𝐿 𝑖 ∈ 𝑈𝑚 (𝜅). Then there exists a weakly filtered A-module M with the following properties: (i) For every 𝑋 ∈ Ob(A), we have M(𝑋) = K𝑟 (𝑋) as 𝑅-modules, namely the 𝑅-module M(𝑋) is a direct sum: M(𝑋) = 𝐶(𝑋 , 𝐿0 ) ⊕ 𝐶(𝑋 , 𝐿1 ) ⊕ · · · ⊕ 𝐶(𝑋 , 𝐿𝑟 ).

(2.30)

(ii) Denote by 𝜇1M the differential of the chain complex M(𝑋). Then the matrix of 𝜇1M with respect to the splitting (2.30) has the following shape: 𝜇1M = (𝑎 𝑖𝑗 )0≤𝑖,𝑗≤𝑟 with 𝑎 𝑖,𝑗 : 𝐶(𝑋 , 𝐿 𝑗 ) −→ 𝐶(𝑋 , 𝐿 𝑖 ), where 𝛼) 𝑎 𝑖,𝑗 = 0 for every 𝑖 > 𝑗. In other words, the matrix of 𝜇1M is upper triangular. 𝛽) 𝑎 𝑖,𝑖 = 𝜇1A : 𝐶(𝑋 , 𝐿 𝑖 ) → 𝐶(𝑋 , 𝐿 𝑖 ). 𝛾) There exist elements 𝑐 𝑞,𝑝 ∈ 𝐶(𝐿 𝑞 , 𝐿 𝑝 ) for all 0 ≤ 𝑝 < 𝑞 ≤ 𝑟, such that for every 𝑖 < 𝑗 the (𝑖, 𝑗)-th entry of the matrix of 𝜇1M is given by

.

𝑎 𝑖,𝑗 ( ) =

(2.31)

Õ

.

𝜇A 𝑑 ( , 𝑐 𝑘 𝑑 ,𝑘 𝑑−1 , . . . , 𝑐 𝑘2 ,𝑘 1 ),

2≤𝑑, 𝑘

where 𝑘 = (𝑘 1 , . . . , 𝑘 𝑑 ) runs over all partitions 𝑖 = 𝑘 1 < 𝑘2 < · · · < 𝑘 𝑑−1 < 𝑘 𝑑 = 𝑗 (the sum in (2.31) is finite because 𝑑 ≤ 𝑗 − 𝑖 ≤ 𝑟). 𝛿) 𝑐 𝑞,𝑝 ∈ 𝐶 ≤𝛼 𝑞,𝑝 (𝐿 𝑞 , 𝐿 𝑝 ), where 𝛼 𝑞,𝑝 = 𝜌 𝑞 − 𝜌 𝑝 + 𝐵 𝑞 𝜉𝑞 ,

(2.32)

where 𝐵 𝑞 is a universal constant in the sense that it depends only on 𝑞, but not on A, the modules K𝑖 or their discrepancy data. (In (ii.𝛿) and in what follows we use the convention that 𝜌0 = 0.) (iii) There exists a quasi-isomorphism of A-modules 𝜎 : K𝑟 → M which shifts action by ≤ 𝜌 𝜎 and has discrepancy ≤ 𝜎 . The latter quantities admit the estimates (2.33)

𝜌 𝜎 ≤ 𝐶 𝑟 𝜉𝑟 ,

𝜖 𝜎𝑑 ≤ 𝐷𝑟,𝑑 𝜒𝑟,𝑑 ,

where the constants 𝐶 𝑟 and {𝐷𝑟,𝑑 } 𝑑∈ℕ are universal in the sense mentioned at point (ii.𝛿) above.

ASTÉRISQUE 426

39

2.6. STRUCTURE THEOREM FOR WEAKLY FILTERED ITERATED CONES

(iv) The first order part 𝜎1 : K𝑟 (𝑋) → M(𝑋) of the quasi-isomorphism 𝜎 is an isomorphism of chain complexes for all 𝑋 ∈ Ob(A), and the matrix corresponding to 𝜎1 with respect to the splitting (2.30) (taken both for K𝑟 (𝑋) and M(𝑋)) is upper triangular with id-maps along its diagonal. (v) The inverse 𝜎1−1 : M(𝑋) → K𝑟 (𝑋) of 𝜎1 is action preserving (i.e. it is filtered and shifts action by ≤ 0). (vi) For every 0 ≤ 𝑗 ≤ 𝑟 the diagonal element Δ 𝑗 = pr𝐶(𝑋 ,𝐿 𝑗 ) ◦ 𝜎1

𝐶(𝑋 ,𝐿 𝑗 )

: 𝐶(𝑋 , 𝐿 𝑗 ) −→ 𝐶(𝑋 , 𝐿 𝑗 )

is the identity map (as follows from point (2.14) above). However, when the domain inherits filtration from K𝑟 (𝑋) and the target from M(𝑋) this map shifts action by ≤ 𝜌 𝜎 . (Note that for 𝑗 ≥ 1, 𝐶(𝑋 , 𝐿 𝑗 ) is in general not a subcomplex of either K𝑟 (𝑋) or of M(𝑋)). For 𝑗 = 0, 𝐶(𝑋 , 𝐿0 ) is a subcomplex of both K𝑟 (𝑋) and of M(𝑋) and the two inherited filtrations on 𝐶(𝑋 , 𝐿0 ) coincide, hence Δ0 = id preserves filtration (i.e. shifts action by ≤ 0). Proof of Theorem 2.14. — We will construct inductively a sequence of weakly filtered modules M𝑖 , 𝑖 = 1, . . . , 𝑟 such that M𝑖 is quasi-isomorphic to K𝑖 and whose differential 𝜇1M𝑖 has a matrix of the type describe by (2.31). The desired module M will then be M𝑟 . In the course of the construction we will successively apply Proposition 2.11, Lemma 2.6 and Lemma 2.5. Fix once and for all ℓ := 𝑟 + 2. We begin the construction with 𝑖 = 1. Put M0 = K0 = L0 , K10 = K1 . Set also 𝜅 0 = 𝜅, so that L1 , K0 ∈ 𝑈𝑚 (𝜅 0 ). Define an auxiliary weakly filtered module (𝜙1 ;𝜌1 +ℓ 𝜅0 ,(1) )

K100 := C𝑜𝑛𝑒(L1 −−−−−−−−−−−−→ K0 ), where (1) is chosen such that (1) ≥ δ 𝜙1 ,

(1) ∈ E(A, K0 ),

By Proposition 2.11 there exists a cycle 𝑐 1 ∈ 𝜃1 , 𝜏1 ∈ hom≤𝜌1 +ℓ 𝜅0

;(1)

(1)

0 𝜖𝑑 ≥ 𝜖K 𝑑+1

≤𝜌 +ℓ 𝜅 K0 1 0 (𝐿1 )

for all 𝑑.

= 𝐶 ≤𝜌1 +ℓ 𝜅0 (𝐿1 , 𝐿0 ) as well as

(L1 , K0 ) with 𝜏1 a cycle and (𝜏1 )1 = · · · = (𝜏1 )ℓ = 0, such that 𝜙1 = 𝜆(𝑐 1 ) + 𝜇mod (𝜃) + 𝜏1 1

in hom≤𝜌1 +ℓ 𝜅0 ; (L1 , K0 ). Define now (1)

(𝜙1 −𝜇mod (𝜃1 );𝜌1 +ℓ 𝜅 0 ,(1) ) 1

M1 := C𝑜𝑛𝑒(L1 −−−−−−−−−−−−−−−−−−−−→ K0 ). (1)

(1)

Note that M1 = max{A, K0 , (1) − 𝜖 1 } = (1) − 𝜖 1 because (1) ∈ E(A, K0 ). For later use we will need to address Assumption 𝑈𝑚 for the module M1 . Indeed, by Lemma 2.9 we have M1 ∈ 𝑈𝑚 (𝜅1 ), where (1)

(1)

(1)

(1)

(1)

(1)

𝜅 1 := max 2𝜅 0 , 2𝑢 A + 𝜖3 − 𝜖 1 , 2𝑢 A + 2𝜖2 − 2𝜖 1 , 𝜅 0 + 𝜖2 − 𝜖1



.

The modules K10 = K1 , K100 and M1 are related by weakly filtered quasiisomorphisms as follows. The identity homomorphism can be viewed as a weakly

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40

filtered quasi-isomorphism 𝐼1 : K1 → K100 which shifts action by ≤ ℓ 𝜅0 and has 𝜙 (1) discrepancy ≤ (𝜖 1 − 𝛿1 1 , 0, . . . , 0, . . . ). Lemma 2.6 provides a quasi-isomorphism 𝜗1 : K100 −→ M1 (1)

which shifts action by ≤ 0 and has discrepancy ≤ (1) − 𝜖 1 . Consider the quasiisomorphism 𝜂1 : K1 → M1 given by the composition 𝜂1 := 𝜗1 ◦ 𝐼1 which shifts action 𝜙 by ≤ ℓ 𝜅 0 and has discrepancy ≤ (1) − 𝛿1 1 . The first order part (𝜂1 )1 : K1 (𝑋) → M1 (𝑋) of the module homomorphism 𝜂1 is an isomorphism of chain complexes for all 𝑋 and its matrix (with respect to the splitting 𝐶(𝑋 , 𝐿0 ) ⊕ 𝐶(𝑋 , 𝐿1 ) of K1 (𝑋) and M1 (𝑋) as 𝑅-modules) is upper triangular with id’s along the diagonal. This follows from the explicit formula of (𝜗1 )1 from the proof of Lemma 2.6. The same formula also shows that (𝜗1 )−1 shift action by ≤ 0 and the same holds 1 −1 −1 for (𝐼1 )1 . It follows that the inverse (𝜂1 )1 of (𝜂1 )1 shifts action by ≤ 0. Next, consider the composition 𝜂1 ◦ 𝜙2 : L2 → M1 . This is a module homomorphism that shifts action by ≤ 𝜌2 + ℓ 𝜅 0 and has discrepancy ≤ 𝜂1 ◦𝜙2 = 𝜂1 ∗ δ 𝜙2 . Define now (𝜂1 ◦𝜙2 ;𝜌2 +ℓ 𝜅0 ,𝜂1 ∗δ 𝜙2 )

K20 = C𝑜𝑛𝑒 L2 −−−−−−−−−−−−−−−−−−→ M1 ,




(𝜂1 ◦𝜙2 ;𝜌2 +ℓ 𝜅1 +ℓ 𝜅0 ,(2) )

K200 = C𝑜𝑛𝑒 L2 −−−−−−−−−−−−−−−−−−→ M1 ,




where (2) is chosen such that (2) ≥ 𝜂1 ∗ δ 𝜙2 ,

(2) ∈ E(A, M1 ),

(2)

1 𝜖𝑑 ≥ 𝜖M 𝑑+1

for all 𝑑.

Applying Proposition 2.11 we can write: 𝜂1 ◦ 𝜙2 = 𝜆(𝑐 2 ) + 𝜇mod (𝜃2 ) + 𝜏2 , 1 ≤𝜌 +ℓ 𝜅 +ℓ 𝜅

where 𝑐 2 ∈ M1 2 1 0 (𝐿2 ) is a cycle and 𝜃2 , 𝜏2 ∈ hom≤𝜌2 +ℓ 𝜅1 +ℓ 𝜅0 ; (L2 , M1 ) with 𝜏2 being a cycle such that (𝜏2 )1 = · · · = (𝜏2 )ℓ = 0. (2)

We define now (𝜂1 ◦𝜙2 −𝜇mod (𝜃2 );𝜌2 +ℓ 𝜅1 +ℓ 𝜅 0 ,(2) ) 1

M2 := C𝑜𝑛𝑒 L2 −−−−−−−−−−−−−−−−−−−−−−−−−→ M1 .




(2)

(1)

(2)

The discrepancy of M2 is ≤ M2 := max{A, M1 , (2) −𝜖 1 } = max{(1) −𝜖1 , (2) −𝜖1 }. By Lemma 2.9 we have M2 ∈ 𝑈𝑚 (𝜅2 ), where (1)

(1)

(2)

(2)

(1)

(1)

𝜅 2 := max 2𝜅1 , 2𝑢 A + 𝜖 3 − 𝜖 1 , 2𝑢 A + 𝜖3 − 𝜖1 , 2𝑢 A + 2𝜖 2 − 2𝜖1 ,



(2)

(2)

(1)

(1)

(2)

(2)

2𝑢 A + 2𝜖 2 − 2𝜖1 , 𝜅0 + 𝜖 2 − 𝜖1 , 𝜅0 + 𝜖 2 − 𝜖 1

.

The modules K2 , K20 , K200 and M2 are related by weakly filtered quasi-isomorphisms 𝜓2

𝐼2

𝜗2

'

'

'

K2 −−→ K20 −−→ K200 −−→ M2 ,

ASTÉRISQUE 426

41

2.6. STRUCTURE THEOREM FOR WEAKLY FILTERED ITERATED CONES

where the shifts in action and discrepancies of these maps are given by 𝜙

𝜓2 ≤ (1) − 𝛿1 1 ,

shift(𝜓2 ) ≤ ℓ 𝜅0 ,

shift(𝜗2 ) ≤ 0,

𝜙

𝜙

𝐼2 ≤ (𝜖 1 − 𝛿1 2 − 𝜖1 + 𝛿1 1 , 0, . . . , 0, . . . ), (2)

shift(𝐼2 ) ≤ ℓ 𝜅 1 ,

(1)

𝜗2 ≤ (2) − 𝜖 1 . (2)

The quasi-isomorphism 𝜓2 is obtained from Lemma 2.5 and 𝜗2 from Lemma 2.6. The quasi-isomorphism 𝐼2 is basically the identity map, relating the same module with two (slightly) different structures of weakly filtered module. Consider now the composition 𝜂2 = 𝜗2 ◦𝐼2 ◦𝜓2 : K2 → M2 . This quasi-isomorphism has the following action shift and discrepancy: shift(𝜂2 ) ≤ ℓ (𝜅 1 + 𝜅0 ),

𝜙

𝜂2 ≤ (2) ∗ (1) − (𝛿1 2 + 𝜖 1 ). (1)

As in the previous step, the first order part (𝜂2 )1 : K2 (𝑋) → M2 (𝑋) of 𝜂2 is an isomorphism of chain complexes and its matrix (with respect to the splitting 𝐶(𝑋 , 𝐿0 ) ⊕ 𝐶(𝑋 , 𝐿1 ) ⊕ 𝐶(𝑋 , 𝐿2 ) of K2 (𝑋) and M2 (𝑋) as 𝑅-modules) is upper triangular with id’s along the diagonal. Moreover, the inverse (𝜂2 )−1 of (𝜂2 )1 shifts action by ≤ 0. 1 These assertions easily follows from the explicit formulas of (𝜓2 )1 and (𝜗2 )1 given in the proofs of Lemmas 2.5 and 2.6 respectively and the fact, already shown in the previous step, that (𝜂1 )1 is a chain isomorphism represented by an upper triangular matrix with id’s along the diagonal. Recall also from the previous step that (𝜂1 )−1 1 shifts action by ≤ 0. An examination of the action shifts shows that each of the maps (𝐼2 )−1 , (𝜓2 )−1 and (𝜗2 )−1 shifts action by ≤ 0, hence the same holds for (𝜂2 )−1 . 1 1 1 1 Continuing as above by induction we obtain, for every 1 ≤ 𝑗 ≤ 𝑟: 1) A weakly filtered module M𝑗 . 2) Two sequences of non-negative real numbers (𝑗) and 𝜂 𝑗 that satisfy: (a) (𝑗) ≥ 𝜂 𝑗−1 ∗ δ 𝜙 𝑗 ,

(𝑗) ∈ E(A, M𝑗−1 ),

(b) 𝜂 𝑗 ≤ (𝑗) ∗ · · · ∗ (1) −

𝜙𝑗 (𝛿1

M𝑗−1

(𝑗)

𝜖 𝑑 ≥ 𝜖 𝑑+1

for all 𝑑.

Í 𝑗−1

(𝑖) 𝜖 ). 𝑖=1 1

+

We use the convention that 𝜂0 = (0, . . . , 0, . . . ). 3) A positive real number 𝜅 𝑗 defined (inductively) by (𝑖)

(𝑖)

(𝑖)

(𝑖)

(𝑖)

(𝑖)

𝜅 𝑗 := max 2𝜅 𝑗−1 , 2𝑢 A + 𝜖3 − 𝜖 1 , 2𝑢 A + 2𝜖2 − 2𝜖1 , 𝜅0 + 𝜖2 − 𝜖 1 ; 1 ≤ 𝑖 ≤ 𝑗 .





(Recall that 𝜅 0 = 𝜅.) ≤𝜌 𝑗 +

4) A cycle 𝑐 𝑗 ∈ M𝑗−1

Í 𝑗−1 𝑖=0

𝜅𝑖

(𝐿 𝑖 ).

5) The module M𝑗 is related to M𝑗−1 by (𝜆(𝑐 𝑗 )+𝜏𝑗 ;𝜌 𝑗 +ℓ

(2.34)

Í 𝑗−1 𝑖=0

𝜅 𝑖 ,(𝑗) )

M𝑗 = C𝑜𝑛𝑒 L𝑗 −−−−−−−−−−−−−−−−−−−−→ M𝑗−1 , Í 𝑗−1

where 𝜏𝑗 ∈ hom≤𝜌 𝑗 +ℓ (

𝑖=0

𝜅 𝑖 );(𝑗)




(L𝑗 , M𝑗−1 ) is a cycle with (𝜏𝑗 )1 = · · · = (𝜏𝑗 )ℓ = 0. (𝑖)

6) The discrepancy of M𝑗 is M𝑗 ≤ max{(𝑖) − 𝜖1 ; 1 ≤ 𝑖 ≤ 𝑗}.

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7) M𝑗 ∈ 𝑈𝑚 (𝜅 𝑗 ). 8) A weakly filtered quasi-isomorphism 𝜂 𝑗 : K𝑗 → M𝑗 which shifts action by ≤ ℓ (𝜅 0 + · · · + 𝜅 𝑗−1 ) and with discrepancy ≤ 𝜂 𝑗 , where the sequences 𝜂 𝑗 is the one from point 2) above. Moreover, the first order part (𝜂 𝑗 )1 is a chain isomorphism represented by an upper triangular matrix with id’s along the diagonal (with respect to the splitting CF(𝑋 , 𝐿0 ) ⊕ · · · ⊕ 𝐶(𝑋 , 𝐿𝑟 )) and its inverse (𝜂 𝑗 )−1 shifts action by ≤ 0. 1 The module M claimed in the statement of the theorem is the module M𝑟 , and the quasi-isomorphism of A-modules is 𝜎 : K𝑟 → M is 𝜂𝑟 . M𝑗

Next, we analyze the differential 𝜇1 module

on the modules M𝑗 . We begin with the

(𝜆(𝑐1 )+𝜏1 ;𝜌1 +ℓ 𝜅0 ,(1) )

M1 = C𝑜𝑛𝑒 L1 −−−−−−−−−−−−−−−−→ K0 .

(2.35)

≤𝜌1 +ℓ 𝜅0

Recall that 𝑐1 ∈ K0




(𝐿1 ) = 𝐶 ≤𝜌1 +ℓ 𝜅0 (𝐿1 , 𝐿0 ). For further use, we will write 𝑐1,0 := 𝑐 1 .

Let 𝑋 ∈ Ob(A). Write M1 (𝑋) = 𝐶(𝑋 , 𝐿1 ) ⊕ 𝐶(𝑋 , 𝐿0 ) as 𝑅-modules. By the definition of the map 𝜆 we have according to this splitting: 𝜇1M1 (𝑏 1 , 𝑏0 ) = 𝜇1A(𝑏1 ), 𝜇1A(𝑏 0 ) + 𝜇2A(𝑏 1 , 𝑐1,0 ) ,




for all 𝑏1 ∈ 𝐶(𝑋 , 𝐿1 ), 𝑏0 ∈ 𝐶(𝑋 , 𝐿0 ).

1 More generally, the higher operations 𝜇M have the following form. Let 1 ≤ 𝑑 ≤ ℓ − 1 𝑑 and 𝑋0 , . . . , 𝑋𝑑−1 ∈ Ob(A). One has , for all 𝑎 𝑖 ∈ 𝐶(𝑋𝑖−1 , 𝑋𝑖 ), 𝑖 = 1, . . . , 𝑑 and for all (𝑏1 , 𝑏0 ) ∈ 𝐶(𝑋𝑑 , 𝐿1 ) ⊕ 𝐶(𝑋𝑑 , 𝐿0 ):

(2.36)

1 𝑎1 , . . . , 𝑎 𝑑−1 , (𝑏1 , 𝑏0 ) 𝜇M 𝑑




A A = 𝜇A 𝑑 (𝑎 1 , . . . , 𝑎 𝑑−1 , 𝑏 1 ), 𝜇𝑑 (𝑎 1 , . . . , 𝑎 𝑑−1 , 𝑏 0 ) + 𝜇 𝑑+1 (𝑎 1 , . . . , 𝑎 𝑑−1 , 𝑏 1 , 𝑐 1,0 ) .




1 Note that the term 𝜏1 in (2.35) does not play any role in the expression for 𝜇M as long 𝑑 as 𝑑 ≤ ℓ − 1, since (𝜏1 )1 = · · · = (𝜏1 )ℓ = 0. Recall also that ℓ = 𝑟 + 2.

We now analyze M2 . Recall that (𝜆(𝑐2 )+𝜏2 ;𝜌2 +ℓ 𝜅1 +ℓ 𝜅0 ,(2) )

M2 := C𝑜𝑛𝑒 L2 −−−−−−−−−−−−−−−−−−−−→ M1 ,

(2.37)

≤𝜌2 +ℓ (𝜅1 +𝜅 0 )

where 𝑐 2 ∈ M1

≤𝜌2 +ℓ (𝜅1 +𝜅0 )

M1




(𝐿2 ). Recall that, as 𝑅-modules, (1)

(𝐿2 ) = 𝐶 ≤𝜌2 −𝜌1 +ℓ 𝜅1 −𝜖1 (𝐿2 , 𝐿1 ) ⊕ 𝐶 ≤𝜌2 +ℓ (𝜅1 +𝜅0 ) (𝐿2 , 𝐿0 ).

Write 𝑐2 = (𝑐 2,1 , 𝑐2,0 ) with respect to this splitting. Let 𝑋 ∈ Ob(A) and write, as 𝑅-modules, (2.38)

M2 (𝑋) = 𝐶(𝑋 , 𝐿2 ) ⊕ M1 (𝑋) = 𝐶(𝑋 , 𝐿2 ) ⊕ 𝐶(𝑋 , 𝐿1 ) ⊕ 𝐶(𝑋 , 𝐿0 ).

ASTÉRISQUE 426

43

2.6. STRUCTURE THEOREM FOR WEAKLY FILTERED ITERATED CONES

By the definition of 𝜆 together with (2.36) we have (2.39)

𝜇1M2 (𝑏2 , 𝑏1 , 𝑏0 ) = 𝜇1A(𝑏 2 ), 𝜇1M1 (𝑏 1 , 𝑏0 ) + 𝜇2M1 (𝑏 2 , 𝑐2 )




= 𝜇1A(𝑏 2 ), 𝜇1A(𝑏1 ) + 𝜇2A(𝑏2 , 𝑐2,1 ), 𝜇1A(𝑏 0 ) + 𝜇2A(𝑏1 , 𝑐1,0 ) + 𝜇2A(𝑏 2 , 𝑐2,0 ) + 𝜇3A(𝑏2 , 𝑐2,1 , 𝑐1,0 ) .




In other words, the matrix of 𝜇1M2 has the following shape: 𝜇A( )

.

(2.40)

𝜇1M2

© 1 = ­­ 0 « 0

𝜇2A( , 𝑐1,0 )

𝜇2A( , 𝑐2,0 ) + 𝜇3A( , 𝑐2,1 , 𝑐1,0 )

.

𝜇1A(

.

.

ª ® ® ¬

𝜇2A(

. , 𝑐2,1) A . 𝜇1 ( )

.)

0

Here the matrix has been calculated with respect to the splitting M2 (𝑋) = 𝐶(𝑋 , 𝐿0 ) ⊕ 𝐶(𝑋 , 𝐿1 ) ⊕ 𝐶(𝑋 , 𝐿2 ) (in contrast to (2.38) and (2.39)) in order to be compatible with (2.30). 2 A similar formula holds also for the higher operations 𝜇M . Let 1 ≤ 𝑑 ≤ ℓ − 2 𝑑 and 𝑋0 , . . . , 𝑋𝑑−1 ∈ Ob(A). Then, for all 𝑎 ∈ 𝐶(𝑋0 , 𝑋1 ) ⊗ · · · ⊗ 𝐶(𝑋𝑑−2 , 𝑋𝑑−1 ):



(2.41)

A A 2 𝜇M ( 𝑎, 𝑏2 , 𝑏1 , 𝑏0 ) = 𝜇A 𝑑 (𝑎, 𝑏 2 ), 𝜇𝑑 (𝑎, 𝑏 1 ) + 𝜇 𝑑+1 (𝑎, 𝑏 2 , 𝑐 2,1 ), 𝑑 A 𝜇A 𝑑 (𝑎, 𝑏 0 ) + 𝜇𝑑+1 (𝑎, 𝑏 1 , 𝑐 1,0 )



A + 𝜇A 𝑑+1 (𝑎, 𝑏 2 , 𝑐 2,0 ) + 𝜇𝑑+2 (𝑎, 𝑏 2 , 𝑐 2,1 , 𝑐 1,0 ) .

Continuing by induction, we obtain the 𝑐 𝑞,𝑝 ∈ 𝐶(𝐿 𝑞 , 𝐿 𝑝 ) for all 0 ≤ 𝑞 < 𝑝 ≤ 𝑟 and the operators 𝑎 𝑖,𝑗 , 𝑖 > 𝑗, as described in (2.31), which form the matrix of the differentials 𝜇1M for the module M = M𝑟 . M𝑗

Note that the 𝜇 𝑘 -operation of the intermediate module M𝑗 involves expressions containing 𝜇A for 𝑑 ≤ 𝑗 + 𝑘 but no higher order 𝜇’s. It is also important to remark 𝑑 M𝑗 that at every step of the construction, the operations 𝜇𝑑 for 𝑑 ≤ 𝑟 + 1 − 𝑗 will depend on the cycles 𝑐 𝑞,𝑝 with 0 ≤ 𝑝 < 𝑞 ≤ 𝑗 but not on the elements 𝜏𝑖 that appear in (2.34). The reason is that (𝜏𝑖 )1 = · · · = (𝜏𝑖 )ℓ = 0 and we have chosen in advance ℓ = 𝑟 + 2. Next, we estimate the action levels 𝛼 𝑞,𝑝 of 𝑐 𝑞,𝑝 from (ii.𝛿) and the action shift and discrepancy of the quasi-isomorphism 𝜎 = 𝜂𝑟 as claimed in (2.33). An inspection of the previous steps in the proof shows that (𝑝)

𝑐 𝑞,𝑝 ∈ 𝐶 ≤𝜌𝑞 −𝜌𝑝 +ℓ (𝜅 𝑝 +···+𝜅 𝑞−1 )−𝜖1 (𝐿 𝑞 , 𝐿 𝑝 ). Thus we need to estimate the 𝜅 𝑗 ’s. This, in turn, would require to estimate the (𝑖) ’s. Note that we can choose at every step of the previous inductive construction the sequence (𝑗) at 2.a) page 41 to satisfy (𝑗)

M𝑗−1

𝜖 𝑑 ≤ 𝜖 𝑑+1 +

𝑑 Õ

𝜂 𝑗−1

𝜖𝑖

𝜙𝑗

M𝑗−1


+ 𝛿𝑖 + 𝜖A 𝑖 + 𝜖𝑖

.

𝑖=1

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(𝑗)

A simple inductive argument now implies the desired estimates for the 𝑑 ’s the 𝜅 𝑗 ’s as well as for the action shift of 𝜂 𝑗 and its discrepancy. Finally, the first statement at point (2.14) follows easily from the induction process defining the maps 𝜂 𝑖 , 𝑖 = 1, . . . , 𝑟, by examining the filtrations induced on CF(𝑋 , 𝐿 𝑗 ) by each of K𝑖 (𝑋) and M𝑖 (𝑋) for 𝑗 ≤ 𝑖 ≤ 𝑟. That 𝐶(𝑋 , 𝐿0 ) is a subcomplex of both K𝑖 (𝑋) and M𝑖 (𝑋) follows from the fact that K𝑖 and M𝑖 are both iterated cones starting with the object L0 . 

2.7. Invariants and measurements for filtered chain complexes As a supplement to the previous material we describe here a number of numerical invariants of filtered chain complexes that will be useful in Chapter 5 when we prove our main geometric results. More details and further results can be found in the expanded version of this paper [BCS]. We begin with basic definitions. Fix a commutative ring R with unity. ⊲ By a filtered chain complex we mean a chain complex (𝐶, 𝑑 𝐶 ) of R-modules endowed with an increasing filtration by sub-chain complexes 𝐶 ≤𝛼 ⊂ 𝐶, indexed by the real numbers 𝛼 ∈ ℝ. ⊲ An R-linear map 𝑓 : 𝐶→ 𝐷 between the filtered chain complexes (𝐶, 𝑑 𝐶 ), (𝐷, 𝑑 𝐷 ) is called filtered if there exists 𝜌 ∈ ℝ such that 𝑓 (𝐶 ≤𝛼 ) ⊂ 𝐷 ≤𝛼+𝜌 for every 𝛼. In that case we also say that 𝑓 shifts action by ≤ 𝜌. In case 𝑓 preserves the filtrations (i.e. it shifts filtration by ≤ 0) we say that 𝑓 is strictly filtered. ⊲ Let (𝐶, 𝑑 𝐶 ) be a filtered chain complex, and 𝑥 ∈ 𝐶. Define 𝐴(𝑥) ∈ ℝ ∪ {−∞, ∞} to be the infimal filtration level of 𝐶 which contains 𝑥, i.e. 𝐴(𝑥) := inf 𝛼 ∈ ℝ ; 𝑥 ∈ 𝐶 ≤𝛼 .





We call 𝐴(𝑥) the action level of 𝑥. Sometimes we will write 𝐴(𝑥; 𝐶) instead of 𝐴(𝑥) in order to keep track of the chain complex 𝐶 that 𝑥 belongs to. ≤𝛼 By our conventions we have 𝐴(0) = −∞ and if 𝛼∈ Ðℝ 𝐶 = {0} then 𝐴(𝑥) = −∞ iff 𝑥 = 0. Also, if the filtration on 𝐶 is exhaustive, i.e. 𝛼∈ℝ 𝐶 ≤𝛼 = 𝐶, then 𝐴(𝑥) < ∞ for every 𝑥 ∈ 𝐶. ⊲ Another measurement relevant to our considerations is the following. Define the “action drop” of the differential 𝑑 𝐶 of the filtered chain complex (𝐶, 𝑑 𝐶 ) as

Ñ

(2.42)

𝛿 𝑑 𝐶 = sup 𝑟 ∈ [0, ∞) ; ∀𝑎 ∈ ℝ , 𝑑 𝐶 (𝐶 ≤𝑎 ) ⊂ 𝐶 ≤𝑎−𝑟 .





2.7.1. Boundary depth and related algebraic notions. — Boundary depth was introduced and studied extensively in symplectic topology (in a slightly different formulation than below) by Usher [Ush11], [Ush13]. Here we introduce variants of this measurement, such as boundary level and homotopical boundary level and explain their relation to boundary depth. Let (𝐶, 𝑑 𝐶 ) be a filtered chain complex and 𝑐 ∈ 𝐶 a boundary.

ASTÉRISQUE 426

45

2.7. INVARIANTS AND MEASUREMENTS FOR FILTERED CHAIN COMPLEXES

Define the boundary level of 𝑐 by 𝐵(𝑐; 𝐶) = inf 𝛼 ∈ ℝ ; ∃ 𝑏 ∈ 𝐶 ≤𝛼 such that 𝑐 = 𝑑 𝐶 𝑏 .



(2.43)



A central measurement in our framework is the following special case. Let (𝐶, 𝑑 𝐶 ) and (𝐷, 𝑑 𝐷 ) be filtered chain complexes. Let 𝜓 : 𝐶 → 𝐷 be a filtered chain map and assume that 𝜓 is null-homotopic. Define the homotopical boundary level 𝐵 ℎ (𝜓) of 𝜓 to be the infimal action shift needed for a chain homotopy between 𝜓 and 0. More precisely: (2.44)

𝐵 ℎ (𝜓) = inf 𝜌 ∈ ℝ ; ∃ an R-linear map ℎ : 𝐶 → 𝐷 which shifts action by ≤ 𝜌 and such that 𝜓 = ℎ𝑑 𝐶 + 𝑑 𝐷 ℎ .



Note that 𝐵 ℎ (𝜓) = 𝐵(𝜓; homR(𝐶, 𝐷)), where we view 𝜓 as a boundary in the chain complex homR(𝐶, 𝐷). The latter chain complex is filtered as follows: for 𝛾 ∈ ℝ, ≤𝛾 homR (𝐶, 𝐷) is the subcomplex consisting of all R-linear maps 𝐶 → 𝐷 that shift action by ≤ 𝛾. The notion of boundary level is closely related to the boundary depth measurement introduced by Usher [Ush11], [Ush13]. The relation is the following. Let (𝐶, 𝑑 𝐶 ) be a filtered chain complex and 𝑐 ∈ 𝐶 a boundary. The boundary depth 𝛽(𝑐; 𝐶) of 𝑐 is defined by the equality 𝐵(𝑐; 𝐶) = 𝐴(𝑐; 𝐶) + 𝛽(𝑐; 𝐶),

(2.45)

where 𝐴(𝑐; 𝐶) is the action level of 𝑐. It is easy to see that 𝛽(𝑐; 𝐶) = inf 𝑟 ≥ 0 ; ∀𝛼 such that 𝑐 ∈ 𝐶 ≤𝛼 , ∃ 𝑏 ∈ 𝐶 ≤𝛼+𝑟 such that 𝑑 𝐶 𝑏 = 𝑐 .





Now let (𝐶, 𝑑 𝐶 ) be a filtered chain complex which is acyclic. Its boundary depth is 𝛽(𝐶) := inf 𝑟 ≥ 0 ; ∀𝛼 and ∀𝑐 ∈ 𝐶 ≤𝛼 with 𝑑 𝐶 (𝑐) = 0, ∃ 𝑏 ∈ 𝐶 ≤𝛼+𝑟 such that 𝑑 𝐶 𝑏 = 𝑐 .





If we assume that (𝐶, 𝑑 𝐶 ) is acyclic (i.e. homotopy equivalent to the trivial chain complex), then we have the inequality (2.46)

𝛽(𝐶) ≤ 𝐵 ℎ (id𝐶 ).

Let (𝐶, 𝑑 𝐶 ) and (𝐷, 𝑑 𝐷 ) be filtered chain complexes and 𝜓 : 𝐶 → 𝐷 a filtered chain map which is null-homotopic. Similarly to the homotopical boundary level we also have the homotopical boundary depth of 𝜓: (2.47)

𝛽 ℎ (𝜓) := 𝛽 𝜓; homR(𝐶, 𝐷) .




More explicitly, 𝛽 ℎ (𝜓) is the infimal 𝑟 ≥ 0 for which 𝜓 is null-homotopic via a chain homotopy that shifts filtration by ≤ 𝐴(𝜓) + 𝑟. As in (2.45) we have 𝐵 ℎ (𝜓) = 𝐴 𝜓; homR(𝐶, 𝐷) + 𝛽 ℎ (𝜓).




Finally, here is another variant of the above measurements. Let A be a weakly filtered 𝐴∞ -category with discrepancy ≤ A (see Chapter 2). Let 𝑚 = (𝜖 1𝑚 = 0, 𝜖2𝑚 , . . . , 𝜖 𝑚 , . . . ) be a sequence of non-negative real numbers, 𝑑 and let M0 , M1 be two weakly filtered A-modules with discrepancy ≤ 𝑚 .

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Let  ℎ be another sequence of non-negative real numbers, and assume that ∈ E(𝑚 , A) (see page 21). ℎ Let hom (M0 , M1 ) be the weakly filtered pre-module homomorphisms M0 → M1 with discrepancy ≤  ℎ (and arbitrary action shift). As explained in § 2.3.1, it is a chain complex when endowed with the differential 𝜇mod of the dg-category of A-modules. 1 ℎ

ℎ

Moreover, this chain complex is filtered by hom≤𝜌; (M0 , M1 ), 𝜌 ∈ ℝ. Now let 𝜓 : M0 → M1 be a weakly filtered module homomorphism with disℎ crepancy ≤  ℎ , and assume that 𝜓 is a boundary in hom (M0 , M1 ) (i.e. 𝜓 is chain homotopic to 0 via a chain homotopy of pre-module maps with discrepancy ≤  ℎ ). Then we can define
ℎ 𝛽 ℎ (𝜓;  ℎ ) := 𝛽 𝜓; hom (M0 , M1 ) and similarly define 𝐵 ℎ (𝜓;  ℎ ). Further variants of the boundary level/depth measurements and their properties can be found in the expanded version [BCS]. 2.7.2. Homotopies of chain isomorphisms. — Here we prove a simple algebraic approximation lemma which says that altering a chain isomorphism by a chain homotopy yields an injective map provided that the chain homotopy shifts action by a small enough amount. 𝐷 be filtered chain complexes, and assume that Lemma 2.15. — Let (𝐶, 𝑑 𝐶 ) and (𝐷, Ñ Ð 𝑑 ) ≤𝛼 ≤𝛼

the filtration on 𝐶 is exhaustive (i.e. 𝛼∈ℝ 𝐶 = 𝐶) and separated (i.e. Let 𝑓 , 𝑔 : 𝐶 → 𝐷 be chain maps with the following properties: ⊲

𝑔 is an isomorphism.

⊲

𝑔 and 𝑔 −1 are strictly filtered.

⊲

𝑓 − 𝑔 is null-homotopic and 𝐵 ℎ ( 𝑓 − 𝑔) < min{𝛿 𝑑 𝐶 , 𝛿 𝑑 𝐷 }.

𝛼∈ℝ

𝐶

= 0).

Then 𝑓 is strictly filtered and moreover 𝑓 is injective. In our geometric applications 𝐷 = 𝐶, 𝑔 will be the identity, and 𝑓 will be the com𝑓1

𝑓2

position of two chain morphisms 𝐶 → 𝐶 0 → 𝐶 that are constructed geometrically. The lemma shows in this case that the middle complex 𝐶 0 contains 𝐶 as a retract. Results of this sort are familiar in symplectic topology since [CR03]. Proof of Lemma 2.15. — Since the filtration on 𝐶 is both exhaustive and separated, we have −∞ < 𝐴(𝑥) < ∞ for every 𝑥 ≠ 0, and 𝐴(0) = −∞. Set 𝜌 := 𝐵 ℎ ( 𝑓 − 𝑔) + 𝜖, where 𝜖 > 0 is small enough such that 𝜌 < min{𝛿 𝑑 𝐶 , 𝛿 𝑑 𝐷 }. Write 𝑓 = 𝑔 + 𝜂𝑑 𝐶 + 𝑑 𝐷 𝜂, where 𝜂 : 𝐶 → 𝐷 is R-linear and shifts action by ≤ 𝜌. Since 𝜌 < min{𝛿 𝑑 𝐶 , 𝛿 𝑑 𝐷 } and 𝑔 is strictly filtered we have 𝐴 𝑓 (𝑥) = 𝐴 𝑔(𝑥) + 𝜂𝑑 𝐶 (𝑥) + 𝑑 𝐷 𝜂(𝑥) ≤ 𝐴(𝑥),




hence 𝑓 is strictly filtered.

ASTÉRISQUE 426




for all 𝑥 ∈ 𝐶,

2.7. INVARIANTS AND MEASUREMENTS FOR FILTERED CHAIN COMPLEXES

47

For the injectivity of 𝑓 , assume that 𝑓 (𝑥) = 0 for some 𝑥 ≠ 0. Then 𝑔(𝑥) = − 𝜂𝑑 𝐶 (𝑥) + 𝑑 𝐷 𝜂(𝑥) ,




and using again inequality 𝜌 < min{𝛿 𝑑 𝐶 , 𝛿 𝑑 𝐷 } we obtain that 𝐴 𝑔(𝑥) = 𝐴 𝜂𝑑 𝐶 (𝑥) + 𝑑 𝐷 𝜂(𝑥) < 𝐴(𝑥).







The last inequality together with the assumption that 𝑔 −1 is strictly filtered imply 𝐴(𝑥) = 𝐴 𝑔 −1 𝑔(𝑥) ≤ 𝐴 𝑔(𝑥) < 𝐴(𝑥).









A contradiction.

Under additional assumptions we can obtain a somewhat stronger result. Before we state it, here are a couple of relevant notions. The filtration 𝐶 ≤𝛼 ⊂ 𝐶, 𝛼 ∈ ℝ induces a topology on 𝐶 which is generated by the cosets of Ñ 𝐶 ≤𝛼 , 𝛼 ∈ ℝ, as basic open subsets. The assumption that the filtration is separated (i.e. 𝛼∈ℝ 𝐶 ≤𝛼 = 0) which implies that 𝐶 is Hausdorff in this topology. The filtration on 𝐶 is called complete if the obvious map 𝐶 −→ lim(𝐶/𝐶 ≤𝛼 ) ←−− 𝛼

is surjective. This assumption implies that the previously mentioned topology on 𝐶 turns 𝐶 into a complete topological space (in the sense that every Cauchy sequence converges).

Lemma 2.16. — Let (𝐶, 𝑑 𝐶 ), (𝐷, 𝑑 𝐷 ), 𝑓 , 𝑔 be as in Lemma 2.15 and assume in addition

that the filtration on 𝐶 is complete. Then 𝑓 is a strictly filtered isomorphism and moreover 𝑓 −1 is also strictly filtered. Proof. — In view of Lemma 2.15 we only need to show that 𝑓 is an isomorphism and that 𝑓 −1 is strictly filtered. We will use a well-known inversion trick, that has already been used in a similar setting in [Ush11], [Ush13]. Fix 0 𝜆0 for every 𝑖 ≥ 1. As usual we set 𝜈(0) = ∞. Let (𝐶, 𝑑 𝐶 ) be a finite dimensional chain complex over Λ. Fix a basis Gof 𝐶 over Λ and let 𝐴 : G → ℝ be a function. Similarly to § 2.2.3 we will use 𝐴 to define a filtration on 𝐶 by Λ0 -modules. Extend 𝐴 to a function 𝐴 : 𝐶 → ℝ ∪ {−∞}, by 𝐴

Õ



𝜆 𝑗 𝑒 𝑗 = max − 𝜈(𝜆 𝑗 ) + 𝐴(𝑒 𝑗 ) ,





where 𝑒 𝑗 are the elements of the basis G, 0 ≠ 𝜆 𝑗 ∈ Λ, 𝐴(𝑒 𝑗 ) is the pre-determined value of 𝐴 on the generator 𝑒 𝑗 , and 𝜈 is the preceding valuation. Define now 𝐶 ≤𝛼 := 𝑥 ∈ 𝐶 ; 𝐴(𝑥) ≤ 𝛼 .





It is easy to see that 𝐶 ≤𝛼 ⊂ 𝐶, 𝛼 ∈ ℝ, is an increasing filtration of 𝐶 by Λ0 -modules (though not by vector spaces over Λ). Since 𝐴(𝑥) = −∞ iff 𝑥 = 0, this filtration is separated. Moreover, it is exhaustive and complete. From now on we will make the following standing assumption: 𝐴(𝑑 𝐶 𝑥) ≤ 𝐴(𝑥), for all 𝑥 ∈ 𝐶. In other words, we assume that each 𝐶 ≤𝛼 ⊂ 𝐶, 𝛼 ∈ ℝ, is a subcomplex of 𝐶 (over Λ0 ). It is important to note that the function 𝐴, as defined above, coincides with the action level of the preceding filtration on 𝐶, as defined at the beginning of Section 2.7. Thus no confusion should arise by denoting them both by 𝐴. We will make use of the following definition from [UZ16].

Definition 2.17. — A subspace 𝑉 ⊂ Ker(𝑑 𝐶 ) ⊂ 𝐶 is called 𝛿-robust if for all 𝑣 ∈ 𝑉 and 𝑤 ∈ 𝐶 such that 𝑣 = 𝑑 𝐶 (𝑤), we have 𝐴(𝑤) ≥ 𝐴(𝑣) + 𝛿.

2.7.3. Remark. — According to the above definition, a complement 𝑊 in Ker(𝑑 𝐶 ) to Im(𝑑 𝐶 ) is a 𝛿-robust subspace for all 𝛿 > 0. Hence if 𝑉 ⊂ Im(𝑑 𝐶 ) is 𝛿-robust then 𝑉 ⊕ 𝑊 is also 𝛿-robust. We will call a 𝛿-robust subspace 𝑉 ⊂ Im(𝑑 𝐶 ) a proper 𝛿-robust subspace.

Proposition 2.18. — Let (𝐶, 𝑑 𝐶 ) be a chain complex as above, and let 𝑓 : 𝐶 → 𝐶 be a chain map. Assume that 𝑑 𝐶 splits as a sum 𝑑 𝐶 = 𝑑0 + 𝑑1 such that 𝑑0 is a Λ-linear differential which (like 𝑑 𝐶 ) also preserves the given filtration on 𝐶. Furthermore, assume that dimΛ 𝐻∗ (𝐶, 𝑑0 ) ≥ dimΛ 𝐻∗ (𝐶, 𝑑 𝐶 ) .







If 𝐵 ℎ ( 𝑓 − id𝐶 ) < 𝛿 𝑑1 , then dimΛ Im( 𝑓 ) ≥ dimΛ 𝐻∗ (𝐶, 𝑑0 ) .




ASTÉRISQUE 426




2.7. INVARIANTS AND MEASUREMENTS FOR FILTERED CHAIN COMPLEXES

49

The proposition follows directly from the following two lemmas.

Lemma 2.19. — Let (𝐶, 𝑑 𝐶 ) be a chain complex as above, and assume that its differential splits as 𝑑 𝐶 = 𝑑0 + 𝑑1 with 𝑑0 satisfying the same assumptions as in Proposition 2.18. Then dimΛ 𝐻∗ (𝐶, 𝑑0 ) − dimΛ 𝐻∗ (𝐶, 𝑑 𝐶 )







is even .

Furthermore, denote the latter number by 2𝑘 and assume that 𝑘 ≥ 0. Then (𝐶, 𝑑𝐶 ) admits a proper 𝛿 𝑑1 -robust subspace of dimension at least 𝑘.

Lemma 2.20. — Let (𝐶, 𝑑 𝐶 ) be a chain complex as in Lemma 2.19 and 𝑓 : 𝐶 → 𝐶 be a chain map. Let 0 < 𝜖 < 𝛿 and suppose that 𝐵 ℎ ( 𝑓 − id𝐶 ) = 𝛿 − 𝜖. Then 𝑓 is injective on each (resp. proper) 𝛿-robust subspace, and maps it to a (resp. proper) 𝜖-robust subspace.

Proof of Proposition 2.18. — By Lemma 2.19, there exists a proper 𝛿 𝑑1 -robust subspace 𝑉 in (𝐶, 𝑑 𝐶 ) of dimension 𝑘 (where 𝑘 is given by that lemma). By Lemma 2.20, 𝑓 (𝑉) will be a proper 𝜖-robust subspace of dimension 𝑘. Consider a subspace 𝑉 0 ⊂ 𝐶 of dimension 𝑘 such that 𝑑 𝐶 (𝑉 0) = 𝑉, and a complement 𝑊 in Ker(𝑑 𝐶 ) to Im(𝑑 𝐶 ). Then 𝑑 𝐶 ( 𝑓 (𝑉 0)) = 𝑓 (𝑉), showing that dim 𝑑 𝐶 ( 𝑓 (𝑉 0)) = 𝑘, and 𝑓 (𝑊) will again be a complement in Ker(𝑑 𝐶 ) to Im(𝑑 𝐶 ). (Note that 𝑓 (𝑊) ∩ 𝑑 𝐶 (𝐶) = 0 because, by assumption, 𝑓 − id𝐶 is null-homotopic, so 𝑓 induces an isomorphism in homology.) Now, by Lemma 2.20 again, 𝑓 (𝑊) will have the correct dimension. Finally the three subspaces 𝑓 (𝑉), 𝑓 (𝑉 0), 𝑓 (𝑊) are direct summands of 𝐶 whence dimΛ Im( 𝑓 ) ≥ dimΛ 𝐻∗ (𝐶, 𝑑 𝐶 ) + 2𝑘,









finishing the proof. Proof of Lemma 2.19. — The identities dimΛ (𝐶) = dimΛ 𝐻∗ (𝐶, 𝑑 𝐶 ) + 2 dimΛ Im(𝑑𝐶 ) ,







dimΛ (𝐶) = dimΛ 𝐻∗ (𝐶, 𝑑0 ) + 2 dimΛ Im(𝑑0 ) ,







show that dimΛ (𝐻∗ (𝐶, 𝑑0 )) − dimΛ (𝐻∗ (𝐶, 𝑑 𝐶 )) is even. Moreover we obtain (2.48)

dimΛ Im(𝑑 𝐶 ) = dimΛ Im(𝑑0 ) + 𝑘.







From [UZ16, Proposition 7.4], it is immediate to construct a projection 𝜋 : 𝐶 → Im(𝑑0 ), that restricts to the identity on Im(𝑑0 ) and satisfies 𝐴(𝜋(𝑥)) ≤ 𝐴(𝑥) for all 𝑥 ∈ 𝐶. From (2.48) we now have that dim(Ker(𝜋 Im(𝑑 𝐶 ))) ≥ 𝑘. We claim that 𝑉 = Ker 𝜋

Im(𝑑 𝐶 )




is 𝛿 𝑑1 -robust. Indeed, if 𝑣 ∈ 𝑉 , 𝑤 ∈ 𝐶, and 𝑣 = 𝑑 𝐶 𝑤, then writing 𝑑 𝐶 𝑤 = 𝑑0 𝑤 + 𝑑1 𝑤, and using 𝜋(𝑣) = 0 we obtain 𝑑0 𝑤 = 𝜋(𝑑0 𝑤) = −𝜋(𝑑1 𝑤), whence 𝑣 = (id −𝜋)(𝑑1 𝑤). Therefore
𝐴(𝑣) = 𝐴 (id −𝜋)(𝑑1 𝑤) ≤ 𝐴(𝑑1 𝑤) ≤ 𝐴(𝑤) − 𝛿 𝑑1 . This implies 𝐴(𝑤) ≥ 𝐴(𝑣) + 𝛿 𝑑1 , concluding the proof.



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Proof of Lemma 2.20. — Let 𝑉 ⊂ 𝐶 be a 𝛿-robust subspace. We write 𝑓 = id𝐶 +𝑑 𝐶 ℎ − ℎ𝑑 𝐶 , where 𝐴(ℎ(𝑥)) ≤ 𝐴(𝑥) + (𝛿 − 𝜖), for all 𝑥 ∈ 𝐶. If 𝑣 ∈ 𝑉 is such that 𝑓 (𝑣) = 0, we would have 𝑣 + 𝑑 𝐶 (ℎ(𝑣)) = 0, which would yield 𝑤 = −ℎ(𝑣), with 𝑣 = 𝑑𝑤 and 𝐴(𝑤) ≤ 𝐴(𝑣) + 𝛿 − 𝜖. On the other hand 𝛿-robustness implies 𝐴(𝑤) ≥ 𝐴(𝑣) + 𝛿. A contradiction. If 𝑓 (𝑣) = 𝑑 𝐶 𝑧, we would have 𝑣 + 𝑑 𝐶 ℎ(𝑣) = 𝑑 𝐶 𝑧,




which would yield 𝑤 = 𝑧 − ℎ(𝑣), with 𝑣 = 𝑑𝑤. Therefore by 𝛿-robustness we obtain 𝐴(𝑣) + 𝛿 ≤ 𝐴 𝑧 − ℎ(𝑣) ≤ max 𝐴(ℎ(𝑣)), 𝐴(𝑧) .








Since 𝐴(ℎ(𝑣)) ≤ 𝐴(𝑣) + 𝛿 − 𝜖, we get 𝐴(𝑣) + 𝛿 ≤ 𝐴(𝑧)

and

𝐴 𝑓 (𝑣) ≤ max 𝐴(𝑣), 𝐴(ℎ(𝑣)) ≤ 𝐴(𝑣) + 𝛿 − 𝜖 ≤ 𝐴(𝑧) − 𝜖.








We conclude that 𝐴(𝑧) ≥ 𝐴( 𝑓 (𝑣)) + 𝜖, which finishes the proof.

ASTÉRISQUE 426



CHAPTER 3 FLOER THEORY AND FUKAYA CATEGORIES

We set up the variant of Floer theory that will be used in this paper. In particular, we discuss how to choose the auxiliary parameters of this theory so that the Fukaya category becomes a weakly filtered 𝐴∞ -category. Let (𝑀, 𝜔) be a symplectic manifold, either closed or convex at infinity. We always assume 𝑀 to be connected. Denote by L𝑎𝑔 we (𝑀) the collection of all closed connected Lagrangian submanifolds 𝐿 ⊂ 𝑀 that are weakly exact. Recall that 𝐿 ⊂ 𝑀 is weakly exact if for every ∫ 𝐷 𝐴 ∈ 𝐻2 (𝑀, 𝐿) we have 𝐴 𝜔 = 0. 5 Let C ⊂ L𝑎𝑔 we be a collection of weakly exact Lagrangians. Unless explicitly stated otherwise, we henceforth make the following mild assumption on C, whenever 𝑀 is not compact. There exists an open domain 𝑈0 ⊂ 𝑀 with compact closure, such that all Lagrangians 𝐿 ⊂ C lie inside 𝑈0 . For further use, also fix another open domain with compact closure 𝑈1 ⊃ 𝑈 0 as well as an 𝜔-compatible almost complex structure 𝐽conv which is compatible with the convexity of 𝑀 outside of 𝑈 1 . Fix a base ring 𝑅 of characteristic 2 (e.g. 𝑅 = ℤ2 ) and let Λ be the Novikov ring over 𝑅 as defined in (2.1). Denote by F𝑢𝑘( C) the Fukaya category, with coefficients in Λ, whose objects are 𝐿 ∈ C. We mostly follow here the implementation of the Fukaya category due to Seidel [Sei08] with several modifications that will be explained shortly. As in [Sei08], for every pair of Lagrangians 𝐿0 , 𝐿1 ∈ C we choose a Floer datum D𝐿0 ,𝐿1 = (𝐻 𝐿0 ,𝐿1 , 𝐽 𝐿0 ,𝐿1 ) consisting of a Hamiltonian function 𝐻 𝐿0 ,𝐿1 : [0, 1]× 𝑀 → ℝ and a time-dependent 𝜔compatible almost complex structure 𝐽 𝐿0 ,𝐿1 = {𝐽𝑡𝐿0 ,𝐿1 } 𝑡∈[0,1] . In case 𝑀 is not compact we require that outside of 𝑈1 we have 𝐻 𝐿0 ,𝐿1 ≡ 0 and 𝐽𝑡𝐿0 ,𝐿1 ≡ 𝐽conv . Denote by O(𝐻 𝐿0 ,𝐿1 ) the set of orbits 𝛾 : [0, 1] → 𝑀 of the Hamiltonian flow 𝐻 𝐿0 ,𝐿1 𝜙𝑡 generated by 𝐻 𝐿0 ,𝐿1 such that 𝛾(0) ∈ 𝐿0 and 𝛾(1) ∈ 𝐿1 . The Floer complex 5. The group 𝐻2𝐷 (𝑀, 𝐿) ⊂ 𝐻2 (𝑀, 𝐿) is by definition the image of the Hurewicz homomorphism 𝜋2 (𝑀, 𝐿) → 𝐻2 (𝑀, 𝐿).

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CF(𝐿0 , 𝐿1 ; D𝐿0 ,𝐿1 ) is a free Λ-module generated by the set O(𝐻 𝐿0 ,𝐿1 ): CF(𝐿0 , 𝐿1 ; D𝐿0 ,𝐿1 ) =

(3.1)

Ê

Λ𝛾.

𝛾∈O(𝐻 𝐿0 ,𝐿1 )

We work here in an ungraded setting. The differential 𝜇1 on the Floer complex is defined by counting solutions 𝑢 of the Floer equation: 𝑢 : ℝ × [0, 1] → 𝑀, 𝜕𝑠 𝑢 +

(3.2)

𝐽𝑡𝐿0 ,𝐿1 (𝑢)𝜕𝑡 𝑢 ∫ ∞∫ 1

𝐸(𝑢) :=

𝑢(ℝ × 0) ⊂ 𝐿0 , 𝑢(ℝ × 1) ⊂ 𝐿1 ,

= −∇𝐻𝑡𝐿0 ,𝐿1 (𝑢),

|𝜕𝑠 𝑢| 2 d𝑡 d𝑠 < ∞.

−∞

0

where (𝑠, 𝑡) ∈ ℝ × [0, 1]. Here, 𝐻𝑡𝐿0 ,𝐿1 (𝑥) := 𝐻 𝐿0 ,𝐿1 (𝑡, 𝑥) and ∇𝐻𝑡𝐿0 ,𝐿1 is the gradient of the function 𝐻𝑡𝐿0 ,𝐿1 : 𝑀 → ℝ with respect to the Riemannian metric 𝑔𝑡 ( , ) = 𝜔( , 𝐽𝑡𝐿0 ,𝐿1 ) associated to 𝜔 and 𝐽𝑡𝐿0 ,𝐿1 . Quantity 𝐸(𝑢) in the last line of (3.2) is the energy of a solution 𝑢 and we consider only finite energy solutions. (Note also that the norm |𝜕𝑠 𝑢| in the definition of 𝐸(𝑢) is calculated with respect to the metric 𝑔𝑡 .) Solutions 𝑢 of (3.2) are also called Floer trajectories.

..

.

.

For 𝛾− , 𝛾+ ∈ O(𝐻 𝐿0 ,𝐿1 ) consider the space of parametrized Floer trajectories 𝑢 connecting 𝛾− to 𝛾+ : (3.3)

M(𝛾− , 𝛾+ ; D𝐿0 ,𝐿1 ) = 𝑢 ; 𝑢 solves (3.2) and lim 𝑢(𝑠, 𝑡) = 𝛾± (𝑡) .





𝑠→±∞

Note that ℝ acts on this space by translations along the 𝑠-coordinate. This action is generally free, with the only exception being 𝛾− = 𝛾+ and the stationary solution 𝑢(𝑠, 𝑡) = 𝛾− (𝑡) at 𝛾− . Whenever 𝛾− ≠ 𝛾+ , we denote by (3.4)

M∗ (𝛾− , 𝛾+ ; D𝐿0 ,𝐿1 ) := M(𝛾− , 𝛾+ ; D𝐿0 ,𝐿1 )/ℝ

the quotient space (i.e. the space of non-parametrized solutions). In the case 𝛾− = 𝛾+ we define M∗ (𝛾− , 𝛾− ; D𝐿0 ,𝐿1 ) in the same way except that we omit the stationary solution at 𝛾− . For a generic choice of Floer datum D𝐿0 ,𝐿1 the space M∗ (𝛾− , 𝛾+ ; D𝐿0 ,𝐿1 ) is a smooth manifold (possibly with several components having different dimensions). Moreover, its 0-dimensional component M0∗ (𝛾− , 𝛾+ ; D𝐿0 ,𝐿1 ) is compact hence a finite set. Define now 𝜇1 : CF(𝐿0 , 𝐿1 ; D𝐿0 ,𝐿1 ) → CF(𝐿0 , 𝐿1 ; D𝐿0 ,𝐿1 ) by (3.5)

𝜇1 (𝛾− ) :=

ÕÕ 𝛾+

𝑇 𝜔(𝑢) 𝛾+ ,

for all 𝛾− ∈ O(𝐻 𝐿0 ,𝐿1 ),

𝑢

and extending linearly over Λ. Here, the outer sum runs over all 𝛾+ ∈ O(𝐻 𝐿0 ,𝐿1 ) and the inner sum over all solutions 𝑢 ∈ M0∗ (𝛾− , 𝛾+ ; D𝐿0 ,𝐿1 ). The term 𝜔(𝑢) is∫ a shorthand notation for the symplectic area of a Floer trajectory 𝑢, namely 𝜔(𝑢) := ℝ×[0,1] 𝑢 ∗ 𝜔.

ASTÉRISQUE 426

CHAPTER 3. FLOER THEORY AND FUKAYA CATEGORIES

53

It is well known that 𝜇1 is a differential and we denote the homology of CF(𝐿0 , 𝐿1 ; D𝐿0 ,𝐿1 ) by HF(𝐿0 , 𝐿1 ; D𝐿0 ,𝐿1 ) – the Floer homology of (𝐿0 , 𝐿1 ). This homology is independent of the choice of the Floer datum in the sense that for every two regular choices of Floer data D𝐿0 ,𝐿1 , D𝐿0 0 ,𝐿1 there is a canonical isomorphism 𝜓 D, D0 : HF(𝐿0 , 𝐿1 ; D𝐿0 ,𝐿1 ) −→ HF(𝐿0 , 𝐿1 ; D𝐿0 0 ,𝐿1 ) which form a directed system. Therefore we can regard this collection of Λ-modules as one and denote it by HF(𝐿0 , 𝐿1 ). The canonical isomorphisms 𝜓 D, D0 do not preserve action-filtrations in general, hence there is no meaning to 𝐻(CF≤𝛼 (𝐿0 , 𝐿1 )) without specifying the Floer datum. The higher operations 𝜇𝑑 , 𝑑 ≥ 2, follow the same scheme as in [Sei08], with the main difference being that we work over the Novikov ring Λ. More precisely, we first make a choice of strip-like ends along the compactification of the moduli-spaces R𝑑+1 , 𝑑 ≥ 2, of disks with (𝑑 + 1)-boundary punctures. For every 𝑟 ∈ R𝑑+1 denote by 𝑆𝑟 the punctured disk corresponding to 𝑟 (thus 𝑆𝑟 is the actual punctured Riemann surface corresponding to the parameter 𝑟 ∈ R𝑑+1 ). Denote the punctures by 𝜁 𝑖 , 𝑖 = 0, . . . , 𝑑, going in clockwise direction. The puncture 𝜁 0 will be called the exit and 𝜁 1 , . . . , 𝜁 𝑑 the entry punctures. We denote the arc along 𝜕𝑆𝑟 connecting 𝜁 𝑖 to 𝜁 𝑖+1 by 𝐶 𝑖 , with the convention that 𝜁 𝑑+1 := 𝜁 0 . 6 Next we make a choice of perturbation data D𝐿0 ,...,𝐿𝑑 = (𝐾 𝐿0 ,...,𝐿𝑑 , 𝐽 𝐿0 ,...,𝐿𝑑 ) for every tuple of 𝑑 + 1 Lagrangians 𝐿0 , . . . , 𝐿 𝑑 ∈ C. The first item 𝐾 𝐿0 ,...,𝐿𝑑 = {𝐾 𝑟𝐿0 ,...,𝐿𝑟 } 𝑟∈R𝑑+1 is a family of 1-forms parametrized by 𝑟 ∈ R𝑑+1 , with values in the space of Hamiltonian functions 𝑀 → ℝ. The second one is a family 𝐽 𝐿0 ,...,𝐿𝑑 = {𝐽𝑟𝐿0 ,...,𝐿𝑑 } 𝑟∈R𝑑+1 of 𝜔-compatible domain-dependent almost complex structures on 𝑀, parametrized 𝐿0 ,...,𝐿 𝑑 } by 𝑟 ∈ R𝑑+1 . In other words for every 𝑟 ∈ R𝑑+1 , 𝐽𝑟𝐿0 ,...,𝐿𝑑 is itself a family {𝐽𝑟,𝑧 𝑧∈𝑆𝑟 of 𝜔-compatible almost complex structure on 𝑀, parametrized by 𝑧 ∈ 𝑆𝑟 . The perturbation data are required to satisfy several additional conditions. The first one is that along each of the strip-like ends the perturbation data coincides with the Floer data associated to the pair of Lagrangians corresponding to that end. More precisely, along the strip-like end corresponding to the puncture 𝜁 𝑖 of 𝑆𝑟 we have (3.6)

𝐾 𝑟𝐿0 ,...,𝐿𝑟 = 𝐻𝑡𝐿𝑖−1 ,𝐿𝑖 d𝑡,

𝐽 𝐿0 ,...,𝐿𝑑 = 𝐽𝑡𝐿𝑖−1 ,𝐿𝑖 ,

𝑖 = 1, . . . , 𝑑 + 1,

where we have used here the convention that 𝐿 𝑑+1 = 𝐿0 . Here (𝑠, 𝑡) are the conformal coordinates corresponding to the strip-like ends. The second condition is that along the arc 𝐶 𝑖 we have (3.7)

𝐾 𝐿0 ,...,𝐿𝑑 (𝜉)

𝐿𝑖

= 0,

for all 𝜉 ∈ 𝑇(𝐶 𝑖 ), 𝑖 = 0, . . . , 𝑑.

6. Of course, 𝜁 𝑖 and 𝐶 𝑖 all depend on 𝑟 but we suppress this from the notation.

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The choices of strip-like ends and perturbation data along R𝑑+1 are required to be compatible with gluing and splitting, or in the language of [Sei08] “consistent”. This means essentially that these choices extend smoothly over the compactification R𝑑+1 of the space of boundary-punctured disks. In turn, this requires that for every 𝑑, the choices of strip-like ends and perturbation data done over R𝑑+1 are compatible with those that appear on all the strata of the boundary 𝜕R𝑑+1 of the compactification R𝑑+1 of R𝑑+1 . We refer the reader to [Sei08, Chapter 9] for the precise definitions and implementation. In case 𝑀 is not compact we add the following conditions on the perturbation data. For every 𝑟 ∈ R𝑑+1 and 𝜉 ∈ 𝑇(𝑆𝑟 ) the Hamiltonian function 𝐾 𝑟𝐿0 ,...,𝐿𝑑 (𝜉) is required to vanish outside of 𝑈1 and 𝐽𝑟𝐿0 ,...,𝐿𝑑 ≡ 𝐽conv outside of 𝑈1 . Once we have made consistent choices of strip-like ends and perturbation data we define the higher operations 𝜇𝑑 for 𝐿0 , . . . , 𝐿 𝑑 ∈ C as follows. For 𝑟 ∈ R𝑑+1 , 𝑧 ∈ 𝑆𝑟 and 𝜉 ∈ 𝑇𝑧 (𝑆𝑟 ) define 𝑌𝑟,𝑧 (𝜉) to be the Hamiltonian vector field 𝐿0 ,...,𝐿 𝑑 (𝜉) : 𝑀 → ℝ. Consider now the following Floer equation: of the function 𝐾 𝑟,𝑧 𝑢 : 𝑆𝑟 → 𝑀, 𝐷𝑢𝑧 +

𝑢(𝐶 𝑖 ) ⊂ 𝐿 𝑖 , 𝑖 = 0, . . . , 𝑑,

𝐿0 ,...,𝐿 𝑑 𝐽𝑟,𝑧 (𝑢)

𝐿0 ,...,𝐿 𝑑 ◦ 𝐷𝑢𝑧 ◦ 𝑗𝑟 = 𝑌𝑟,𝑧 (𝑢) + 𝐽𝑟,𝑧 ◦ 𝑌𝑟,𝑧 (𝑢) ◦ 𝑗𝑟 ,

(3.8) 𝐸(𝑢) :=

∫ 𝑆𝑟

|𝐷𝑢 − 𝑌𝑟 | 2𝐽 𝜎 < ∞.

Here 𝑗𝑟 stands for the complex structure on 𝑆𝑟 . The last quantity in (3.8) is the energy of a solution 𝑢 and we consider only solutions of finite energy. The definition of 𝐸(𝑢) involves an area form 𝜎 on 𝑆𝑟 and the norm | | 𝐽 on the space of linear maps 𝑇𝑧 (𝑆𝑟 ) → 𝑇𝑢(𝑧) (𝑀) which is induced by 𝑗𝑟 , 𝐽 := 𝐽𝑟𝐿0 ,...,𝐿𝑑 and 𝜎. See [MS12, Section 2.2, page 20] for the definition. Note that 𝐸(𝑢) does not depend on 𝜎.

.

Given orbits 𝛾−1 , . . . , 𝛾−𝑑 , 𝛾+ with 𝛾−𝑖 ∈ O(𝐻 𝐿𝑖−1 ,𝐿𝑖 ) and 𝛾+ ∈ O(𝐻 𝐿0 ,𝐿𝑑 ) define the space of so called Floer polygons connecting 𝛾−1 , . . . , 𝛾−𝑑 to 𝛾+ to be the space of all pairs (𝑟, 𝑢) with 𝑟 ∈ R𝑑+1 and 𝑢 : 𝑆𝑟 → 𝑀 such that 1) 𝑢 is a solution of (3.8); 2) lim𝑠→∞ 𝑢(𝑠, 𝑡) = 𝛾−𝑖 (𝑡) for 1 ≤ 𝑖 ≤ 𝑑 on the strip-like end corresponding to puncture 𝜁 𝑖 , where (𝑠, 𝑡) ∈ (−∞, 0] × [0, 1] are the conformal coordinates on the strip-like end of 𝜁 𝑖 ; 3) lim𝑠→∞ 𝑢(𝑠, 𝑡) = 𝛾+ (𝑡) for 1 ≤ 𝑖 ≤ 𝑑 on the strip-like end corresponding to puncture 𝜁 0 , where (𝑠, 𝑡) ∈ [0, ∞) × [0, 1] are the conformal coordinates on the striplike end of 𝜁 0 . We denote this space by M(𝛾−1 , . . . , 𝛾−𝑑 , 𝛾+ ; D𝐿0 ,...,𝐿𝑑 ). For generic choices of Floer and perturbation data this space is a smooth manifold and its 0-dimensional component M0 (𝛾−1 , . . . , 𝛾−𝑑 , 𝛾+ ; D𝐿0 ,...,𝐿𝑑 )

ASTÉRISQUE 426

55

3.1. UNITS

is compact hence a finite set. 7 Define now 𝜇𝑑 (𝛾−1 , . . . , 𝛾−𝑑 ) =

(3.9)

ÕÕ

𝑇 𝜔(𝑢) 𝛾+ ∈ CF(𝐿0 , 𝐿 𝑑 ; D𝐿0 ,𝐿𝑑 ),

𝛾+ (𝑟,𝑢)

where the first sum goes over all 𝛾+ ∈ O(𝐻 𝐿0 ,𝐿𝑑 ) and the second sum goes over all pairs (𝑟, 𝑢) ∈ M0 (𝛾−1 , . . . , 𝛾−𝑑 , 𝛾+ ; D𝐿0 ,...,𝐿𝑑 ). The term 𝜔(𝑢) stands for the symplectic area of 𝑢, 𝜔(𝑢) :=

∫ 𝑆𝑟

𝑢 ∗ 𝜔.

Extending 𝜇𝑑 multi-linearly over Λ we obtain an operation: 𝜇𝑑 : CF(𝐿0 , 𝐿1 ; D𝐿0 ,𝐿1 ) ⊗ · · · ⊗ CF(𝐿 𝑑−1 , 𝐿 𝑑 ; D𝐿𝑑−1 ,𝐿𝑑 ) −→ CF(𝐿0 , 𝐿 𝑑 ; D𝐿0 ,𝐿𝑑 ). With all the operations above F𝑢𝑘( C) becomes an 𝐴∞ -category. The proof of this is essentially the same as the one in [Sei08], the only difference is that one needs to keep track of the areas appearing as exponents in the variable 𝑇 of the Novikov ring.

3.1. Units We now explain briefly how to construct homology units in F𝑢𝑘( C). More details can be found in [Sei08, Chapter 8]. Denote by 𝑆 = 𝐷\𝜁 0 the unit disk punctured at one boundary point 𝜁 0 ∈ 𝜕𝐷. Fix a strip-like end around 𝜁 0 making 𝜁 0 an exit puncture and let (𝑠, 𝑡) be the conformal coordinates associated to this strip-like end. Let 𝐿 ∈ C and D𝐿,𝐿 be a regular Floer datum for the pair (𝐿, 𝐿). Pick a regular perturbation datum D𝑆 = (𝐾, 𝐽), as described earlier with the only difference that 𝐾 and 𝐽 are are defined only on 𝑆 (i.e. there is no dependence on any space like R𝑑+1 ). As before, we require that 𝐷𝑆 coincides with the Floer datum D𝐿,𝐿 along the strip-like ends in the sense of (3.6). For 𝑧 ∈ 𝑆, 𝜉 ∈ 𝑇𝑧 (𝑆) define 𝑌𝑧 (𝜉) as before. Given 𝛾 ∈ O(𝐻 𝐿,𝐿 ) consider the space M(𝛾; D𝑆 ) of solutions 𝑢 : (𝑆, 𝜕𝑆) → (𝑀, 𝐿) of 𝐿0 ,...,𝐿 𝑑 , 𝑗 replaced by 𝑆, 𝑌 , 𝐽 and 𝑖 the last two lines of equation (3.8), with 𝑆𝑟 , 𝑌𝑟,𝑧 , 𝐽𝑟,𝑧 𝑟 𝑧 𝑧 respectively, and such that along the strip-like end at 𝜉0 we have lim𝑠→∞ 𝑢(𝑠, 𝑡) = 𝛾(𝑡). Define now an element 𝑒 𝐿 ∈ CF(𝐿, 𝐿; D𝐿,𝐿 ) by (3.10)

𝑒 𝐿 :=

Õ

Õ

𝛾∈O(𝐻 𝐿,𝐿 )

𝑢

𝑇 𝜔(𝑢) 𝛾,

where the second sum runs over all solutions 𝑢 in the 0-dimensional component M0 (𝛾; D𝑆 ) of M(𝛾; D𝑆 ). By standard theory 𝑒 𝐿 is a cycle and its homology class in HF(𝐿, 𝐿) is independent of the choice of the Floer and perturbation data. Moreover, [𝑒 𝐿 ] ∈ HF(𝐿, 𝐿) is a unit for the product induced by 𝜇2 . 7. Recall that 𝑑 ≥ 2 hence we do not divide here by any reparametrization group.

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3.2. Families of Fukaya categories The Fukaya category F𝑢𝑘( C) depends on all the choices made – strip-like ends, Floer and perturbation data. We fix once and for all a consistent choice of striplike ends and denote by 𝐸 the space of all consistent choices of perturbation data (compatible with the fixed choice of strip-like ends). The space 𝐸 can be endowed with a natural topology (and a structure of a Fréchet manifold), induced from a larger space in which one allows perturbation data in appropriate Sobolev spaces (see [Sei08, Chapter 9]). The subspace 𝐸reg ⊂ 𝐸 of regular perturbation data is residual hence a dense subset. The space 𝐸 contains a distinguished subspace N ⊂ 𝐸 consisting of all consistent 0 choices of perturbation data D = (𝐾, 𝐽) with 𝐾 ≡ 0. Fix a subset 𝐸reg ⊂ 𝐸reg whose 0 closure 𝐸reg contains N. 0 For 𝑝 ∈ 𝐸reg we denote by F𝑢𝑘( C; 𝑝) the associated Fukaya category with choice 0 , of perturbation data 𝑝. We thus obtain a family of 𝐴∞ -categories {F𝑢𝑘( C; 𝑝)} 𝑝∈𝐸reg 0 . It is well known that this is a coherent system of parametrized by 𝑝 ∈ 𝐸reg 𝐴∞ -categories (see [Sei08, Chapter 10]), in particular they are all mutually quasiequivalent. In what follows we will sometimes use the following notation. Given 𝐿0 , 𝐿1 ∈ C 0 and 𝑝 ∈ 𝐸reg we write CF(𝐿0 , 𝐿1 ; 𝑝) for CF(𝐿0 , 𝐿1 ; D𝐿0 ,𝐿1 ), where D𝐿0 ,𝐿1 is the Floer 0 . datum prescribed by the choice 𝑝 ∈ 𝐸reg 3.3. Weakly filtered structure on Fukaya categories We start by defining filtrations on the Floer complexes of pairs of Lagrangians in C. We follow here the general recipe from § 2.2.3. Denote by 𝜈 : Λ → ℝ ∪ {∞} the standard valuation defined by



𝜈 𝑎 0 𝑇 𝜆0 +

(3.11)

∞ Õ



𝑎 𝑖 𝑇 𝜆 𝑖 = 𝜆0 ,

𝑖=1

where 𝑎0 ≠ 0 and 𝜆 𝑖 > 𝜆0 for every 𝑖 ≥ 1. As usual we set 𝜈(0) = ∞. Let 𝐿0 , 𝐿1 ∈ C be two Lagrangians and D𝐿0 ,𝐿1 = (𝐻 𝐿0 ,𝐿1 , 𝐽 𝐿0 ,𝐿1 ) a Floer datum. We define an “action functional” A : CF(𝐿0 , 𝐿1 ; D𝐿0 ,𝐿1 ) −→ ℝ ∪ {−∞} 𝜆 𝑖 ∈ Λ with 𝜆 < 𝜆 for all 𝑖 ≥ 1, and 𝑎 ≠ 0. as follows. Let 𝑃(𝑇) = ∞ 0 𝑖 0 𝑖=0 𝑎 𝑖 𝑇 Let 𝛾 ∈ O(𝐻 𝐿0 ,𝐿1 ) be a Hamiltonian orbit. We first define:

Í

A 𝑃(𝑇)𝛾 := −𝜈(𝑃(𝑇)) +




∫

1

𝐻𝑡𝐿0 ,𝐿1

𝛾(𝑡) d𝑡 = −𝜆0 +




0

∫

1

𝐻𝑡𝐿0 ,𝐿1 𝛾(𝑡) d𝑡.




0

Í𝑙

Now let 𝑘=1 𝑃𝑘 (𝑇)𝛾𝑘 ∈ CF(𝐿0 , 𝐿1 ; D𝐿0 ,𝐿1 ) be a general non-trivial element, where the 𝛾𝑘 ’s are mutually distinct. We extend the definition of A to such an element by A 𝑃1 (𝑇)𝛾1 + · · · + 𝑃𝑙 (𝑇)𝛾𝑙 := max A 𝑃𝑘 (𝑇)𝛾𝑘 ; 𝑘 = 1, . . . , 𝑙 .




Finally, we put A(0) = −∞.

ASTÉRISQUE 426








57

3.3. WEAKLY FILTERED STRUCTURE ON FUKAYA CATEGORIES

We now define a filtration on CF(𝐿0 , 𝐿1 ; D𝐿0 ,𝐿1 ) by 𝐶𝐹 ≤𝛼 (𝐿0 , 𝐿1 ; D𝐿0 ,𝐿1 ) := 𝑥 ∈ CF(𝐿0 , 𝐿1 ; D𝐿0 ,𝐿1 ) ; A(𝑥) ≤ 𝛼 .



(3.12)



Before we go on, a quick remark regarding the Hamiltonian functions 𝐻 𝐿0 ,𝐿1 in the Floer data is in order. We do not assume that these functions are normalized (e.g. by requiring them to have zero mean when 𝑀 is closed, or to be compactly supported when 𝑀 is open). This means that if we replace 𝐻 𝐿0 ,𝐿1 by 𝐻 𝐿0 ,𝐿1 + 𝑐(𝑡) for some family of constants 𝑐(𝑡), we get the same chain complex as CF(𝐿0 , 𝐿1 ; D𝐿0 ,𝐿1 ) but with a shifted action-filtration. Returning to (3.12), it is easy to see that 𝐶𝐹 ≤𝛼 (𝐿0 , 𝐿1 ; D𝐿0 ,𝐿1 ) is a Λ0 -module (though not a Λ-module). The fact that this filtration is preserved by 𝜇1 and moreover, that it provides F𝑢𝑘( C) with a structure of a weakly filtered 𝐴∞ -category are the subject of the following proposition. 0 Proposition 3.1. — There exists a choice 𝐸reg ⊂ 𝐸reg \ N with 𝐸0reg ⊃ N and such that

0 and F𝑢𝑘( C; 𝑝) be the corresponding Fukaya category. Then the following holds. Let 𝑝 ∈ 𝐸reg there exist a sequence of non-negative real numbers (𝑝) = (𝜖1 (𝑝) = 0, 𝜖 2 (𝑝), . . . , 𝜖 𝑑 (𝑝), . . . ) and 𝑢(𝑝), 𝜁(𝑝), 𝜅(𝑝) ∈ ℝ+ , depending on 𝑝, such that: (i) With the filtrations described above on the Floer complexes, F𝑢𝑘( C; 𝑝) becomes a weakly filtered 𝐴∞ -category with discrepancy ≤ (𝑝).

(ii) F𝑢𝑘( C; 𝑝) is h-unital in the weakly filtered sense and there is a choice of homology units with discrepancy ≤ 𝑢(𝑝). (iii) F𝑢𝑘( C; 𝑝) ∈ 𝑈 𝑒 (𝜁(𝑝)). (iv) Let 𝐿 ∈ C and denote by L its Yoneda module. Then L ∈ 𝑈𝑚 (𝜅(𝑝)). (v) For every 𝑝0 ∈ N ⊂ 𝐸 (see page 56) we have lim 𝜖 𝑑 (𝑝) = 0, for all 𝑑 ≥ 2,

𝑝→𝑝0

lim 𝑢(𝑝) = lim 𝜁(𝑝) = lim 𝜅(𝑝) = 0.

𝑝→𝑝 0

𝑝→𝑝 0

𝑝→𝑝 0

Proof. — We will only give a sketch of the proof, as most of the ingredients are standard in the theory (see e.g. [Sei08]). 0 will be given The precise definition of the set of choices of perturbation data 𝐸reg in the course of the proof.

We begin by showing that the filtration (3.12) is preserved by 𝜇1 . Let 𝐿0 , 𝐿1 ∈ Cand D𝐿0 ,𝐿1 = (𝐻 𝐿0 ,𝐿1 , 𝐽 𝐿0 ,𝐿1 ) be a Floer datum. Let 𝛾− , 𝛾+ ∈ O(𝐻 𝐿0 ,𝐿1 ) be two generators of CF(𝐿0 , 𝐿1 ; D𝐿0 ,𝐿1 ) and let 𝑢 ∈ M0 (𝛾− , 𝛾+ ; D𝐿0 ,𝐿1 ) be an element of the 0-dimensional component of Floer trajectories connecting 𝛾− to 𝛾+ . By (3.5), the contribution of 𝑢 to 𝜇1 (𝛾− ) is 𝑇 𝜔(𝑢) 𝛾+ . We now have the following standard energy-area identity for solutions 𝑢 ∈ M(𝛾− , 𝛾+ ; D𝐿0 ,𝐿1 ) of the Floer equation (3.13)

𝐸(𝑢) = 𝜔(𝑢) +

∫ 0

1

𝐻𝑡𝐿0 ,𝐿1 𝛾− (𝑡) d𝑡 −




∫

1

𝐻𝑡𝐿0 ,𝐿1 𝛾+ (𝑡) d𝑡.




0

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CHAPTER 3. FLOER THEORY AND FUKAYA CATEGORIES

It immediately follows that A(𝑇

𝜔(𝑢)

𝛾+ ) = −𝜔(𝑢) +

∫

1

𝐻𝑡𝐿0 ,𝐿1

𝛾+ (𝑡) d𝑡 ≤




0

∫

1

𝐻𝑡𝐿0 ,𝐿1 𝛾− (𝑡) d𝑡 = A(𝛾− ).




0

This shows that 𝜇1 preserves the filtration (3.12) on CF(𝐿0 , 𝐿1 ; D𝐿0 ,𝐿1 ). The next step is to analyze the behavior of the higher operations 𝜇𝑑 , 𝑑 ≥ 2, with respect to our filtration. Let 𝐿0 , . . . , 𝐿 𝑑 ∈ C and D𝐿0 ,...,𝐿𝑑 be the corresponding perturbation data. Let 𝛾−𝑖 ∈ O(𝐻 𝐿𝑖−1 ,𝐿𝑖 ), 𝛾+ ∈ O(𝐻 𝐿0 ,𝐿𝑑 ), and (𝑟, 𝑢) ∈ M0 (𝛾−1 , . . . , 𝛾−𝑑 , 𝛾+ ; D𝐿0 ,...,𝐿𝑑 ). The contribution of 𝑢 to 𝜇𝑑 (𝛾−1 , . . . , 𝛾−𝑑 ) is 𝑇 𝜔(𝑢) 𝛾+ . Similarly to (3.13) we have the following energy-area identity (3.14) 𝐸(𝑢) = 𝜔(𝑢) −

∫

1

𝐿 ,𝐿 𝑑

𝐻𝑡 0

𝛾+ (𝑡) d𝑡 +




𝑑 ∫ Õ 𝑗=1

0

1

𝐿 𝑗−1 ,𝐿 𝑗

𝐻𝑡

𝑗

𝛾− (𝑡) 𝑣 +




∫

0

𝑅𝐾

𝐿0 ,...,𝐿 𝑑

(𝑢),

𝑆𝑟

𝐿 ,...,𝐿

for solutions 𝑢 of (3.8), where 𝑅 𝐾 0 𝑑 is the curvature 2-form on 𝑆𝑟 associated to the perturbation form 𝐾 𝐿0 ,...,𝐿𝑑 . In local conformal coordinates (𝑠, 𝑡) ∈ 𝑆𝑟 it can be written as follows. Write 𝐾 𝐿0 ,...,𝐿𝑑 = 𝐹𝑠,𝑡 d𝑠 + 𝐺 𝑠,𝑡 d𝑡 for some functions 𝐹𝑠,𝑡 , 𝐺 𝑠,𝑡 : 𝑀 → ℝ. Then 𝐿0 ,...,𝐿 𝑑

𝑅 𝐾𝑠,𝑡

(3.15)



= −

 𝜕𝐹𝑠,𝑡 𝜕𝐺 𝑠,𝑡 + − {𝐹𝑠,𝑡 , 𝐺 𝑠,𝑡 } d𝑠 ∧ d𝑡, 𝜕𝑡 𝜕𝑠

where {𝐹𝑠,𝑡 , 𝐺 𝑠,𝑡 } := −𝜔(𝑋 𝐹𝑠,𝑡 , 𝑋 𝐺 𝑠,𝑡 ) is the Poisson bracket of the functions 𝐹𝑠,𝑡 , 𝐺 𝑠,𝑡 . We now need to bound the term

∫ 𝑆𝑟

𝑅𝐾

𝐿0 ,...,𝐿 𝑑

(𝑢) from (3.14) independently of (𝑟, 𝑢).

𝐿 ,...,𝐿 R𝑑+1 , the curvature 𝑅 𝐾 0 𝑑

To this end, first note that for any given 𝑟 ∈ vanishes identically along the strip-like ends of 𝑆𝑟 by assumption on the perturbation 1-form. Next, let S𝑑+1 be the universal family of disks with 𝑑 + 1 boundary punctures (see [Sei08, Chapter 9], see also [BC14, Section 3.1]). This is a fiber bundle over R𝑑+1 whose fiber over 𝑟 ∈ R𝑑+1 is the surface 𝑆𝑟 . The space S𝑑+1 admits a partial compactification S𝑑+1 over R𝑑+1 and can be endowed with a smooth structure. Since the perturbation data D𝐿0 ,...,𝐿𝑑 was chosen consistently, the forms 𝐾 𝐿0 ,...,𝐿𝑑 extend to the partial compactification 𝑆 𝑑+1 over R𝑑+1 . Now let W ⊂ 𝑆 𝑑+1 be the union of all the strip-like ends corresponding to all the surfaces parametrized by 𝑟 ∈ R𝑑+1 . Then S𝑑+1 \ Int W is compact. It follows that for all (𝑟, 𝑢) ∈ M0 (𝛾−1 , . . . , 𝛾−𝑑 , 𝛾+ ; D𝐿0 ,...,𝐿𝑑 ) we have

∫ 𝐾 𝐿0 ,...,𝐿 𝑑 𝑅 (𝑢) ≤ 𝜖 𝑑 (𝐾 𝐿0 ,...,𝐿𝑑 ),

(3.16)

𝑆𝑟

(𝐾 𝐿0 ,...,𝐿𝑑 )

where 𝜖 𝑑 depends only on the 𝐶 1 -norm of 𝐾 𝐿0 ,...,𝐿𝑑 (defined in the S𝑑+1 as well as 𝑀 directions). Moreover, we have 𝜖 𝑑 (𝐾 𝐿0 ,...,𝐿𝑑 ) → 0 as 𝐾 𝐿0 ,...,𝐿𝑑 → 0 in the 𝐶 1 -topology (along S𝑑+1 \ Int W and 𝑀).

ASTÉRISQUE 426

3.3. WEAKLY FILTERED STRUCTURE ON FUKAYA CATEGORIES

59

A few words are in order for the case when 𝑀 is not compact. In that case the arguments above continue to work due to our choice of perturbation data. More precisely, recall that we had two open domains 𝑈0 , 𝑈1 ⊂ 𝑀 with compact closure, with 𝑈 0 ⊂ 𝑈1 , and with the following properties: all Lagrangians 𝐿 ∈ L lie in 𝑈0 and outside of 𝑈1 we have D𝐿0 ,...,𝐿𝑑 = (0, 𝐽conv ) for all 𝑟 ∈ R𝑑+1 . This implies that the Floer equations (3.2) and (3.8) become homogeneous at the points where 𝑢(𝑧) ∈ 𝑀 \ 𝑈1 . Since (𝑀, 𝜔, 𝐽conv ) is convex at infinity, the maximum principle implies that all solutions 𝑢 lie within one compact domain of 𝑀. Thus the estimate (3.16) follows by bounding the 𝐶 1 -norm of 𝐾 𝐿0 ,...,𝐿𝑑 only along that compact domain. Coming back to the estimate (3.16), it follows from (3.14) that A(𝑇 𝜔(𝑢) 𝛾+ ) ≤ A(𝛾−1 ) + · · · + A(𝛾−𝑑 ) + 𝜖 𝑑 (𝐾 𝐿0 ,...,𝐿𝑑 ).

(3.17)

In order to obtain a weakly filtered structure on F𝑢𝑘( C; 𝑝) we need to bound from above 𝜖 𝑑 (𝐾 𝐿0 ,...,𝐿𝑑 ) uniformly in 𝐿0 , . . . , 𝐿 𝑑 ∈ C, so that the ultimate discrepancy 𝜖 𝑑 (𝑝) 0 . This is easily done by restricting the set 𝐸0 depends only on the choice of 𝑝 ∈ 𝐸reg reg to choices of perturbation data 𝑝 = { D𝐿0 ,...,𝐿𝑑 } 𝐿0 ,...,𝐿𝑑 ∈ C for which the 𝐶 1 -norms of the forms 𝐾 𝐿0 ,...,𝐿𝑑 are uniformly bounded (in 𝐿0 , . . . , 𝐿 𝑑 ). Since 𝐸reg ⊂ 𝐸 is residual it 0 still has N in its closure. follows that the restricted set of choices 𝐸reg 0 This concludes the proof that F𝑢𝑘( C; 𝑝)𝑝∈𝐸reg is a family of weakly filtered 𝐴∞ categories, and that the bounds on their discrepancies (𝑝) have the property that for all 𝑝0 ∈ N we have lim𝑝→𝑝0 𝜖 𝑑 (𝑝) = 0 for every 𝑑 ≥ 2. We now turn to the statements about the unitality of the categories F𝑢𝑘( C; 𝑝) and 𝐿,𝐿 , 𝐽 𝐿,𝐿 ) be the Floer 0 . Fix 𝐿 ∈ C and let D their Yoneda modules. Let 𝑝 ∈ 𝐸reg 𝐿,𝐿 = (𝐻 datum of (𝐿, 𝐿) prescribed by 𝑝. Recall that a homology unit 𝑒 𝐿 ∈ CF(𝐿, 𝐿; D𝐿,𝐿 ) can be defined by (3.10). Let 𝑆 = 𝐷 \ {𝜁 0 } and D𝑆 = (𝐾, 𝐽) as in Section 3.1. Let 𝛾 ∈ O(𝐻 𝐿,𝐿 ) and 𝑢 ∈ M0 (𝛾; D𝑆 ). According to (3.10) the contribution of 𝛾 and 𝑢 to 𝑒 𝐿 is 𝑇 𝜔(𝑢) 𝛾. The energy-area identity for 𝑢 gives 𝐸(𝑢) = 𝜔(𝑢) −

∫

1

𝐻𝑡𝐿,𝐿 𝛾(𝑡) d𝑡 −




0

∫

𝑅 𝐾 (𝑢),

𝑆

𝑅 𝐾 (𝑢)

where is the curvature associated to the 1-form 𝐾 from the perturbation datum D𝑆 and is defined in a similar way as in (3.15). Note that we can choose the perturbation datum D𝑆 = (𝐾, 𝐽) such that the 𝐶 1 -norm of the 1-form 𝐾 is of the same 𝐿,𝐿 𝐿,𝐿 order size as the 𝐶 1 -norm ∫ of 𝐻 (i.e. k𝐾k 𝐶 1 ≤ 𝐶 k𝐻 k 𝐶 1 for some constant 𝐶). 𝐾 0 𝐿,𝐿 By doing that we obtain | 𝑆 𝑅 (𝑢)| ≤ 𝐶 k𝐻 k 𝐶 1 for some constant 𝐶 0. It follows that A(𝑒 𝐿 ) ≤ 𝐶 0 k𝐻 𝐿,𝐿 k 𝐶 1 . By restricting all the Hamiltonians 𝐻 𝐿,𝐿 , for all 𝐿 ∈ C, to have a uniformly bounded 𝐶 1 -norm we obtain one constant 𝑢(𝑝) (that depends on the choice 𝑝) such that for all every 𝐿 ∈ C we have A(𝑒 𝐿 ) ≤ 𝑢(𝑝). Moreover, 𝑢(𝑝) → 0 as 𝑝 → 𝑝0 ∈ N in the 𝐶 1 topology. This proves the statement about the discrepancy of the units in F𝑢𝑘( C; 𝑝). We now turn to proving statements (3.1) and (3.1) of Proposition 3.1 and the corresponding claims on 𝜁(𝑝) and 𝜅(𝑝) from statement (3.1).

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Let 𝐿, 𝐿0 ∈ C. Choose 𝑆 = 𝐷 \ {𝜁 0 }, D𝑆 and define 𝑒 𝐿 as explained above. Denote by D𝐿,𝐿,𝐿0 be the perturbation datum of the triple (𝐿, 𝐿, 𝐿0) as prescribed by 𝑝. Consider also a disk 𝑆0 = 𝐷 \ {𝜁00 , 𝜁01 , 𝜁02 } with three boundary punctures, ordered clockwise along 𝜕𝐷. We fix strip-like ends near these three punctures such that 𝜁00 , 𝜁01 are entries and 𝜁02 is an exit. Consider a 1-parametric family ({𝑆00𝜏 } 𝜏∈(0,1] , 𝑗𝜏 ) of surfaces (endowed with complex structures) obtained by performing gluing 𝑆 and 𝑆0 at the points 𝜁 0 , 𝜁00 Ý respectively. We construct this family so that 𝑆00𝜏 → 𝑆 𝑆0 as 𝜏 → 0 and 𝑆100 = ℝ ×[0, 1] is the standard strip. Next, we choose a generic family { D𝜏 } 𝜏∈(0,1] of perturbation data over the family {𝑆00𝜏 } 𝜏∈(0,1] such that 1) for 𝜏 → 0, D𝜏 converges to D𝑆 on the 𝑆 component and D𝐿,𝐿,𝐿0 on the 𝑆00 component. 2) D1 = D𝐿,𝐿 . As the family { D𝜏 } 𝜏∈(0,1] is generic, none of the elements in the D𝜏 , 𝜏 < 1 is invariant under reparametrization by any non-trivial automorphism 𝜎 ∈ Aut(𝑆𝜏 ). Let 𝛾, 𝜆 ∈ O(𝐻 𝐿,𝐿 ) and consider the space M(𝛾, 𝜆; { D𝜏 }) of all pairs (𝜏, 𝑢), with 𝜏 ∈ (0, 1] and 𝑢 : 𝑆𝜏 → 𝑀 a solution of the Floer equation (3.8) with the obvious modifications: namely, the lower part of 𝜕𝑆𝜏 is mapped by 𝑢 to 𝐿 and the upper one to 𝐿0, 𝑢 converges to 𝛾 at the entry 𝜁01 and to 𝜆 at the exit 𝜁02 , (𝑆𝑟 , 𝑗𝑟 ) is replaced by (𝑆𝜏 , 𝑗𝜏 ), and 𝐽𝑟,𝑧 and 𝑌𝑟,𝑧 are replaced by the corresponding structures from D𝜏 . 0

Assume that 𝛾 ≠ 𝜆, and consider the 0-dimensional component M0 (𝛾, 𝜆; { D𝜏 }). This is compact 0-dimensional manifold hence a finite set. It gives rise to a map Φ : CF(𝐿, 𝐿0; D𝐿,𝐿0 ) −→ CF(𝐿, 𝐿0; D𝐿,𝐿0 ),

(3.18)

Φ(𝛾) :=

ÕÕ

𝑇 𝜔(𝑢) 𝜆,

for all 𝛾 ∈ O(𝐻 𝐿,𝐿 ), 0

𝜆 (𝜏,𝑢)

where the outer sum is over all 𝜆 ∈ O(𝐻 𝐿,𝐿 ) with 𝜆 ≠ 𝛾 and the second sum is over all (𝜏, 𝑢) ∈ M0 (𝛾, 𝜆; { D𝜏 }). We extend the formula in the second line of (3.18) linearly over Λ. We claim that the following formula holds: 0

(3.19)

𝜇2 (𝑒 𝐿 , 𝑥) = 𝑥 + 𝜇1 ◦ Φ(𝑥) + Φ ◦ 𝜇1 (𝑥),

for all 𝑥 ∈ CF(𝐿, 𝐿0; D𝐿,𝐿0 ),

i.e. Φ is a chain homotopy between the map 𝜇2 (𝑒 𝐿 , ·) and the identity. To prove this, let 𝛾, 𝛾+ ∈ O(𝐻 𝐿,𝐿 ) and consider the 1-dimensional component M1 (𝛾, 𝛾+ ; { D𝜏 }) of the space M(𝛾, 𝛾+ ; { D𝜏 }). This space admits a compactification M1 (𝛾, 𝛾+ ; { D𝜏 }) which is a 1-dimensional manifold with boundary. The elements in the boundary of this space fall into four types: 1) Elements corresponding to the splitting of 𝑆00 into 𝑆 and 𝑆0 at 𝜏 = 0. These can be written as pairs (𝑢𝑆 , 𝑢𝑆0 ) with 𝑢𝑆 ∈ M0 (𝛾0; D𝑆 ) for some 𝛾0 ∈ O(𝐻 𝐿,𝐿 ) and 𝑢𝑆0 ∈ M0 (𝛾0 , 𝛾, 𝛾+ ; D𝐿,𝐿,𝐿0 ). 0

2) Elements corresponding to splitting of 𝑆𝜏 at some 0 < 𝜏0 < 1 into a Floer strip 𝑢0 followed by a solution 𝑢1 : 𝑆𝜏0 → 𝑀 of the Floer equation for the perturbation datum D𝜏0 . More precisely, these can be written as (𝜏0 , 𝑢0 , 𝑢1 ) with 0 < 𝜏0 < 1, 0 𝑢0 ∈ M∗ (𝛾, 𝛾0; D𝐿,𝐿0 ) and 𝑢1 ∈ M(𝛾0 , 𝛾+ ; D𝜏0 ) for some 𝛾0 ∈ O(𝐻 𝐿,𝐿 ).

ASTÉRISQUE 426

3.3. WEAKLY FILTERED STRUCTURE ON FUKAYA CATEGORIES

61

3) The same as 2) only that the splitting occurs in reverse order, namely first an element of M(𝛾, 𝛾0; D𝜏0 ) followed by an element of M∗ (𝛾0 , 𝛾+ ; D𝐿,𝐿0 ). 4) Elements corresponding to 𝜏 = 1. These are 𝑢 : ℝ ×[0, 1] → 𝑀 that belong to the space M0 (𝛾, 𝛾+ ; D𝐿,𝐿0 ) or in other words elements of the 0-dimensional component of the space M(𝛾, 𝛾+ ; D𝐿,𝐿0 ) of parametrized Floer trajectories connecting 𝛾 to 𝛾+ . The latter space has a 0-dimensional component if and only if 𝛾 = 𝛾+ in which case that component contains only the stationary trajectory at 𝛾. Summing up, this type of boundary point occurs if and only if 𝛾 = 𝛾+ and 𝑢 is the stationary solution at 𝛾. Summing up over all these four possibilities (for every given area of solutions 𝑢) yields formula (3.19). Note that the first term (i.e. the summand 𝑥) on the right-hand side of (3.19) comes exactly from the boundary points of type 4). To conclude the proof we only need to estimate the shift in action (or filtration) of the chain homotopy Φ. This is done in a similar way to the argument used above to estimate (𝑝), namely by using an energy-area identity as in (3.14). Indeed we can choose the perturbation data D𝑆 and D𝜏 , 0 < 𝜏 < 1, to be of the same size 0 order (in the 𝐶 1 -norm) as the Hamiltonian 𝐻 𝐿,𝐿 , hence the curvature term in the 0 0 energy-area identity can be bounded by a constant 𝐶(𝐻 𝐿,𝐿 ) that depends on 𝐻 𝐿,𝐿 0 0 and such that 𝐶(𝐻 𝐿,𝐿 ) → 0 as 𝐻 𝐿,𝐿 → 0 in the 𝐶 1 -topology. By taking all the 0 Hamiltonians 𝐻 𝐿,𝐿 for all 𝐿, 𝐿0 ∈ C to be uniformly bounded in the 𝐶 1 -topology we obtain a uniform bound 𝜅(𝑝) on the action shift of the chain homotopy Φ that holds for all pairs 𝐿, 𝐿0 ∈ C and such that 𝜅(𝑝) → 0 as 𝑝 → 𝑝0 ∈ N. This shows that the Yoneda module L satisfies Assumption 𝑈𝑚 (𝜅(𝑝)). By taking 𝐿0 = 𝐿 it also follows immediately that 𝜇2 (𝑒 𝐿 , 𝑒 𝐿 ) = 𝑒 𝐿 + 𝜇1 (𝑐) for some chain 𝑐 with A(𝑐) ≤ 𝜅(𝑝), hence F𝑢𝑘( C; 𝑝) ∈ 𝑈 𝑒 (𝜅(𝑝)) (so we can actually take 𝜁(𝑝) = 𝜅(𝑝)).  3.3.1. Remark. — In some variants of Floer theory it is common to normalize the Hamiltonian functions involved in the definition of the Floer complexes. For example, when the ambient manifold is closed one often normalizes the Hamiltonian functions to have zero mean, and for open manifolds one requires the Hamiltonian functions to have compact support. This solves the ambiguity of adding constants to the Hamiltonian functions and consequently provides a “canonical” way to define action filtrations. This especially makes sense when one aims to construct invariants of Hamiltonian diffeomorphisms (or flows) by means of filtered Floer homology. See e.g. the theory of spectral numbers [Sch00], [Oh06], [Oh05a], [Oh05b], [EP03], see also [Vit92] for an earlier approach via generating functions. While we could have normalized the Hamiltonian functions in the Floer and perturbation data, we have opted not to do so. At first glance, this might seem to 0 have odd implications. For example, suppose that 𝑝1 , 𝑝2 ∈ 𝐸reg are two choices of perturbation data such that 𝑝2 is obtained by adding a (different) constant to each of the Hamiltonian functions (or forms) in the perturbation data from 𝑝1 . Clearly, the Fukaya categories F𝑢𝑘( C; 𝑝 1 ) and F𝑢𝑘( C; 𝑝 2 ) are precisely the same, but they have completely different (and generally unrelated) weakly filtered structures. Our justification for not imposing any normalization on the Hamiltonian functions is that their role is purely auxiliary, and moreover, ideally we would like to make

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them arbitrarily small. More specifically, the focus of our study is the collections of Lagrangians C and its Fukaya category, whereas the Hamiltonian functions in the Floer data serve only as perturbations whose sole purpose is technical, namely to set up the Floer theory so that it fits into a (infinite dimensional) Morse theoretic framework. In reality, we view the Hamiltonian perturbations as quantities that can be taken arbitrarily small and consider families of Fukaya categories parametrized by choices of perturbations that tend to 0. (See e.g. Proposition 3.1.) In fact, our theory would become simpler and cleaner if we could set up the Fukaya category without appealing to any perturbations at all. If this were possible (which means that all the Floer trajectories and polygons are unperturbed pseudoholomorphic curves) our Fukaya categories would be genuinely filtered rather than only “weakly filtered”. (See [FOOO09a], [FOOO09b] for a “perturbation-less” construction of an 𝐴∞ -algebra associated to a single Lagrangian.) Another point related to the matter of normalization is that when extending our theory to Lagrangian cobordisms (see Section 3.4) we are forced to work with noncompactly supported Hamiltonian perturbations. While one could have attempted a different sort of normalization in that case (suited for the class of non-compactly supported perturbations used for cobordisms), we will not do that for the very same reasons as those for not doing it for F𝑢𝑘( C; 𝑝).

3.4. Extending the theory to Lagrangian cobordisms Most of the theory developed in the previous subsections of Chapter 3 extends to Lagrangian cobordisms. We will briefly go over the main points here and refer the reader to [BC14], [BC13] for more details. Let (𝑀, 𝜔) be a symplectic manifold as at the beginning of Chapter 3. We fix a e := ℝ2 × 𝑀 endowed collection C of Lagrangians in 𝑀 as in Chapter 3. Consider 𝑀 with the split symplectic structure 𝜔 e := 𝜔ℝ2 ⊕ 𝜔, where 𝜔ℝ2 is the standard symplectic structure of ℝ2 . Fix a strip 𝐵 = [𝑎, 𝑏] × ℝ ⊂ ℝ2 in the plane. Consider the collection e C of all e that have the following (𝐿1 , . . . , 𝐿𝑟 ) in 𝑀 Lagrangian cobordisms 𝑉 : (𝐿01 , . . . , 𝐿0𝑠 ) additional properties. We assume that 𝑉 is cylindrical (with horizontal ends) outside of 𝐵 × 𝑀 and that all of its ends 𝐿0𝑖 , 𝐿 𝑗 are Lagrangian submanifolds from the collection C. Moreover, we assume that the ends of 𝑉 are all located along horizontal ends whose 𝑦-coordinates are in ℤ. Finally, we further assume that 𝑉 is weakly exact as a e Lagrangian submanifold of 𝑀. The Lagrangians (𝐿01 , . . . , 𝐿0𝑠 ) are referred to as the positive ends and (𝐿1 , . . . , 𝐿𝑟 ) are the negative ends. Note that the values of 𝑟 and 𝑠 are allowed to vary arbitrarily. We also allow 𝑠 or 𝑟 to be 0 in which case 𝑉 is a null cobordism, i.e. a cobordism with only negative ends (if 𝑠 = 0) or only positive ends (if 𝑟 = 0). The case 𝑟 = 𝑠 = 0 means e that 𝑉 is a closed Lagrangian submanifold of 𝑀. One can associate a Fukaya category F𝑢𝑘 cob ( e C) to the collection e C. This is an 𝐴∞ category (or rather a family of such categories, depending on auxiliary choices) whose

ASTÉRISQUE 426

63

3.4. EXTENDING THE THEORY TO LAGRANGIAN COBORDISMS

objects are the elements of e C. The precise construction is detailed in [BC14]. The main ingredients in the construction are completely analogous to the case F𝑢𝑘( C), the main 0 e𝑉 ,𝑉 0 , e differences being the following. The Floer datum D𝑉 ,𝑉 0 = ( 𝐻 𝐽 𝑉 ,𝑉 ) of a pair of cobordisms 𝑉 , 𝑉 0 has a special form at infinity. Namely, there is a compact subset 𝐶𝑉 ,𝑉 0 ⊂ 𝐵 × 𝑀 such that outside of 𝐶𝑉 ,𝑉 0 we have

e𝑉 ,𝑉 (𝑡, 𝑧, 𝑝) = ℎ(𝑧) + 𝐻 𝑉 ,𝑉 (𝑡, 𝑝), 𝐻 0

0

where 𝑧 ∈ ℝ2 , 𝑝 ∈ 𝑀, 𝐻 𝑉 ,𝑉 : [0, 1] × 𝑀 → ℝ is a Hamiltonian function on 𝑀 and ℎ : ℝ2 → ℝ is the so called profile function whose purpose is to generate a Hamiltonian perturbation at infinity which disjoins 𝑉 0 from 𝑉 at infinity while keeping both of them cylindrical and horizontal at infinity. Note that the profile function ℎ is not (and in fact cannot be) compactly supported. We use the same function ℎ for the perturbation data of all pairs of Lagrangians 𝑉 , 𝑉0 ∈ e C. We remark also that ℎ can be taken to be arbitrarily small in the 𝐶 1 topology. Precise details on the construction of ℎ can be found in [BC14, Section 3]. 0 The almost complex structures e 𝐽 𝑉 ,𝑉 appearing in the Floer data have also a special form whose purpose is to retain compactness of the space of Floer trajectories. We will not repeat its definition here, since its particular form does not have any significance to the weakly filtered structure on F𝑢𝑘 cob ( e C) that we want to achieve. The only relevant thing is that with this choice of Floer data, there is a compact subset 𝐶𝑉0 ,𝑉 0 ⊂ 𝐵 × 𝑀 0

e𝑉 ,𝑉 ) lie inside 𝐶𝑉 ,𝑉 0 and moreover all Floer trajectories for such that all orbits O( 𝐻 0 the pair (𝑉 , 𝑉 ) lie inside 𝐶𝑉 ,𝑉 0 . The perturbation data used for the definition of F𝑢𝑘 cob ( e C) are analogous to those used for F𝑢𝑘( C) with the following differences. For a given tuple V = (𝑉0 , . . . , 𝑉𝑑 ) e V, e with 𝑉𝑗 ∈ e C the perturbation data D V = ( 𝐾 𝐽 V) is chosen so that 0

(3.20)

eV 𝐾

𝑆𝑟

e V, = ℎ · 𝑑𝑎 𝑟 + 𝐾 0

where 𝑎 𝑟 : 𝑆𝑟 → [0, 1] are the so called transition functions which depend smoothly on 𝑟 ∈ R𝑑+1 . See [BC14, Section 3.1] for their precise definition. The 1-form

e V ∈ Ω1 𝑆𝑟 , 𝐶 ∞ ( 𝑀) e 𝐾 0




is chosen so that it has the following two additional properties. e V satisfies condition (3.7). The first one is that 𝐾 0 The second one is that there is a compact subset 𝐶 V ⊂ 𝐵 × 𝑀 that contains all the subsets 𝐶𝑉0 𝑖 ,𝑉𝑗 (mentioned earlier) such that the Hamiltonian vector fields 𝑋 𝐾0 (𝜉), e V(𝜉) : 𝑀 e → ℝ are vertical for all 𝑟 ∈ R𝑑+1 and 𝜉 ∈ 𝑇(𝑆𝑟 ) generated by the function 𝐾 0 outside of 𝐶 V. By “vertical” we mean that eV

𝐷𝜋(𝑋 𝐾0

e V(𝜉)

) = 0,

e → ℝ2 is the projection. Note that due to the ℎ · 𝑑𝑎 𝑟 term in the where 𝜋 : 𝑀 e V this form does not satisfy condition (3.7). However, this will perturbation form 𝐾 not play any role for the purposes of establishing a weakly filtered 𝐴∞ -category.

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The almost complex structures e 𝐽 V from D V are also chosen to have restricted form, similarly to the ones appearing in the Floer data. We refer the reader to [BC14, Section 3.2] for the details. With these choices made it can be proved that there exists e such that for all (𝑟, 𝑢) ∈ M( e a compact subset 𝐶 V ⊂ 𝑀 𝛾1 , . . . , e 𝛾𝑑 ; D V) we have image (𝑢) ⊂ 𝐶 0V. See [BC14, Section 3.3] and in particular Lemma 3.3.2 there. Of course, apart from the above the perturbation data D V are assumed to be consistent and also compatible with the Floer data along the strip-like ends of the 𝑆𝑟 ’s. With these choices made we can define the 𝐴∞ -category F𝑢𝑘 cob ( e C) by the same recipe as in the previous sections of Chapter 3, in particular by formula (3.9). This 𝐴∞ -category is ℎ-unital, and a choice of homology units 𝑒𝑉 ∈ CF(𝑉 , 𝑉; D𝑉 ,𝑉 ) can be constructed by the same recipe as in Section 3.1 (see also [BC14, Remark 3.5.1] for an alternative approach). Similarly to F𝑢𝑘( C) our category F𝑢𝑘 cob ( e C) depends on the various choices made, namely a choice of strip-like ends and perturbation data. Note that part of the choices made for the perturbation data is the choice of a profile function and the choice of transition functions. We now fix the same choice of strip-like ends as for F𝑢𝑘( C) and denote the space e We denote the subspace of regular choices of choices of perturbation data by 𝐸. e ereg we denote by F𝑢𝑘cob ( e of perturbation data by 𝐸reg . For e 𝑝 ∈ 𝐸 C; e 𝑝 ) the category corresponding to e 𝑝. Next we endow F𝑢𝑘 cob ( e C; e 𝑝 ) with a weakly filtered structure. This is done in precisely the same way as for F𝑢𝑘( C; 𝑝). More precisely we define the action filtration on the Floer complexes CF(𝑉0 , 𝑉1 ; D𝑉0 ,𝑉1 ) by the same recipe as in Section 3.3. With these filtrations fixed, we now have the following:

Proposition 3.2. — The statement of Proposition 3.1 holds for the 𝐴∞ -categories F𝑢𝑘 cob ( e C; e 𝑝 ),

0 ereg e 𝑝∈𝐸 ,

0 0 ereg ereg is defined in an analogous way as 𝐸reg where 𝐸 ⊂𝐸 ⊂ 𝐸reg (see page 56).

The proof of this Proposition is essentially the same as that of Proposition 3.2 with straightforward modifications related to the special form of the perturbation data e 𝑝. For technical reasons we will need in the following also enlargements of the categories F𝑢𝑘 cob ( e C; e 𝑝 ) which will be denoted F𝑢𝑘 cob ( e C1/2 ; e 𝑝 ). These are defined in e the same was as F𝑢𝑘 cob ( C; e 𝑝 ) only that the collection of objects e Cis extended to allow Lagrangian cobordisms 𝑉 with ends from Cbut these ends are now allowed to lie over horizontal rays with 𝑦-coordinate in 21 ℤ (rather than only ℤ). This larger collection of objects is denoted by e C1/2 . The perturbation data, the 𝐴∞ -operations as well as the weakly filtered structures are defined in an analogous way as for F𝑢𝑘 cob ( e C; e 𝑝 ). e We denote the space of choices of perturbation data for these categories by 𝐸1/2 0 0 ereg,1/2 . Similarly to 𝐸reg ereg and the space of regular such choices by 𝐸 and 𝐸 we also e0 ereg,1/2 . An obvious analogue of Proposition 3.2 continues ⊂ 𝐸 have the space 𝐸 reg,1/2 e0 to hold for the family of categories F𝑢𝑘 cob ( e C1/2 ; e 𝑝 ), e 𝑝∈𝐸 . reg,1/2

ASTÉRISQUE 426

65

3.5. THE MONOTONE CASE

The relation between F𝑢𝑘 cob ( e C) and F𝑢𝑘 cob ( e C1/2 ) is simple. Any regular choice e of perturbation data for F𝑢𝑘 cob ( C1/2 ) can be used, by restriction to smaller class of objects, for F𝑢𝑘 cob ( e C). Thus, with the right choices of perturbation data we obtain a full and faithful embedding F𝑢𝑘 cob ( e C) → F𝑢𝑘 cob ( e C1/2 ). We shall give now a more precise description of this.

e1/2 → 𝐸 e with There is an obvious restriction map 𝑟 : 𝐸 ereg,1/2 ) ⊂ 𝐸 ereg 𝑟( 𝐸

and

0 e0 ereg 𝑟(𝐸 )⊂𝐸 reg,1/2

e0 e (the space of perturbations with ) contains N and such that the closure of 𝑟(𝐸 reg,1/2 0 ereg perturbation form 0, similarly to N on page 56). We will replace from now on 𝐸 0 0 e e with 𝑟(𝐸reg,1/2 ) and continue to denote the latter by 𝐸reg . There is also a (non-unique) right inverse to 𝑟 which is an extension map e1/2 ) −→ 𝐸 e1/2 𝑗 : 𝑟( 𝐸 0 ) ⊂ 𝐸 e ⊂ 𝑁 e1/2 and such that 𝑗( 𝐸 ereg e0 with 𝑗( 𝑁) . The map 𝑗 induces an obvious reg,1/2 family of extension functors

(3.21)

0 ereg 𝒥 : F𝑢𝑘 cob ( C; e 𝑝 ) −→ F𝑢𝑘 cob e C1/2 ; 𝑗( e 𝑝) , e 𝑝∈𝐸 .




These are 𝐴∞ -functors which are full and faithful (on the chain level). Note also that these functors 𝒥 are filtered, i.e. they have discrepancy ≤ 0. 0 ) and continue to denote the latter e0 ereg From now on we replace 𝐸 with 𝑗( 𝐸 reg,1/2 e0 by 𝐸 . With these conventions made, the maps reg,1/2 𝑟

0 e0 ereg :𝐸 →𝐸 reg,1/2

e0 𝐸 reg,1/2

and

𝑗

0 ereg 𝐸

0 0 ereg ereg :𝐸 →𝐸

become bijections, inverse one to the other. Therefore, whenever no confusion arises we omit 𝑗 and 𝑟 from the notation and denote 𝑗( e 𝑝 ) by e 𝑝 keeping in mind that e 𝑝 is e a regular choice of perturbation data for F𝑢𝑘 cob ( C) which admits an extension, still denoted by e 𝑝 , to a regular choice of perturbation data for F𝑢𝑘 cob ( e C1/2 ). e An important property of the extension map 𝑗 is the following. For every e 𝑝0 ∈ N we have F𝑢𝑘 cob ( e C1/2 ;𝑗( e 𝑝 ))

(3.22)

lim 𝑑

e 𝑝 →e 𝑝0

= 0,

for all 𝑑.

This follows easily from Proposition 3.2 together with the fact that 𝑗 is continuous, 0 ) contains N ereg e ⊂N e1/2 and that the closure of 𝑗( 𝐸 e1/2 . that 𝑗( N) 3.5. The monotone case The theory developed earlier in the paper continues to work in the more general setting of monotone Lagrangian submanifolds. We will assume henceforth all symplectic manifolds as well as Lagrangian submanifolds to be connected.

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Let (𝑀, 𝜔) be a symplectic manifold and 𝐿 ⊂ 𝑀 a Lagrangian submanifold. Recall that 𝐿 is called monotone if the following two conditions are satisfied: 1) There exists a constant 𝜌 > 0 such that 𝜔(𝐴) = 𝜌 · 𝜇(𝐴),

for all 𝐴 ∈ 𝐻2𝐷 (𝑀, 𝐿).

Here 𝐻2𝐷 (𝑀, 𝐿) ⊂ 𝐻2 (𝑀, 𝐿) is the image of the Hurewicz homomorphism 𝜋2 (𝑀, 𝐿) → 𝐻2 (𝑀, 𝐿) and 𝜇 is the Maslov index of 𝐿. 2) The minimal Maslov number 𝑁𝐿 of 𝐿, defined by 𝑁𝐿 := min 𝜇(𝐴) ; 𝐴 ∈ 𝐻2𝐷 (𝑀, 𝐿), 𝜇(𝐴) > 0





satisfies 𝑁𝐿 ≥ 2. (We use the convention that min ∅ = ∞.) A basic invariant of monotone Lagrangians 𝐿 is the Maslov-2 disk count, d 𝐿 ∈ Λ0 . This element is defined as d 𝐿 := 𝑑𝑇 𝑎 , where 𝑑 ∈ ℤ2 is the number of 𝐽-holomorphic disks (for generic 𝐽) of Maslov index 2 whose boundaries go through a given point in 𝐿, and 𝑎 = 2𝜌 > 0 is the area of each of these disks. Note that if there are no 𝐽-holomorphic disks of Maslov 2 at all then d 𝐿 = 0 by definition. It is well known that d 𝐿 is independent of the choices made in the definition (the almost complex structure 𝐽 and the point on 𝐿 through which we count the disks recall that 𝐿 is assumed to be connected). We refer the reader to [BC12, Section 2.5.1] for the precise definition of the coefficient 𝑑 in d 𝐿 and its properties. 8 In different forms this invariant has appeared in [Oh93], [Oh95], [Che97], [FOOO09a], [Aur07]. Under additional assumptions on 𝐿, one can define a version of this invariant also over other base rings (such as ℤ and ℂ) sometimes taking additional structures (like local systems) into account (see e.g. [Aur07], [BC12]), but we will not need that in the sequel. Fix an element d ∈ Λ0 of the form d = 𝑑𝑇 𝑎 , 𝑑 ∈ ℤ2 , 𝑎 > 0. Denote by L𝑎𝑔 mon,d (𝑀) the class of closed monotone Lagrangians 𝐿 ⊂ 𝑀 with d 𝐿 = d. Let C ⊂ L𝑎𝑔 mon,d (𝑀) be a collection of Lagrangians. Then one can define the Fukaya categories F𝑢𝑘( C; 𝑝) in the same way as described earlier and the theory developed above in Chapter 3 carries over without any modifications. (The main difference in the monotone case is that HF(𝐿, 𝐿) might not be isomorphic to 𝐻∗ (𝐿), and in fact may even vanish. This however will not affect any of our considerations. Apart from that, the monotone case poses some grading issues for Floer complexes, but in this paper we work in an ungraded framework.) Before we go on, we mention another basic measurement for monotone Lagrangians that will be relevant in the sequel. Given a monotone Lagrangian 𝐿 ⊂ 𝑀 define its minimal disk area 𝐴𝐿 by (3.23)

𝐴𝐿 = min 𝜔(𝐴) ; 𝐴 ∈ 𝐻2𝐷 (𝑀, 𝐿), 𝜔(𝐴) > 0 .





Turning to cobordisms, the theory continues to work if we restrict to monotone (𝐿1 , . . . , 𝐿𝑟 ) Lagrangian cobordisms 𝑉 ⊂ ℝ2 × 𝑀. Note that if 𝑉 : (𝐿01 , . . . , 𝐿0𝑠 ) 8. Note that the definition in that paper is done over ℤ so the 𝑑 above is obtained by reducing mod 2.

ASTÉRISQUE 426

3.6. INCLUSION FUNCTORS

67

is monotone then its ends 𝐿0𝑖 and 𝐿0𝑗 are automatically monotone too. Moreover, as observed by Chekanov [Che97], if 𝑉 is a monotone Lagrangian cobordism then one can define the Maslov-2 disk count d 𝑉 in the same way as above (i.e. for closed Lagrangian submanifolds) and d 𝑉 continues to be invariant of the choices made in its definition. Furthermore, if 𝑉 is connected then d 𝑉 = d 𝐿0𝑖 = d 𝐿 𝑗 ,

for all 𝑖, 𝑗.

Given d = 𝑑𝑇 𝑎 ∈ Λ0 and a collection C ⊂ L𝑎𝑔 mon,d (𝑀), denote by e C the collection of connected monotone Lagrangian cobordisms 𝑉 all of whose ends are in C. Note that by the preceding discussion each 𝑉 ∈ e C must have d 𝑉 = d. Therefore we omit d e from the notation of Cand C. This also keeps the notation consistent with the weakly exact case. From now, unless explicitly indicated, we treat uniformly both the weakly exact case as well as the monotone one. In particular the class of admissible Lagrangians will be denoted by L𝑎𝑔 ∗ (𝑀), where ∗ = we in the weakly exact case, and ∗ = (mon, d) in the monotone case. We will use similar notation L𝑎𝑔 ∗ (ℝ2 × 𝑀) for the admissible classes of cobordisms.

3.6. Inclusion functors Let 𝛾 ⊂ ℝ2 be an embedded curve with horizontal ends, i.e. 𝛾 is the image of a proper embedding ℝ ↩→ ℝ2 whose image outside of a compact set coincides with two horizontal rays having 𝑦-coordinates in 21 ℤ. In [BC14, Section 4.2] we associated to 𝛾 a family of mutually quasi-isomorphic 𝐴∞ -functors I𝛾 : F𝑢𝑘( C) −→ F𝑢𝑘 cob ( e C1/2 ) which we called inclusion functors. They all have the same action on objects which is given by I𝛾 (𝐿) = 𝛾 × 𝐿 for every 𝐿 ∈ C. Here is a more precise description of this family of functors. Denote by Hprof the space of profile functions (see Section 3.4, see also [BC14, Section 3] for the precise definition). The construction of the inclusion functors from [BC14] involves 0 (𝛾) ⊂ H the following ingredients. First, we restrict to a special subset Hprof prof which 1 contains arbitrarily 𝐶 -small profile functions. Apart from being profile functions, these functions ℎ : ℝ2 → ℝ have the following additional properties: 1) ℎ 𝛾 is a Morse function with an odd number of critical points 𝑂 1 , . . . , 𝑂 𝑙 ∈ 𝛾, where 5 ≤ 𝑙 = odd. Moreover, in a small Darboux-Weinstein neighborhood of 𝛾, ℎ is constant along each cotangent fiber. Thus 𝜙𝑡ℎ (𝛾) ∩ 𝛾 = {𝑂 1 , . . . , 𝑂 𝑙 } for every 𝑡. 2) The image, (𝜙1ℎ )−1 (𝛾), of 𝛾 under the inverse of the time-1 map of the Hamiltonian diffeomorphism generated by ℎ is as depicted in Figure 1.

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Figure 1. The curves 𝛾 and (𝜙1ℎ )−1 (𝛾).

We refer the reader to [BC14, Section 4] for more details. In that paper such functions were called extended profile functions. The word “extended” indicates that these functions are adapted to cobordisms with ends along rays having 𝑦-coordinates in 21 ℤ rather than just ℤ. Next, there is a map 0 0 e0 𝜄 𝛾 : 𝐸reg × Hprof (𝛾) −→ 𝐸 , reg,1/2

(3.24) and an 𝐴∞ -functor

I𝛾;𝑝,ℎ : F𝑢𝑘( C; 𝑝) → F𝑢𝑘 cob ( e C1/2 ; 𝜄 𝛾 (𝑝, ℎ)),

(3.25)

0 × H0 (𝛾) such that for all (𝑝, ℎ) the following holds: defined for every (𝑝, ℎ) ∈ 𝐸reg prof 1) For 𝐿0 , 𝐿1 ∈ C, let ⊲ D𝐿0 ,𝐿1 = (𝐻 𝐿0 ,𝐿1 , 𝐽 𝐿0 ,𝐿1 ) be the Floer datum of (𝐿0 , 𝐿1 ) prescribed by 𝑝 and ⊲ D𝛾×𝐿0 ,𝛾×𝐿1 = (𝐻 𝛾×𝐿0 ,𝛾×𝐿1 , 𝐽 𝛾×𝐿0 ,𝛾×𝐿1 ) the one prescribed by 𝜄 𝛾 (𝑝, ℎ).

Let 1 ≤ 𝑗 = odd ≤ 𝑙. Then for a small neighborhood U𝑗 of 𝑂 𝑗 we have 𝐻 𝛾×𝐿0 ,𝛾×𝐿1 (𝑧, 𝑚) = ℎ(𝑧) + 𝐻 𝐿0 ,𝐿1 (𝑚) for all (𝑧, 𝑚) ∈ U𝑗 × 𝑀. Moreover, we have 𝑂 𝑗 × 𝑥 ∈ O(𝐻 𝛾×𝐿0 ,𝛾×𝐿1 ) for every orbit 𝑥 ∈ O(𝐻 𝐿0 ,𝐿1 ) and 1 ≤ 𝑗 = odd ≤ 𝑙. In the following we will denote 𝑥 (𝑗) := 𝑂 𝑗 × 𝑥. Furthermore, we may assume that these are all the orbits in O(𝐻 𝛾×𝐿0 ,𝛾×𝐿1 ), i.e. O(𝐻 𝛾×𝐿0 ,𝛾×𝐿1 ) =

Ð

𝑗

𝑂 𝑗 × O(𝐻 𝛾×𝐿0 ,𝛾×𝐿1 ) ,




where the union runs over all 1 ≤ 𝑗 = odd ≤ 𝑙. 2) I𝛾;𝑝,ℎ (𝐿) = 𝛾 × 𝐿 for every 𝐿 ∈ C. 3) The first order term (I𝛾;𝑝,ℎ )1 is the chain map (3.26)

(I𝛾;𝑝,ℎ )1 : CF(𝐿0 , 𝐿1 ; D𝐿0 ,𝐿1 ) −→ CF(𝛾 × 𝐿0 , 𝛾 × 𝐿1 ; D𝛾×𝐿0 ,𝛾×𝐿1 ),

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defined by the formula (I𝛾;𝑝,ℎ )1 (𝑥) = 𝑥 (1) + 𝑥 (3) + · · · + 𝑥 (𝑙) , for all 𝑥 ∈ O(𝐻𝐿0 ,𝐿1 ). 4) The higher terms of I𝛾;𝑝,ℎ vanish: (I𝛾;𝑝,ℎ )𝑑 = 0 for every 𝑑 ≥ 2. 5) The homological functor associated to I𝛾;𝑝,ℎ is full and faithful.

e1/2 as ℎ → 0, 𝑝 → 𝑝0 . (The limits here 6) For every 𝑝0 ∈ N we have lim 𝜄 𝛾 (𝑝, ℎ) ∈ N 1 are in the 𝐶 -topology.) 7) Let 𝑝0 ∈ N. The weakly filtered 𝐴∞ -categories F𝑢𝑘 cob ( e C1/2 ; 𝜄(𝑝, ℎ)) have disF𝑢𝑘 cob ( e C

;𝜄(𝑝,ℎ))

1/2 crepancy ≤ F𝑢𝑘cob ( C1/2 ;𝜄(𝑝,ℎ)) , where for every 𝑑, lim 𝜖 𝑑 = 0 as 𝑝 → 𝑝0 and ℎ → 0. (The limits here are in the 𝐶 1 -topology.) 8) In case the ends of 𝛾 are along rays with 𝑦-coordinates in ℤ the map 𝜄 𝛾 and 0 and F𝑢𝑘 ereg e 𝑝 ) respectively. functors I𝛾;𝑝,ℎ can be assumed to have values in 𝐸 cob ( C; e More precisely, the map 𝜄 𝛾 factors as a composition

e

𝜄0𝛾

𝑗

0 0 0 ereg e0 𝐸reg × Hprof (𝛾) −−→ 𝐸 −→ 𝐸 reg,1/2

and the functors I𝛾;𝑝,ℎ factor as the composition of the following two 𝐴∞ -functors: 0 I𝛾;𝑝,ℎ

(3.27)

𝒥

F𝑢𝑘( C; 𝑝) −−−−−→ F𝑢𝑘 cob ( e C; 𝜄0𝛾 (𝑝, ℎ)) −−→ F𝑢𝑘 cob ( e C1/2 ; 𝜄 𝛾 (𝑝, ℎ)).

0 The map 𝜄0𝛾 and the 𝐴∞ -functor I𝛾;𝑝,ℎ have the same properties as described above for 𝜄 𝛾 and I𝛾;𝑝,ℎ respectively, with obvious modifications). The map 𝑗 and functor 𝒥 are the ones introduced in (3.21). We refer the reader to [BC14, Section 4.2] for a more detailed construction of these functors.

3.6.1. Additional properties relative to a given cobordism. — Suppose we fix in advance a Lagrangian cobordism 𝑊 ∈ e C with the following properties. Let 𝐾 1 , . . . , 𝐾 𝑟 ∈ C be the negative ends of 𝑊. Let 𝜋 : ℝ2 × 𝑀 → ℝ2 be the projection. Assume that 𝛾 intersects 𝜋(𝑊) only along the projection of the horizontal cylindrical negative part of 𝑊 (corresponding to its negative ends) with one intersection point corresponding to each end. Assume further that the intersection of 𝛾 and 𝜋(𝑊) is transverse and denote the intersection points by 𝑄 1 , . . . , 𝑄 𝑟 ∈ ℝ2 , where 𝑄 𝑗 corresponds to the 𝑗 0th negative end of 𝑊. Then we can restrict to profile functions ℎ that have 𝑂 2𝑗+1 = 𝑄 𝑗 0 for every 1 ≤ 𝑗 ≤ 𝑘 and redefine the spaces Hprof and Hprof by adding this restriction 0 to their definitions. For simplicity, we will continue to denote these spaces by Hprof and Hprof . Now, in addition to the previous list of properties, the map 𝜄 𝛾 can be assumed to have also the following property: let 𝐿 ∈ C be a Lagrangian, and denote by ⊲ D𝐿,𝐾 𝑗 = (𝐻 𝐿,𝐾 𝑗 , 𝐽 𝐿,𝐾 𝑗 ) the Floer datum of (𝐿, 𝐾 𝑗 ) prescribed by 𝑝, ⊲ D𝛾×𝐿,𝑉 = (𝐻 𝛾×𝐿,𝑉 , 𝐽 𝛾×𝐿,𝑉 ) the Floer datum of (𝛾 × 𝐿, 𝑉) prescribed by 𝜄 𝛾 (𝑝, ℎ). Then we may assume that for small neighborhoods U𝑗 of 𝑄 𝑗 we have 𝐻 𝛾×𝐿,𝑉 (𝑧, 𝑚) = ℎ(𝑧) + 𝐻 𝐿,𝐾 𝑗 (𝑚)

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for every (𝑧, 𝑚) ∈ U𝑗 × 𝑀. Moreover, we may assume that O(𝐻 𝛾×𝐿,𝑉 ) =

Ð𝑘

𝑗=1

𝑄 𝑗 × O(𝐻 𝐿,𝐾 𝑗 ) .




3.6.2. The weakly filtered structure of the inclusion functors. — The next proposition shows that the inclusion functors are weakly filtered and gives more information on their discrepancies. 0 × H0 (𝛾), has the Proposition 3.3. — The family of 𝐴∞ -functors I𝛾;𝑝,ℎ , (𝑝, ℎ) ∈ 𝐸reg prof

following properties: (i) I𝛾;𝑝,ℎ is weakly filtered (see Section 2.3 for the definition). I𝛾;𝑝,ℎ

(ii) I𝛾;𝑝,ℎ has discrepancy ≤ I𝛾;𝑝,ℎ , where 𝜖 𝑑 I𝛾;𝑝,ℎ 𝜖1

= 0 for every 𝑑 ≥ 2 and

≤ max ℎ(𝑂 𝑘 ) ; 1 ≤ 𝑘 = odd ≤ 𝑙 .





I𝛾;𝑝,ℎ

Note that 𝜖 1 → 0 as ℎ → 0 in the 𝐶 0 -topology. (iii) I𝛾;𝑝,ℎ is homologically unital. 0 (iv) For every 𝐿 ∈ C denote by 𝑒 𝛾×𝐿 = (I𝛾;𝑝,ℎ )1 (𝑒 𝐿 ) ∈ CF(𝛾 × 𝐿, 𝛾 × 𝐿; D𝛾×𝐿,𝛾×𝐿 ) the image of the homology unit 𝑒 𝐿 ∈ CF(𝐿, 𝐿; D𝐿,𝐿 ) under the functor I𝛾;𝑝,ℎ . The collection of 0 elements {𝑒 𝛾×𝐿 } 𝐿∈ C can be extended to a collection of homology units

e E = {𝑒𝑉0 }𝑉 ∈ eC for F𝑢𝑘 cob ( e C1/2 , 𝜄 𝛾 (𝑝, ℎ)) with discrepancy ≤ e 𝑢 0(𝑝, ℎ), where e 𝑢 0(𝑝, ℎ) → 0 as 𝑝 → 𝑝0 ∈ N 1 and ℎ → 0 in the 𝐶 -topologies. (v) With respect to the collection of homology units e E above, F𝑢𝑘 cob ( e C1/2 ; 𝜄 𝛾 (𝑝, ℎ)) 𝑒 belongs to 𝑈 (e 𝜁(𝑝, ℎ)), where e 𝜁(𝑝, ℎ) → 0 as 𝑝 → 𝑝0 ∈ N and ℎ → 0 in the 𝐶 1 -topologies. (vi) Let V be the Yoneda module of 𝑉 ∈ e C1/2 . Then, with respect to the collection of e homology units E above we have V ∈ 𝑈𝑚 e 𝜅 (𝑝, ℎ) ,




where e 𝜅(𝑝, ℎ) → 0 as 𝑝 → 𝑝0 ∈ N and ℎ → 0 in the 𝐶 1 -topologies. In case the ends of 𝛾 have 𝑦-coordinates in ℤ an obvious analogue holds for the family of 0 functors I𝛾;𝑝,ℎ from (3.27). The proof of this proposition is straightforward, given the precise definition of the functors I𝛾;𝑝,ℎ which is described in detail in [BC14, Section 4.2]. 3.7. Weakly filtered iterated cones coming from cobordisms Let 𝑉 ∈ e C be a Lagrangian cobordism and denote by 𝐿0 , . . . , 𝐿𝑟 ∈ C its negative ends. (In contrast to Section 3.4 as well as [BC14], in this section we index the negative ends from 0 to 𝑟 rather than from 1 to 𝑟.) 0 and ℎ ∈ H0 (𝛾) be such Let 𝛾 ⊂ ℝ2 be the curve depicted in Figure 2. Let 𝑝 ∈ 𝐸reg prof that 𝑙 := #(𝜙1ℎ )−1 (𝛾) ∩ 𝛾 = 2𝑟 + 5.

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3.7. WEAKLY FILTERED ITERATED CONES COMING FROM COBORDISMS

Figure 2. The curves 𝛾, (𝜙1ℎ )−1 (𝛾) and the cobordism 𝑉.

Denote by Vthe Yoneda module of 𝑉, which we view here as an 𝐴∞ -module over the category F𝑢𝑘 cob ( e C1/2 ; 𝜄 𝛾 (𝑝, ℎ)). Consider now the pullback module ∗ M𝑉;𝛾,𝑝,ℎ := I𝛾;𝑝,ℎ V,

(3.28)

which is a F𝑢𝑘( C; 𝑝)-module. Since I𝛾;𝑝,ℎ is a weakly filtered functor the module M𝑉;𝛾,𝑝,ℎ is weakly filtered.

Proposition 3.4. — The weakly filtered module M𝑉;𝛾,𝑝,ℎ has the following properties. (i) For every 𝑁 ∈ C and 𝛼 ∈ ℝ we have ≤𝛼 M𝑉;𝛾,𝑝,ℎ (𝑁) = CF≤𝛼−ℎ(𝑂3 ) (𝑁 , 𝐿0 ; 𝑝) ⊕ CF≤𝛼−ℎ(𝑂5 ) (𝑁 , 𝐿1 ; 𝑝) ⊕ · · ·

· · · ⊕ CF≤𝛼−ℎ(𝑂2𝑟+3 ) (𝑁 , 𝐿𝑟 ; 𝑝), where the last equality is of Λ0 -modules (but not necessarily of chain complexes). Here CF(𝑁 , 𝐿 𝑖 ; 𝑝) stands for CF(𝑁 , 𝐿 𝑖 ; D𝑁 ,𝐿𝑖 ), where D𝑁 ,𝐿𝑖 is the Floer datum prescribed 0 . by 𝑝 ∈ 𝐸reg (ii) M𝑉;𝛾,𝑝,ℎ has discrepancy ≤ M𝑉;𝛾,𝑝,ℎ , where M𝑉;𝛾,𝑝,ℎ

(3.29)

𝜖𝑑

F𝑢𝑘 cob ( e C1/2 ;𝜄 𝛾 (𝑝,ℎ))

≤ (𝑑 − 1) max ℎ(𝑂 𝑘 ) ; 1 ≤ 𝑘 = odd ≤ 2𝑟 + 5 + 𝜖 𝑑





.

Proof. — The second statement follows immediately from Proposition 3.3 and Lemma 2.2 together with the fact that the higher terms of I𝛾;𝑝,ℎ vanish. The first statement can be verified by a straightforward calculation.  3.7.1. Remark. — An inspection of the arguments from [BC14, Section 4.4] shows 𝑉;𝛾,𝑝,ℎ in (3.29) can be slightly improved by that the estimate for the discrepancy 𝜖 M 𝑑 replacing the “max” term from (3.29) with max{ℎ(𝑂 𝑘 ) ; 3 ≤ 𝑘 = odd ≤ 2𝑟 + 3}. We

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will not go into details on that since this improvement will not play any role in our applications. Recall from [BC14, Section 4.4] that the module M𝑉;𝛾,𝑝,ℎ is naturally isomorphic to an iterated cone with attaching objects corresponding to the ends 𝐿0 , . . . , 𝐿𝑟 of 𝑉. More precisely, denote by L𝑗 the Yoneda module corresponding to 𝐿 𝑗 . Then 𝜙𝑟

𝜙 𝑟−1

M𝑉;𝛾,𝑝,ℎ  C𝑜𝑛𝑒 L𝑟 −−→ C𝑜𝑛𝑒 L𝑟−1 −−−−→ C𝑜𝑛𝑒 · · · 𝜙2

𝜙1








· · · C𝑜𝑛𝑒 L2 −−→ C𝑜𝑛𝑒 L1 −−→ L0 ···

,

where 𝜙 𝑗 is a module homomorphism between L𝑗 and the intermediate iterated cone involving the attachment of only the first 𝑗 + 1 objects L0 , . . . , L𝑗 . As we will see shortly, the module homomorphisms 𝜙 𝑗 are weakly filtered (and obviously the L𝑖 ’s too) and consequently the iterated cone M𝑉;𝛾,𝑝,ℎ can be endowed with a weakly filtered structure by the algebraic recipe of Sections 2.4 and 2.6. At the same time, we have just seen that M𝑉;𝛾,𝑝,ℎ has another weakly filtered structure as it is the pull back module by an inclusion functor, as described in Proposition 3.4. Our goal now is to compare these two weakly filtered structures and show that they are essentially the same. Consider the following collection of curves 𝛾1 , . . . , 𝛾𝑟 ⊂ ℝ2 with horizontal ends, as depicted in Figure 3 next page. We assume that 𝛾𝑟 = 𝛾, the curve involved in the definition of M𝑉;𝛾,𝑝,ℎ . 0 (𝛾 ) and such We also choose profile functions ℎ1 , . . . , ℎ 𝑟 : ℝ2 → ℝ with ℎ 𝑗 ∈ Hprof 𝑗 that the following holds (see Figure 3): 1) ℎ 𝑟 = ℎ. 2) (𝜙1ℎ )−1 (𝛾) ∩ 𝛾 = {𝑂 1 , . . . , 𝑂 2𝑟+5 } ℎ𝑗

𝑗

𝑗

𝑗

3) (𝜙1 )−1 (𝛾 𝑗 ) ∩ 𝛾 𝑗 = {𝑂 1 , . . . , 𝑂 2𝑗+5 }, where 𝑂 𝑘 = 𝑂 𝑘 for all 1 ≤ 𝑘 ≤ 2𝑗 + 3. Thus 𝑗 𝑗 only the last two intersection points 𝑂 2𝑗+4 , 𝑂2𝑗+5 do not belong to the 𝛾𝑙 ’s for 𝑙 > 𝑗. 1 }, where 𝑦2𝑗+3 is the 4) ℎ 𝑗 coincides with ℎ over the half-plane {𝑦 ≤ 𝑦2𝑗+3 + 100 𝑦-coordinate of 𝑂 2𝑗+3 . We denote the space of all tuples of profile functions (ℎ1 , . . . , ℎ 𝑟 ) satisfying these 0 (𝛾 , . . . , 𝛾 ) and denote elements of this space by 𝜐 = (ℎ , . . . , ℎ ). conditions by Hprof 1 𝑟 1 𝑟

With this notation, it is possible as in (3.24) to choose maps 0 0 e0 𝜄 𝛾𝑗 : 𝐸reg × Hprof (𝛾 𝑗 ) −→ 𝐸 , reg,1/2

𝑗 = 1, . . . , 𝑟,

0 × H0 (𝛾 , . . . , 𝛾 ) the choice of data satisfying the following. For every (𝑝, 𝜐) ∈ 𝐸reg 𝑟 prof 1 0 e 𝜄 𝛾𝑗 (𝑝, ℎ 𝑗 ) ∈ 𝐸 has the properties listed for 𝜄 𝛾 (𝑝, ℎ) on page 68 but with 𝛾 replaced reg,1/2

e1/2 as by 𝛾 𝑗 and ℎ by ℎ 𝑗 . (Consequently, for every e 𝑝0 ∈ N we have lim 𝜄 𝛾𝑗 (𝑝, ℎ 𝑗 ) ∈ N 𝑝 → 𝑝0 and 𝜐 = (ℎ1 , . . . , ℎ 𝑟 ) → (0, . . . , 0).) Moreover, we require that for every 𝑗 0 , 𝜐 = (ℎ , . . . , ℎ ) ∈ H0 (𝛾 , . . . , 𝛾 ) the data prescribed by 𝜄 (𝑝, ℎ ) is and 𝑝 ∈ 𝐸reg 1 𝑟 𝑟 𝛾𝑗 𝑗 prof 1 compatible with that prescribed by 𝜄 𝛾𝑗−1 (𝑝, ℎ 𝑗−1 ) (in the obvious sense, similar to ℎ 𝑗 being compatible with ℎ 𝑗−1 ). By Section 3.6, the curves 𝛾 𝑗 and the maps 𝜄 𝛾𝑗 induce a

ASTÉRISQUE 426

3.7. WEAKLY FILTERED ITERATED CONES COMING FROM COBORDISMS

73

Figure 3. A closer look at the curves 𝛾, 𝛾 𝑗−1 , and 𝛾 𝑗 near the (𝑗 − 1)-th and 𝑗-th ends of 𝑉.

family of inclusion functors I𝛾𝑗 ;𝑝,ℎ 𝑗 : F𝑢𝑘( C; 𝑝) −→ F𝑢𝑘 e C1/2 ; 𝜄 𝛾𝑗 (𝑝, ℎ 𝑗 ) ,




0 , 𝜐 = (ℎ , . . . , ℎ ) ∈ H0 (𝛾 , . . . , 𝛾 ). We will use below the parametrized by 𝑝 ∈ 𝐸reg 1 𝑟 𝑟 prof 1 notation I𝛾𝑗 ;𝑝,𝜐 := I𝛾𝑗 ;𝑝,ℎ 𝑗 ,

where ℎ 𝑗 is the 𝑗-th entry in the tuple 𝜐 since it reflects better the parameters (𝑝, 𝜐) parametrizing this family of functors. We will also write 𝜄 𝛾𝑗 (𝑝, 𝜐) for 𝜄 𝛾𝑗 (𝑝, ℎ 𝑗 ) sometimes. Consider now the pullback F𝑢𝑘( C; 𝑝)-modules M𝑉;𝛾𝑗 ,𝑝,𝜐 = I𝛾∗𝑗 ;𝑝,𝜐 V,

(3.30)

𝑗 = 1, . . . , 𝑟.

We endow each of these modules with its weakly filtered structure as defined at the beginning of Section 3.7 and further described by Proposition 3.4 (where 𝑙 = 2𝑗 + 5, and 𝛾 should be replaced by 𝛾 𝑗 and ℎ by ℎ 𝑗 ). Next, for every 0 ≤ 𝑗 ≤ 𝑟 denote by L𝑗 the Yoneda module associated to 𝐿 𝑗 , endowed with its weakly filtered structure induced from F𝑢𝑘( C; 𝑝). Finally, recall that for a weakly filtered module Mand 𝜈 ∈ ℝ, 𝑆 𝜈M stands for the weakly filtered module obtained from M by an action-shift of 𝜈 (see § 2.3.4). 0 × H0 (𝛾 , . . . , 𝛾 ) there exist weakly filtered Proposition 3.5. — For every (𝑝, 𝜐) ∈ 𝐸reg 𝑟 prof 1

module homomorphisms 𝜙1 : L1 −→ L0

and

𝜙 𝑗 : L𝑗 −→ 𝑆 ℎ(𝑂3 ) M𝑉;𝛾𝑗−1 ,𝑝,𝜐

(𝑗 = 2, . . . , 𝑟)

such that the following holds for every 1 ≤ 𝑗 ≤ 𝑟: (i) 𝜙 𝑗 shifts action by ≤ 0.

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(ii) The discrepancy of 𝜙 𝑗 is ≤ δ 𝜙 𝑗 , where (3.31)

F𝑢𝑘 cob ( e C1/2 ;𝜄 𝛾 𝑗 (𝑝,𝜐))

𝜙𝑗

𝛿 𝑑 := (𝑑 − 1) max ℎ(𝑂 𝑘 ) + 𝜖 𝑑

+ ℎ(𝑂 2𝑗+3 ) − ℎ(𝑂 3 ).

1≤𝑘≤2𝑗+3 𝑘 odd

(iii) For every 1 ≤ 𝑗 ≤ 𝑟, 𝑆 ℎ(𝑂3 ) M𝑉;𝛾𝑗 ,𝑝,𝜐 = C𝑜𝑛𝑒(𝜙 𝑗 ; 0, δ 𝜙 𝑗 ) as weakly filtered module. (See Section 2.4 for our conventions for weakly filtered cones.) In other words, the weakly filtered module 𝑆 ℎ(𝑂3 ) M𝑉;𝛾𝑗 ,𝑝,𝜐 coincides with the weakly filtered mapping cone over 𝜙 𝑗 . Recalling that M𝑉;𝛾,𝑝,ℎ = M𝑉;𝛾𝑟 ,𝑝,𝜐 , the above proposition implies that 𝜙𝑟

(3.32)

𝜙 𝑟−1

𝑆 ℎ(𝑂3 ) M𝑉;𝛾,𝑝,ℎ = C𝑜𝑛𝑒(L𝑟 −−→ C𝑜𝑛𝑒(L𝑟−1 −−−−→ C𝑜𝑛𝑒(· · · 𝜙2

𝜙1

· · · C𝑜𝑛𝑒(L2 −−→ C𝑜𝑛𝑒(L1 −−→ L0 ))···))), where 𝜙 𝑗 := (𝜙 𝑗 ; 0, δ 𝜙 𝑗 ) and the cones in (3.32) are endowed with the filtrations as defined in Section 2.4. In other words, up to a small action-shift, M𝑉;𝛾,𝑝,ℎ can be viewed as a weakly filtered iterated cone by the very same recipe described at the beginning of Section 2.6 (with 𝜌 𝑗 = 0 and K𝑗 = 𝑆 ℎ(𝑂3 ) M𝑉;𝛾𝑗 ,𝑝,𝜐 ). Consequently, in our geometric applications we can use Theorem 2.14 for K𝑟 = 𝑆 ℎ(𝑂3 )) M𝑉;𝛾,𝑝,ℎ . Proof of Proposition 3.5. — The proof is based on two main ingredients. The first one is the theory developed in [BC14, Sections 4.2, 4.4] from which it follows that, ignoring action-filtrations, we have M𝑉;𝛾𝑗 ,𝑝,𝜐 = C𝑜𝑛𝑒(L𝑗 → M𝑉;𝛾𝑗−1 ,𝑝,𝜐 ). The second one comprises direct action-filtration calculations for the modules M𝑉;𝛾𝑗 ,𝑝,𝜐 and the homomorphisms 𝜙 𝑗 . Before we go on, we should remark a notational difference between [BC14] and the present paper. In [BC14] the negative ends of the cobordism 𝑉 are indexed from 1 to 𝑟, whereas in the present text the indexing runs between 0 and 𝑟. This results in several other indexing differences between the two texts. For example, the curves 𝛾 𝑗 in the present text are the same as 𝛾 𝑗+1 in [BC14]. In the present text, the number of ℎ𝑗 intersection points between 𝜙1 (𝛾 𝑗 ) and 𝛾 𝑗 is 2𝑗 + 5, whereas in [BC14] this number is 2𝑗 + 3, etc. We start by adding to the collection of curves 𝛾1 , . . . , 𝛾𝑟 another curve 𝛾0 , defined in the same way as the 𝛾 𝑗 ’s only that it is adapted to the 𝐿0 -end of 𝑉 in the sense that the negative end of 𝛾0 goes above the 𝐿0 -end and below the 𝐿1 end. We also choose 0 (𝛾 ) satisfying the same conditions as the ℎ ’s (see page 72) only for 𝑗 = 0. ℎ0 ∈ Hprof 0 𝑗 We write  (𝜙1∗ ℎ0 )−1 (𝛾0 ) ∩ 𝛾0 = 𝑂 1 , 𝑂2 , 𝑂3 , 𝑂40 , 𝑂50 . To simplify the notation we also extend the tuple 𝜐 = (ℎ1 , . . . , ℎ 𝑟 ) to contain also ℎ0 and write 𝜐 = (ℎ 0 , . . . , ℎ 𝑟 ). As before we have an inclusion functor associated to 𝛾0 , 𝑝, ℎ 0 and we consider the pullback module M𝑉;𝛾0 ,𝑝,ℎ0 := I𝛾∗0 ;𝑝,ℎ0 V.

ASTÉRISQUE 426

75

3.7. WEAKLY FILTERED ITERATED CONES COMING FROM COBORDISMS

We will denote this module also by M𝑉;𝛾0 ,𝑝,𝜐 to be consistent with the previous notation. We first claim that there exist module homomorphisms 𝜙 𝑗 : L𝑗 → M𝑉;𝛾𝑗−1 ,𝑝,ℎ 𝑗−1 for all 1 ≤ 𝑗 ≤ 𝑟, such that 𝜙𝑗

M𝑉;𝛾𝑗 ,𝑝,ℎ 𝑗 = C𝑜𝑛𝑒 L𝑗 −−→ M𝑉;𝛾𝑗−1 ,𝑝,ℎ 𝑗−1 ,

(3.33)




where at the moment we ignore the action filtrations. This statement is not explicitly stated in [BC14, Section 4.4.2], but it follows easily from the arguments in that paper. More specifically, what is stated explicitly in [BC14, Section 4.4.2] is that there exists an exact triangle – in the derived category 𝐷F𝑢𝑘( C; 𝑝) – of the form L𝑗 −→ M𝑉;𝛾𝑗−1 ,𝑝,ℎ 𝑗−1 −→ M𝑉;𝛾𝑗 ,𝑝,ℎ 𝑗 . Here however, we claim a stronger statement, namely that (3.33) holds at the chain level. We will now explain how to deduce (3.33) from the theory developed in [BC14]. In doing that we will mostly follow the notation from that paper. By [BC14, Proposition 4.4.1] for every 0 ≤ 𝑗 ≤ 𝑟 we have the following: d1) 𝐴∞ -categories B𝑗 and B0𝑗 (depending on 𝛾 𝑗 , 𝑝 and ℎ 𝑗 ). Quasi-isomorphisms of 𝐴∞ -categories: 𝑒 𝑗 : F𝑢𝑘( C; 𝑝) → B𝑗 , 𝑝 𝑗 : B𝑗 → B0𝑗 , 𝜎 𝑗 : B0𝑗 → F𝑢𝑘( C; 𝑝) and 𝑞 𝑗 : B0𝑗 → B0𝑗−1 , for 𝑗 ≥ 1, all with vanishing higher order terms. Moreover, they satisfy: (3.34) 𝜎 𝑗 ◦ 𝑝 𝑗 ◦ 𝑒 𝑗 = id,

for all 𝑗 ≥ 0,

and

𝑞 𝑗 ◦ 𝑝 𝑗 ◦ 𝑒 𝑗 = 𝑝 𝑗−1 ◦ 𝑒 𝑗−1

for all 𝑗 ≥ 1.

A B𝑗 -module M𝑗 and a B0𝑗 -module M0𝑗 such that M𝑉;𝛾𝑗 ,𝑝,ℎ 𝑗 = 𝑒 ∗𝑗 M𝑗 ,

𝑝 ∗𝑗 M0𝑗 = M𝑗 , ∀𝑗 ≥ 0, 𝜑𝑗

(3.35)

M0𝑗 = C𝑜𝑛𝑒 𝜎∗𝑗 L𝑗 −−→ 𝑞 ∗𝑗 M0𝑗−1 ,




∀𝑗 ≥ 1,

for some module homomorphism 𝜑 𝑗 . (This homomorphism was denoted by 𝜙 𝑗 in [BC14, Proposition 4.4.1]. We have denoted it here by 𝜑 𝑗 since 𝜙 𝑗 is already used for a slightly different homomorphism.) d2) For 𝑗 = 0 we have M00 = 𝜎0∗ L0 . We now pull back the second line of (3.35) by the functor 𝑝 𝑗 ◦ 𝑒 𝑗 . The desired equality (3.33) now follows by using (3.34) together with the fact that 𝐴∞ -functors pull back mapping cones to mapping cones (at the chain level). Note that for 𝑗 = 0, pulling back the equality from point d2) above yields: M𝑉;𝛾0 ,𝑝,ℎ0 = L0 . We now turn to the weakly filtered setting. Throughout the rest of the proof it is useful to keep in mind that ℎ 𝑗 (𝑂 𝑘 ) = ℎ(𝑂 𝑘 ) for every 0 ≤ 𝑗 ≤ 𝑟 and 1 ≤ 𝑘 ≤ 2𝑗 + 3. We claim that in the weakly filtered setting the correct version of (3.33) is (𝜙 𝑗 ;0,δ

(3.36)

𝜙𝑗

)

𝑆 ℎ(𝑂3 ) M𝑉;𝛾𝑗 ,𝑝,𝜐 = C𝑜𝑛𝑒 L𝑗 −−−−−−−−→ 𝑆 ℎ(𝑂3 ) M𝑉;𝛾𝑗−1 ,𝑝,𝜐 , 𝑆

ℎ(𝑂3 )




for all 1 ≤ 𝑗 ≤ 𝑟,

M𝑉;𝛾0 ,𝑝,𝜐 = L0 .

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Of course, by Lemma 2.4, the first line of (3.36) is equivalent to: (𝜙 𝑗 ;0,δ

(3.37)

𝜙𝑗

+ℎ(𝑂3 ))

M𝑉;𝛾𝑗 ,𝑝,𝜐 = C𝑜𝑛𝑒 L𝑗 −−−−−−−−−−−−−→ M𝑉;𝛾𝑗−1 ,𝑝,𝜐 ,

for all 1 ≤ 𝑗 ≤ 𝑟.




To prove (3.37) one needs to go over the arguments in the proof of [BC14, Proposition 4.4.1] and take action-filtrations into consideration. An inspection of these arguments shows that the categories B𝑗 , B0𝑗 and functors 𝑒 𝑗 , 𝑝 𝑗 , 𝜎 𝑗 , 𝑞 𝑗 are all weakly filtered, and so are the modules M0𝑗 and M𝑗 . Moreover, we have: d1) The discrepancies of both B𝑗 and B0𝑗 are ≤ 

F𝑢𝑘 cob ( e C1/2 ;𝜄 𝛾 𝑗 (𝑝,ℎ 𝑗 ))

.

d2) Both functors 𝑝 𝑗 and 𝑞 𝑗 are filtered, i.e. have discrepancies ≤ 0. 𝑒𝑗

𝑗

d3) 𝑒 𝑗 has discrepancy ≤ 𝑒 𝑗 , where 𝜖 1 = max{ℎ 𝑗 (𝑂 𝑘 ) ; 1 ≤ 𝑘 = odd ≤ 2𝑗 + 5} 𝑒𝑗 and 𝜖 𝑑 = 0 for all 𝑑 ≥ 2. d4) 𝑝 𝑗 ◦ 𝑒 𝑗 has discrepancy ≤ 𝑝 𝑗 ◦𝑒 𝑗 , where 𝑝 𝑗 ◦𝑒 𝑗

𝜖1 𝑝 𝑗 ◦𝑒 𝑗

and 𝜖 𝑑

= max ℎ(𝑂 𝑘 ) ; 1 ≤ 𝑘 = odd ≤ 2𝑗 + 3





= 0 for all 𝑑 ≥ 2. 𝜎𝑗

𝜎𝑗

d5) 𝜎 𝑗 has discrepancy ≤ 𝜎 𝑗 , where 𝜖 1 = −ℎ(𝑂 2𝑗+3 ) and 𝜖 𝑑 = 0 for all 𝑑 ≥ 2. d6) The module homomorphism 𝜑 𝑗 : 𝜎∗𝑗 L𝑗 → 𝑞 ∗𝑗 M0𝑗−1 shifts action by ≤ 0 and has discrepancy ≤ 𝜑 𝑗 , where 𝜑𝑗

F𝑢𝑘 cob ( e C1/2 ;𝜄 𝛾 𝑗 (𝑝,𝜐))

𝜖𝑑 = 𝜖𝑑

+ ℎ(𝑂2𝑗+3 ).

d7) The modules M0𝑗 and M0𝑗 have discrepancies ≤ 

F𝑢𝑘 cob ( e C1/2 ;𝜄 𝛾 𝑗 (𝑝,ℎ 𝑗 ))

.

The equalities (or identifications) from (3.35) hold also in the weakly filtered sense, where the cone over 𝜑 𝑗 on the second line of (3.35) is now taken over (𝜑 𝑗 ; 0, 𝜑 𝑗 ). d8) M00 = 𝑆−ℎ(𝑂3 ) 𝜎0∗ L0 as weakly filtered modules. To conclude the proof of (3.37) we pull back the weakly filtered version of the second line of (3.35) by 𝑝 𝑗 ◦ 𝑒 𝑗 and use Lemmas 2.7, 2.2 and 2.3 (recall that 𝑝 𝑗 , 𝑒 𝑗 do not have higher order terms). The assertion that 𝑆 ℎ(𝑂3 ) M𝑉;𝛾0 ,𝑝,𝜐 = L0 follows in a similar way. 

ASTÉRISQUE 426

CHAPTER 4 QUASI-EXACT AND QUASI-MONOTONE COBORDISMS

For reasons that will become apparent when we introduce shadow metrics in Chapter 6 we need to extend some of the theory from Chapter 3, especially from Section 3.7, to the cases of quasi-exact and quasi-monotone cobordisms. Quasi-exact cobordisms form a larger class than the usual weakly-exact cobordisms considered earlier in the paper but, from the point of view of 𝐽-holomorphic machinery, they behave in the the same way except that only for particular classes of almost complex structures 𝐽. The same applies to quasi-monotone cobordisms versus monotone ones. 4.1. Quasi-exact cobordisms Fix a symplectic manifold (𝑀, 𝜔), as at the beginning of Section 3 and denote by L𝑎𝑔 we (𝑀) the class of weakly-exact Lagrangian submanifolds 𝐿 ⊂ 𝑀. As before, we write

e 𝜔 ( 𝑀, e) = (ℝ2 × 𝑀, 𝜔ℝ2 ⊕ 𝜔) e → ℝ2 the projection. and denote by 𝜋 : 𝑀 We begin with a simple definition that will be useful in the following.

Definition 4.1. — Let 𝑉 ⊂ ℝ2 × 𝑀 be a Lagrangian cobordism and let 𝐾𝑉 ⊂ ℝ2 be a subset with compact closure. We say that 𝑉 is cylindrical over ℝ2 \ 𝐾𝑉 if over ℝ2 \ 𝐾𝑉

the cobordism 𝑉 is equal to a disjoint union with terms 𝛾𝑘 × 𝐿 𝑘 where 𝛾𝑘 are pairwise disjoint, unbounded, connected, and embedded curves in the plane, horizontal at infinity, and 𝐿 𝑘 ⊂ 𝑀 are Lagrangians. Next, we introduce quasi-exact Lagrangian cobordisms (with weakly exact ends).

e be a Lagrangian cobordism with ends in L𝑎𝑔 we (𝑀). Definition 4.2. — Let 𝑉 ⊂ 𝑀 We say that 𝑉 is quasi-exact if there is a compact subset 𝐾𝑉 ⊂ ℝ2 and an 𝜔 e-compatible almost complex structure 𝐽𝑉 such that: 1) 𝑉 is cylindrical over ℝ2 \ Int (𝐾𝑉 ).

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2) 𝜋 is (𝐽𝑉 , 𝑖)-holomorphic over ℝ2 \ Int (𝐾𝑉 ).

e 𝑉). 3) There are no non-constant 𝐽𝑉 -holomorphic disks 𝑢 : (𝐷, 𝜕𝐷) → ( 𝑀, Sometimes we will say that (𝑉 , 𝐽𝑉 , 𝐾𝑉 ) is quasi-exact. A pair (𝐽𝑉 , 𝐾𝑉 ) as above will be called quasi-exact admissible for 𝑉. Sometimes the focus will be on the subset 𝐾𝑉 , and we will say that 𝐾𝑉 is quasi-exact admissible for 𝑉 if there exists 𝐽𝑉 such that (𝑉 , 𝐽𝑉 , 𝐾𝑉 ) is quasi-exact. We denote by L𝑎𝑔 𝑞,we (ℝ2 ×𝑀) the collection of quasi-exact Lagrangian cobordisms 𝑉 ⊂ ℝ2 × 𝑀. 4.1.1. Remarks

e is quasi-exact then 𝑉 must have at least one (non-void) end. 1) If 𝑉 ⊂ 𝑀 Indeed, if 𝑉 has no ends at all, then 𝑉 is a closed Lagrangian submanifold of ℝ2 × 𝑀 and so it can be displaced by a (compactly supported) Hamiltonian diffeomorphism. By standard results, for every 𝜔 e-compatible almost complex structure e 𝐽 there exists a non-constant 𝐽𝑉 -holomorphic disk with boundary on 𝑉, contradicting the quasiexactness of 𝑉. e ∫ 2) If 𝑉 ⊂ 𝑀 is quasi-exact, then necessarily 𝑀 is weakly-exact in the sense that

𝜔 = 0 for every 𝐴 ∈ 𝐻2𝑆 (𝑀), where 𝐻2𝑆 (𝑀) ⊂ 𝐻2 (𝑀) is the image of the Hurewicz e In particular homomorphism 𝜋2 (𝑀) → 𝐻2 (𝑀). Obviously the same holds also for 𝑀. e has non-constant pseudo-holomorphic spheres, for any compatible neither 𝑀 nor 𝑀 almost complex structure. Indeed, by point (4.1.1) above, 𝑉 has at least one (non-void) end, say 𝐿. By assumption 𝐿 ⊂ 𝑀 is weakly-exact, hence so is 𝑀. 𝑆

e → ℝ2 is (e 3) The condition that 𝜋 : 𝑀 𝐽, 𝑖)-holomorphic over a subset 𝑆 ⊂ ℝ2 is equivalent to e 𝐽 being fiberwise split over 𝑆. The space of 𝜔 e-compatible almost 2 e complex structures 𝐽 that are fiberwise split over 𝑆 ⊂ ℝ is path-connected (and, in fact, contractible). 4) One can also define quasi-exact cobordisms with ends being quasi-exact Lagrangians (not just weakly-exact). We will not pursue this degree of generality here.

4.1.2. Examples. — Here are several examples of quasi-exact cobordisms. 1) Weakly-exact cobordisms.

e 𝑉). 2) Cobordisms 𝑉 ⊂ ℝ2 × 𝑀, where dimℝ 𝑀 = 2 and 𝜇 = 0 on 𝜋2 ( 𝑀, 3) More generally, cobordisms 𝑉 ⊂ ℝ2 × 𝑀 with 𝜇(𝐴) ≤ 1 − e 𝑉) with 𝜔 𝐴 ∈ 𝜋2 ( 𝑀, e(𝐴) > 0.

1 2

dimℝ (𝑀), for all

4) As will be seen in Proposition 6.2 in Section 6.1, compositions of quasi-exact cobordisms (along a pair of matching ends) are quasi-exact.

ASTÉRISQUE 426

4.2. EXTENDING THE RESULTS FROM SECTION 3.7 TO QUASI-EXACT COBORDISMS

79

4.2. Extending the results from Section 3.7 to quasi-exact cobordisms

e be a quasi-exact Lagrangian cobordism with ends in L𝑎𝑔 we (𝑀). Fix Let 𝑉 ⊂ 𝑀 a quasi-exact admissible pair (𝐽𝑉 , 𝐾𝑉 ). Let 𝛾 ⊂ ℝ2 be a plane curve with horizontal ends, e.g. as depicted in Figure 2, page 71. Assume in addition that: 1) 𝛾 ⊂ ℝ2 \ 𝐾𝑉 . 2) 𝛾 intersects 𝜋(𝑉) only along the horizontal rays associated to the ends of 𝑉 (be they on the negative or positive side of 𝑉) and 𝛾 intersects each such ray at most once. Moreover these intersections are transverse. Fix 𝑝 and ℎ as at the beginning of Section 3.7. Denote by C the collection of weaklyexact Lagrangians in 𝑀 and by e C the collection of weakly-exact Lagrangian cobordisms in ℝ2 × 𝑀. As in Section 3.7 we have the Fukaya categories F𝑢𝑘( C; 𝑝) and F𝑢𝑘 cob ( e C1/2 ; 𝜄 𝛾 (𝑝, ℎ)). Note that, unless 𝑉 is weakly-exact, 𝑉 is not an object of the latter category. Consider now the (full) subcategory F𝑢𝑘 cob, C,𝛾 ⊂ F𝑢𝑘 cob ( e C1/2 ; 𝜄 𝛾 (𝑝, ℎ)) whose objects are 𝛾 × 𝑁 with 𝑁 ∈ C. We will define now a F𝑢𝑘 cob, C,𝛾 -module Vqe associated to 𝑉, constructed in an analogous way to the Yoneda module V from Section 3.7. More precisely, we set Vqe (𝛾 × 𝑁) = CF(𝛾 × 𝑁 , 𝑉; D𝛾×𝑁 ,𝑉 ) and define the higher 𝐴∞ -module operations Vqe

𝜇𝑑

as for a Yoneda module (associated to 𝑉) but with the following modifications

ee for the Floer and perturbations data D = D𝛾×𝑁1 ,...,𝛾×𝑁𝑑 ,𝑉 = ( 𝐾, 𝐽 ): (P1) We force the transition functions 𝑎 𝑟 : 𝑆𝑟 → [0, 1] to be identically 0 on the arc 𝜕𝑉 𝑆𝑟 corresponding to 𝑉. See (3.20) on page 63 for how the transition functions are incorporated into the perturbation data. See also [BC14, pp. 1757–1759 and 1762–1764], for more details.

ee (P2) The almost complex structures e 𝐽 in the perturbation data D = ( 𝐾, 𝐽 ) are such e that 𝐽 𝜕𝑉 𝑆𝑟 = 𝐽𝑉 . As usual we make the preceding choices of perturbation data to be consistent with the compactification of the spaces R𝑑+1 , 𝑑 ≥ 2, of punctured disks (in other words, the perturbation data can be chosen to be consistent with breaking and gluing). Before we proceed, here are a few important remarks explaining why this type of perturbation data makes sense at all, and why it does not collide with other aspects of the construction coming from [BC14]. First note that as we are only aiming at e involved in defining a module Vqe over F𝑢𝑘 cob, C,𝛾 , the Floer polygons 𝑢 : 𝑆𝑟 → 𝑀 Vqe

the definition of 𝜇𝑑 map the last arc along the boundary of 𝑆𝑟 to 𝑉, and all other arcs to Lagrangians of the type 𝛾 × 𝑁𝑖 . Moreover, as 𝛾 is transverse to the rays of 𝜋(𝑉), Vqe

then when defining the 𝜇𝑑 -operations we do not need to perform any horizontal perturbation (in the ℝ2 -direction) for strip-like ends corresponding to (𝛾 × 𝑁𝑑 , 𝑉) and (𝛾 × 𝑁1 , 𝑉). Thus vertical perturbations (in the 𝑀-direction) are enough. Therefore we can force the transition functions 𝑎 𝑟 to be 0 along 𝜕𝑉 𝑆𝑟 .

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Recall also that apart from (P2) above, the almost complex structures e 𝐽 in the perturbation data also have a restricted form, as described in [BC14] page 1764, namely they should satisfy that the projection 𝜋 is (e 𝐽𝑧 , (𝜙 𝑎ℎ (𝑧) )∗ (𝑖))-holomorphic for 𝑟

every 𝑟 ∈ R𝑑+1 , 𝑧 ∈ 𝑆𝑟 , over the complement of some compact subset in ℝ2 . We note that the latter condition is compatible with (P2) above because 𝑎 𝑟 = 0 along 𝜕𝑉 𝑆𝑟 and because 𝜋 is (𝐽𝑉 , 𝑖)-holomorphic over the complement of a compact subset 𝐾𝑉 ⊂ ℝ2 . Finally, it is straightforward to see that a consistent choice of perturbation data as described above indeed exists. Vqe

We now claim that with these modification the 𝜇𝑑 -operations are well defined and satisfy the 𝐴∞ -module identities. To see this we need to address the following points: compactness and transversality of the relevant spaces of Floer polygons (defined using the preceding perturbation data), and finally, that the 𝐴∞ -module identities indeed hold. Assuming compactness and transversality, the last point easily follows from the fact that the perturbation data can be chosen in a consistent way. For transversality, the arguments used in [BC14, Sections 3.4 and 4.3] (see also [BCb, Section 4.3.2 and Remark 4.3.5]) can be easily adapted to the present setting. The point is that imposing conditions (P1) and (P2) has no effect on transversality for the spaces of Floer trajectories, since these conditions affect the values of 𝑎 𝑟 and e 𝐽 only along 𝜕𝑉 𝑆𝑟 while in the interior of 𝑆𝑟 we can perform arbitrary perturbations (subject to [BC14, pp. 1762–1764]). We now address compactness. Here there are two separate issues to take care of. The first one is to verify that all Floer polygons (with fixed input and output chords) e The second issue is to control bubbling of holomorphic lie in a compact region of 𝑀. disks and spheres (recall that 𝑉 is not assumed to be weakly-exact anymore but only quasi-exact). The first point can be dealt with by the same arguments as in [BC14, Section 3.3]. Indeed, since 𝛾 is assumed to be transverse to the rays of 𝜋(𝑉) corresponding to the ends, condition (P1) does not interfere with the arguments from [BC14, Section 3.3]. Condition (P2) works well with the the arguments from [BC14, Section 3.3] since 𝜋 is (𝐽𝑉 , 𝑖) holomorphic over ℝ2 \ 𝐾𝑉 . This concludes the argument showing that all e Floer polygons lie within a compact region of 𝑀. Finally, we claim that in our setting no bubbling of holomorphic disks or spheres can occur. Indeed, by condition (P2) e 𝐽 𝜕𝑉 𝑆𝑟 = 𝐽𝑉 and by assumption there are no nonconstant 𝐽𝑉 -holomorphic disks with boundary on 𝑉. Therefore, bubbling of disks cannot occur at the 𝜕𝑉 𝑆𝑟 arc. As all the Lagrangians corresponding to the other arcs of 𝑆𝑟 are weakly-exact (they are of the type 𝛾 × 𝑁 with 𝑁 ∈ C) bubbling of disks cannot occur at these arcs too. Bubbling of holomorphic disks is also impossible since e is a weakly-exact symplectic manifold. by point (4.1.1) of Remark 4.1.1, 𝑀 This concludes the definition of the F𝑢𝑘 cob, C,𝛾 -module Vqe . 4.2.1. Remark. — The module Vqe is, strictly speaking, not a Yoneda module (although it is defined in an analogous way to Yoneda modules). The reason is that

ASTÉRISQUE 426

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81

a quasi-exact (but not weakly-exact) cobordism 𝑉 is not an object of F𝑢𝑘 cob, C,𝛾 , nor of any other 𝐴∞ -category we are considering in this paper. It is possible to set up an 𝐴∞ -category whose objects are quasi-exact cobordisms, by further modifications of the construction above. But this is not needed for the applications in this paper and so we will not pursue this direction here. We continue with extending the constructions from Sections 3.6 and 3.7 to the quasi-exact setting. Let 𝑉 ⊂ ℝ2 × 𝑀 be a cobordism as at the beginning of Section 3.7, only that now we assume that 𝑉 is only quasi-exact. Let 𝛾, 𝑝, ℎ be as in Section 3.7. Consider also the module Vqe as constructed above. Note that the inclusion functor I𝛾;𝑝,ℎ has its image in F𝑢𝑘 cob, C,𝛾 hence can be viewed as a functor I𝛾;𝑝,ℎ : F𝑢𝑘( C; 𝑝) −→ F𝑢𝑘 cob, C,𝛾 . By analogy to (3.28) we define a F𝑢𝑘( C; 𝑝)-module: ∗ M𝑉;𝛾,𝑝,ℎ := I𝛾;𝑝,ℎ Vqe . qe

(4.1)

We define also the modules M𝑉;𝛾𝑗 ,𝑝,𝜐 , 𝑗 = 1, . . . , 𝑟, in the same way as in (3.30), only that we now use the module Vqe instead of the Yoneda module V. qe

Proposition 4.3. — The statements of Propositions 3.4 and 3.5 continue to hold for the qe qe modules M𝑉;𝛾,𝑝,ℎ and M𝑉;𝛾𝑗 ,𝑝,𝜐 that have just been defined. The proof is exactly the same as the proofs of Propositions 3.4 and 3.5. 4.3. Quasi-monotone cobordisms By analogy to the “quasi-exact vs. weakly-exact” case, there is also a similar notion of quasi-monotone cobordisms generalizing monotone ones. Fix d := 𝑑𝑇 𝑎 ∈ Λ0 , where 𝑑 ∈ ℤ2 , 𝑎 > 0. As in Section 3.5 denote by L𝑎𝑔 mon,d (𝑀) the class of closed monotone Lagrangians 𝐿 ⊂ 𝑀 with d 𝐿 = d. Note that existence of a monotone Lagrangian in 𝑀 implies that the ambient manifold 𝑀 is monotone too. In particular, for every 𝐴 ∈ 𝜋2 (𝑀) with 𝜔(𝐴) > 0 we have 𝑐 1 (𝐴) > 0.

Definition 4.4. — Let 𝑉 ⊂ ℝ2 × 𝑀 be a Lagrangian cobordism with ends in L𝑎𝑔 mon,d (𝑀), not all void. We say that 𝑉 is quasi-monotone if there is a compact subset 𝐾𝑉 ⊂ ℝ2 and an 𝜔 e-compatible almost complex structure 𝐽𝑉 such that: 1) 𝑉 is cylindrical over ℝ2 \ Int (𝐾𝑉 ). 2) 𝜋 is (𝐽𝑉 , 𝑖)-holomorphic over ℝ2 \ Int (𝐾𝑉 ).

e 𝑉) we have 𝜇(𝑢) ≥ 2. 3) For all 𝐽𝑉 -holomorphic disks 𝑢 : (𝐷, 𝜕𝐷) → ( 𝑀, As in the quasi-exact case we will call (𝐽𝑉 , 𝐾𝑉 ) quasi-monotone admissible for 𝑉, and sometimes say that (𝑉 , 𝐽𝑉 , 𝐾𝑉 ) is quasi-monotone. We denote the class of quasi-monotone cobordisms 𝑉 as above by L𝑎𝑔 qm,d (ℝ2 × 𝑀). The parameter d indicates the value of d 𝐿𝑖 for the ends 𝐿 𝑖 of 𝑉 ∈ L𝑎𝑔 qm,d (ℝ2 × 𝑀).

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In the following we will need the following lemma, which is valid both in the quasi-exact and quasi-monotone cases.

Lemma 4.5. — Let (𝑉 , 𝐽𝑉 , 𝐾𝑉 ) be quasi-exact (resp. quasi-monotone). Let 𝐽𝑉0 be another 𝜔 e-compatible almost complex structure such that 𝐽𝑉0 = 𝐽𝑉 over 𝐾𝑉 and 𝐽𝑉0 is fiberwise split over ℝ2 \ Int (𝐾𝑉 ). Then (𝑉 , 𝐽𝑉0 , 𝐾𝑉 ) is also quasi-exact (resp. quasi-monotone). Proof. — This is an immediate application of the open mapping theorem, combined with the weak exactness (resp. monotonicity) of the ends of 𝑉. Indeed, by the open mapping theorem we deduce that any 𝐽𝑉0 -holomorphic disk with boundary on 𝑉 has to have its image inside 𝜋−1 (𝐾𝑉 ). As (𝑉 , 𝐽𝑉 , 𝐾𝑉 ) is quasi-exact (resp. quasi-monotone) and 𝐽𝑉 = 𝐽𝑉0 over 𝐾𝑉 this implies the claim.  4.4. Extending the results from Section 3.7 to quasi-monotone cobordisms This is similar to Section 4.2 only that now we have to take care of bubbling of 𝑞𝑚 holomorphic disks. The goal is to construct the modules V𝑞𝑚 and M𝑉;𝛾,𝑝,ℎ analogous qe to Vqe and M𝑉;𝛾,𝑝,ℎ . For brevity, denote by C = L𝑎𝑔 mon,d (𝑀).

e be a quasi-monotone Lagrangian cobordism with ends in C. Fix a quasiLet 𝑉 ⊂ 𝑀 monotone admissible pair (𝐽𝑉 , 𝐾𝑉 ). Let 𝛾 and 𝑝, ℎ be as in Section 4.2. Recall that in the monotone Fukaya category of 𝑀 the choices of the Floer data prescribed by 𝑝 are assumed to satisfy the following additional conditions. Let 𝐾 0 , 𝐾1 ∈ Cand D𝐾0 ,𝐾1 = (𝐻 𝐾0 ,𝐾1 , {𝐽𝑡𝐾0 ,𝐾1 }) be the Floer datum of (𝐾 0 , 𝐾1 ) prescribed by 𝑝. Let 𝜂 ∈ O(𝐻 𝐾0 ,𝐾1 ) be a Hamiltonian chord. Then for both 𝜈 = 0 and 𝜈 = 1, the almost complex structure 𝐽𝜈𝐾0 ,𝐾1 is regular for all 𝐽𝜈𝐾0 ,𝐾1 -holomorphic disks with boundary on 𝐾 𝜈 that have Maslov index 2 and moreover 𝜂(𝜈) ∈ 𝐾 𝜈 is a regular value of the evaluation maps ev𝐾 𝜈 ,𝐴 : (M(𝐴, 𝐽𝜈𝐾0 ,𝐾1 ) × 𝜕𝐷)/Aut(𝐷) → 𝐾 𝜈 , for all Í 𝐴 ∈ 𝜋2 (𝑀, 𝐾 𝜈 ) with 𝜇(𝐴) = 2. (And of course, by assumption the 𝐴 degℤ2 ev𝐾 𝜈 ,𝐴 = 𝑑, where the sum is over all 𝐴 ∈ 𝜋2 (𝑀, 𝐾 𝜈 ) with 𝜇(𝐴) = 2. Here 𝑑 ∈ ℤ2 is the coefficient of 𝑇 𝑎 in d, i.e. d = 𝑑𝑇 𝑎 .) 𝛾×𝑁 ,𝑉 Fix 𝑁 ∈ C. Let 𝐽 𝛾×𝑁 ,𝑉 = {𝐽𝑡 } be a time-dependent 𝜔 e-compatible almost complex satisfying the following properties: 1) For each intersection point 𝑥 ∈ 𝛾 ∩ 𝜋(𝑉), denote by 𝐿 𝑥 ⊂ 𝑀 the Lagrangian 𝛾×𝑁 ,𝑉 corresponding to the end of 𝑉 over 𝑥. We require that 𝐽𝑡 = 𝑖 ⊕ 𝐽𝑡𝑁 ,𝐿𝑥 in 𝑈 𝑥 × 𝑀 for some small neighborhood 𝑈 𝑥 of 𝑥 which is contained in ℝ2 \ 𝐾𝑉 . Here, {𝐽𝑡𝑁 ,𝐿𝑥 } is the choice prescribed by 𝑝 for the pair (𝑁 , 𝐿 𝑥 ). 𝛾×𝑁 ,𝑉

2) 𝐽1

𝛾×𝑁 ,𝑉

is fiberwise split over ℝ2 \ 𝐾𝑉 .

3) 𝐽1 coincides with 𝐽𝑉 over 𝐾𝑉 . As will be seen soon, 𝐽 𝛾×𝑁 ,𝑉 will be used as the almost complex structure for the Floer datum D𝛾×𝑁 ,𝑉 of the pair (𝛾 × 𝑁 , 𝑉). As such, 𝐽 𝛾×𝑁 ,𝑉 needs to satisfy the

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83

usual additional conditions we impose on Floer data for Lagrangian cobordisms, as described in [BC14, Section 3.2, p. 1764]. It is easy to see that almost complex 𝛾×𝑁 ,𝑉 } as described above exist (recall that the space of 𝜔 e-compatible structures {𝐽𝑡 fiberwise split almost complex structures is connected). Note that by Lemma 4.5 𝛾×𝑁 ,𝑉 (𝐽1 , 𝐾𝑉 ) continues to be quasi-monotone admissible for 𝑉. The 𝐴∞ -category F𝑢𝑘 cob, C,𝛾 is constructed in a similar way to what we have done in Section 4.2, only that we work in the monotone framework. To define the module V𝑞𝑚 we take Floer data of the type D𝛾×𝑁 ,𝑉 = 𝐻 𝛾×𝑁 ,𝑉 , {𝐽 𝛾×𝑁 ,𝑉 } ,




where the almost complex structure {𝐽 𝛾×𝑁 ,𝑉 } is as described above. The Hamiltonian term 𝐻 𝛾×𝑁 ,𝑉 is assumed to have the following form: for every 𝑥 ∈ 𝛾 ∩ 𝜋(𝑉) we have 𝐻 𝛾×𝑁 ,𝑉 (𝑧, 𝑢) = 𝜎(𝑥) (𝑧)𝐻 𝑁 ,𝐿𝑥 (𝑢), for 𝑧 ∈ 𝑈 𝑥 , 𝑢 ∈ 𝑀. Here, 𝐻 𝑁 ,𝐿𝑥 is the Hamiltonian term in the Floer datum of (𝑁 , 𝐿 𝑥 ) and 𝜎(𝑥) : 𝑈 𝑥 → [0, 1] is a smooth function with compact support in 𝑈 𝑥 and such that 𝜎(𝑥) ≡ 1 near 𝑥. Outside of the union of the subsets 𝑈 𝑥 , 𝑥 ∈ 𝛾 ∩ 𝜋(𝑉), we set 𝐻 𝛾×𝑁 ,𝑉 to be 0. ee Next, we define in a similar way to Section 4.2 perturbation data D = ( 𝐾, 𝐽 ) for tuples of the type (𝛾 × 𝑁1 , . . . , 𝛾 × 𝑁𝑑 , 𝑉) with the difference that we require now that e 𝐽 𝜕𝑉 𝑆𝑟 coincides with 𝐽𝑉 over 𝐾𝑉 . It is straightforward to see that consistent choices of perturbation data with these additional properties exist. Moreover, there exist such consistent choices which are regular. The latter does not require any new arguments beyond those remarked in the quasi-exact case. The definition of the module V𝑞𝑚 is now done in the same way as for the module Vqe in the quasi-exact case. Beyond the arguments for the weakly-exact and quasiexact cases, there is only one point that needs to be analyzed – bubbling of disks and spheres within spaces of Floer polygons of dimensions ≤ 1 and its effect on the 𝑞𝑚 operations. 𝐴∞ -module identities for the 𝜇V 𝑑 To this ends, suppose that bubbling of a holomorphic disk or sphere occurs in a sequence of Floer polygons whose index is ≤ 1 (i.e. the dimension of the space of these polygons is ≤ 1). We claim that this can happen only if the Floer polygons are in fact Floer strips (i.e. the polygons are 2-gons with boundaries on two Lagrangians), the incoming and exit chords coincide and moreover, after removing the bubbles we are left with a “constant” Floer strip, namely a degenerate Floer strip whose image is that common Hamiltonian chord. Indeed, if bubbling of disks occurs along 𝑉 then by quasi-monotonicity each such bubble has Maslov index ≥ 2. If bubbling of a holomorphic disk occurs along one of the 𝛾 × 𝑁𝑖 ’s, then by the monotonicity of 𝑁𝑖 we again have that the Maslov index of each such bubble is ≥ 2. Finally, if bubbling of a holomorphic sphere occurs, then the Chern number of such bubbles is ≥ 1 because 𝑀 is a monotone symplectic manifold (see the beginning of Section 4.3). Thus, in all cases the total index of the Floer polygon that remains after removing the bubbles is negative. By transversality this cannot happen unless that polygon is “constant at a chord”. Moreover, if the

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perturbation data are chosen generically, such a limit can occur only if the polygons are strips. We are thus left only with the case when bubbling occurs for Floer strips (in a 1dimensional space) connecting a chord 𝜂 to itself, and after bubbling of a holomorphic disk the remaining Floer strip is “constant” at 𝜂. The holomorphic disk bubble has boundary on one of the Lagrangians involved and passes through 𝜂(0) or 𝜂(1). 𝑞𝑚 Now, the only effect of the last phenomenon on the 𝜇V operation is for 𝑑 = 1, 𝑑 V𝑞𝑚 V𝑞𝑚 namely when trying to show that 𝜇1 ◦ 𝜇1 is 0. Note that the two pairs of Lagrangians involved in this operation are of the type 𝛾 × 𝑁 with 𝑁 ∈ C and 𝑉. By our choices of Floer data, the only Hamiltonian chords 𝜂 between these two Lagrangians are of the type 𝑥 × 𝜂0, where 𝑥 ∈ 𝛾 ∩ 𝜋(𝑉) and 𝜂0 ∈ O(𝐻 𝑁 ,𝐿𝑥 ). Here 𝐿 𝑥 ⊂ 𝑀 is the Lagrangian corresponding to the end of 𝑉 over 𝑥. By our choice of almost complex structures in the Floer data and by applying the open mapping theorem all the holomorphic disks with boundary on either 𝛾 × 𝑁 or on 𝑉 that pass through 𝜂(0) or 𝜂(1) must have constant projection to ℝ2 . Thus these disks lie in 𝑥 × 𝑀 and are in fact either 𝐽0𝑁 ,𝐿𝑥 -holomorphic with boundary on 𝑁 and pass through 𝜂0(0) or are 𝐽1𝑁 ,𝐿𝑥 holomorphic with boundary on 𝐿 𝑥 and pass through 𝜂0(1). By gluing results the outcome of this is that 𝜇1V𝑞𝑚 ◦ 𝜇1V𝑞𝑚 (𝑥 × 𝜂0) =

(4.2)

Õ

(d 𝑁 − d 𝐿𝑥 )(𝑥 × 𝜉0).

𝜉0 ∈O(𝐻 𝑁 ,𝐿 𝑥 )

Since both 𝑁 and all the ends of 𝑉 belong to C = L𝑎𝑔 mon,d (𝑀) we have d 𝑁 = d 𝐿𝑥 , hence (4.2) vanishes. This concludes the construction of the F𝑢𝑘 cob, C,𝛾 -module V𝑞𝑚 . 𝑞𝑚 𝑞𝑚 The construction of the F𝑢𝑘( C; 𝑝)-modules M𝑉;𝛾,𝑝,ℎ and M𝑉;𝛾 ,𝑝,ℎ is done as in (4.1) 𝑗 with V𝑞𝑒 replaced by V𝑞𝑚 . As earlier, Propositions 3.4 and 3.5 continue to hold in the quasi-monotone case (with the same proofs):

Proposition 4.6. — The statements of Propositions 3.4 and 3.5 continue to hold for the 𝑞𝑚

𝑞𝑚

module M𝑉;𝛾,𝑝,ℎ and M𝑉;𝛾𝑗 ,𝑝,𝜐 that has just been defined.

ASTÉRISQUE 426

CHAPTER 5 PROOF OF THE MAIN GEOMETRIC STATEMENTS

We prove here the main geometric results. We will make use of the following variants of the notion of Gromov width. Let (𝑀 2𝑛 , 𝜔) be a symplectic manifold, 𝐿 ⊂ 𝑀 a Lagrangian submanifold and 𝑄 ⊂ 𝑀 a subset. Following [BC07], [BC06] we define the Gromov width 𝛿(𝐿; 𝑄) of 𝐿 relative to 𝑄 as follows. Assume first that 𝐿 ⊄ 𝑄. Define: (5.1)

𝛿(𝐿; 𝑄) = sup 𝜋𝑟 2 ∈ (0, ∞] ; ∃a symplectic embedding 𝑒 : 𝐵(𝑟) → 𝑀



such that 𝑒 −1 (𝐿) = 𝐵ℝ (𝑟) and 𝑒(𝐵(𝑟)) ∩ 𝑄 = ∅ .



Here 𝐵(𝑟) ⊂ ℝ2𝑛 is the standard 2𝑛-dimensional closed ball of radius 𝑟, endowed with the standard symplectic structure from ℝ2𝑛 , and 𝐵ℝ (𝑟) := 𝐵(𝑟) ∩ (ℝ 𝑛 × {0}) is the real part of 𝐵(𝑟). In case 𝐿 ⊂ 𝑄 we set 𝛿(𝐿; 𝑄) := 0. Another variant of the Gromov width is associated to an immersed Lagrangian. b be a smooth closed manifold (possibly disconnected) and let 𝜄 : 𝕃 b → 𝑀 be a Let 𝕃 b Lagrangian immersion with image 𝕃 := 𝜄(𝕃 ). We will measure the “size” of a subset of the double points of 𝕃 relative to a given subset 𝑄 ⊂ 𝑀. Denote by Σ(𝜄) ⊂ 𝕃 the set of points that have more than one preimage under the immersion 𝜄. Let Σ0 ⊂ Σ(𝜄) be a non-empty subset such that each point in Σ0 is a transverse intersection of two branches of the immersion. As before, let 𝑄 ⊂ 𝑀 be a subset. 0 Assume first that Σ0 ⊄ 𝑄. We define the Gromov width 𝛿Σ (𝕃 ; 𝑄) of the self0 intersection set Σ relative to 𝑄 by 0

𝛿Σ (𝕃 ; 𝑄) = sup 𝜋𝑟 2 ∈ (0, ∞] ; ∀𝑥 ∈ Σ0 , ∃ a symplectic embedding 𝑒 𝑥 : 𝐵(𝑟) → 𝑀 with



𝑒 𝑥 (0) = 𝑥, 𝑒 𝑥−1 (𝕃 ) = 𝐵ℝ (𝑟) ∪ 𝑖𝐵ℝ (𝑟), 𝑒 𝑥 (𝐵(𝑟)) ∩ 𝑄 = ∅, and 𝑒 𝑥0 (𝐵(𝑟)) ∩ 𝑒 𝑥00 (𝐵(𝑟)) = ∅ whenever 𝑥 0 ≠ 𝑥 00 .



Here 𝑖𝐵ℝ (𝑟) stands for the imaginary part of the ball, 𝑖𝐵ℝ (𝑟) := 𝐵(𝑟) ∩ ({0} × ℝ 𝑛 ). 0

In case ∅ ≠ Σ0 ⊂ 𝑄 we set 𝛿Σ (𝕃 ; 𝑄) = 0. In case Σ0 = ∅ we set 𝛿 ∅ (𝕃 ; 𝑄) = ∞.

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In what follows, if 𝑄 = ∅, then we omit the set 𝑄 from the notation in both 𝛿(𝐿; 𝑄) 0 and 𝛿Σ (𝕃 ; 𝑄). The next important geometric measurement is the shadow of a cobordism, as defined in [CS19] and already mentioned in the introduction. Let 𝑉 ⊂ ℝ2 × 𝑀 be a Lagrangian cobordism. Denote by 𝜋 : ℝ2 × 𝑀 → ℝ2 the projection. The shadow S(𝑉) of 𝑉 is defined as (5.2)

S(𝑉) = Area ℝ2 \ U ,




where U ⊂ ℝ2 \ 𝜋(𝑉) is the union of all the unbounded connected components of ℝ2 \ 𝜋(𝑉). We now restate here the main geometric results for the convenience of the reader. Recall that L𝑎𝑔 ∗ (𝑀) denotes the collection of closed Lagrangian submanifolds of 𝑀 of class ∗, where ∗ stands either for the weakly exact Lagrangians (∗ = we in short), or for the monotone Lagrangians with given Maslov-2 disk count d ∈ Λ0 (∗ = (mon, d) in short) as introduced in Section 3.5. Similarly, we have the collection L𝑎𝑔 ∗ (ℝ2 × 𝑀) of Lagrangian cobordisms 𝑉 ⊂ ℝ2 × 𝑀 of class ∗, where ∗ is as above. (𝐿1 , . . . , 𝐿 𝑘 ) a weakly 𝐿 the union of the Lagrangians corresponding 𝑖 𝑖=1

Theorem 5.1. — Let 𝐿, 𝐿1 , . . . , 𝐿 𝑘 ∈ L𝑎𝑔 we (𝑀) and 𝑉 : 𝐿 Ð𝑘 exact Lagrangian cobordism. Denote 𝑆 := to the negative ends of 𝑉. Then (5.3)

S(𝑉) ≥ 21 𝛿(𝐿; 𝑆).

For the next two points of the theorem we will use the following. Let 𝑁 ∈ L𝑎𝑔 we (𝑀) be another Ð𝑘 weakly exact Lagrangian submanifold and consider 𝑆 = 𝑖=1 𝐿 ⊂ 𝑀 and 𝑁 ∪ 𝑆 ⊂ 𝑀 as Ý𝑘 Ý Ý𝑘 𝑖 immersed Lagrangians (parametrized by 𝑖=1 𝐿 𝑖 and 𝑁 ( 𝑖=1 𝐿 𝑖 ) respectively). (i) Assume that 𝑁 intersects each of the Lagrangians 𝐿1 , . . . , 𝐿 𝑘 transversely and that 0 𝑁 ∩ 𝐿 𝑖 ∩ 𝐿 𝑗 = ∅ for all 𝑖 ≠ 𝑗. Denote Σ0 := 𝑁 ∩ 𝑆. If S(𝑉) < 12 𝛿Σ (𝑁 ∪ 𝑆) then (5.4)

# (𝑁 ∩ 𝐿) ≥

𝑘 Õ

#(𝑁 ∩ 𝐿 𝑖 ).

𝑖=1

(ii) Assume that the Lagrangians 𝐿1 , . . . , 𝐿 𝑘 intersect pairwise transversely and that no three of them have a common intersection point (i.e. 𝐿 𝑖 ∩ 𝐿 𝑗 ∩ 𝐿𝑟 = Ð ∅ for all distinct indices 𝑖, 𝑗, 𝑟). Let Σ00 be the set of all double points of 𝑆, i.e. Σ00 := 1≤𝑖 𝜆0 for every 𝑖 ≥ 1. As usual we set 𝜈(0) = ∞. All Floer complexes will be taken with coefficients in Λ as in Section 3 and the filtrations on them will be defined by action, according to the recipe from Section 3.3. Given such a Floer complex, say 𝐶, we will denote by 𝐴 : 𝐶 → ℝ ∪ {−∞} the action level, as defined in Section 2.7. (Recall that for 𝑥 ∈ 𝐶 we write 𝐴(𝑥) and 𝐴(𝑥; 𝐶) interchangeably.) Note that 𝐴 coincides with A from Section 3.3, and below we will continue to denote it by 𝐴 (rather than A) to keep compatibility with our general algebraic conventions.

5.1. Proof of Theorem 5.1 We begin with the proof of inequality (5.3). We first assume that the Lagrangians 𝐿, 𝐿1 , . . . , 𝐿 𝑘 intersect pairwise transversely, and treat the general case afterwards. We start by bending the positive end of 𝑉 by 180◦ clockwise in such a way as to get a cobordism 𝑊 without positive ends, and whose negative ends are (𝐿0 , 𝐿1 , . . . , 𝐿 𝑘 ), where 𝐿0 := 𝐿 (see Figure 4). Clearly S(𝑊) = S(𝑉). Fix 𝜖 > 0. Let 𝛾, 𝛾0 be two curves, as depicted in Figure 5, and such that there exists a (not compactly supported) Hamiltonian isotopy, horizontal at infinity, 𝜑𝑡 : ℝ2 → ℝ2 , 𝑡 ∈ [0, 1], with 𝜑0 = id, 𝜑1 (𝛾) = 𝛾0 and with (5.7)

length{𝜑𝑡 } ≤ S(𝑊) + 12 𝜖,

where length{𝜑𝑡 } stands for the Hofer length of the isotopy {𝜑𝑡 }.

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Figure 4. The cobordisms 𝑊 obtained from 𝑉 by bending the positive end.

Figure 5. The curves 𝛾 and 𝛾0 and the cobordism 𝑊. 𝑘 Put 𝑆 = 𝑖=1 𝐿 𝑖 and let 𝑒 : 𝐵(𝑟) → 𝑀 \ 𝑆 be a symplectic embedding as in the definition of 𝛿(𝐿0 ; 𝑆) in (5.1), with

Ð

𝛿(𝐿0 , 𝑆) − 𝜖 ≤ 𝜋𝑟 2 ≤ 𝛿(𝐿0 , 𝑆). Next, let (5.8)

𝐵 := image (𝑒),

𝑞 := 𝑒(0) ∈ 𝐿0 ,

𝐽 𝐵 := 𝑒∗ (𝐽std ),

where the latter is the complex structure on 𝐵 corresponding to the standard complex structure 𝐽std of 𝐵2𝑛 (𝑟) via the embedding 𝑒. Next, we fix a symplectic identification between a small open neighborhood 𝑈 of 𝐿0 in 𝑀 and a neighborhood 𝑈 0 of the zero-section in 𝑇 ∗ (𝐿0 ). Let 𝑓 : 𝐿0 → ℝ be a 𝐶 1 -small Morse function with exactly one local maximum at the point 𝑞 ∈ 𝐿0 . We extend 𝑓 to a function e 𝑓 : 𝑈 0 → ℝ by setting it to be constant along the fibers of the cotangent bundle. Finally, let 𝐻 𝐿𝑓 0 ,𝐿0 : 𝑀 → ℝ be a smooth function such that 𝐻 𝐿𝑓 0 ,𝐿0 𝑈 coincides with e 𝑓 via the identification between 𝑈 and 𝑈 0 that we have just fixed. We now turn to the Fukaya categories relevant for this proof. Let Cbe the collection of Lagrangians 𝐿0 , . . . , 𝐿 𝑘 . We will use the Fukaya categories F𝑢𝑘( C) and F𝑢𝑘 cob ( e C) 0 associated to C. More specifically, we consider regular perturbation data 𝑝 ∈ 𝐸reg 0 (𝛾) as in Section 3.6. and 𝐶 1 -small profile functions ℎ ∈ Hprof

ASTÉRISQUE 426

89

5.1. PROOF OF THEOREM 5.1

We impose two additional restrictions on the admissible choices of perturbation data 𝑝 as follows. The first one is that the datum D𝐿0 ,𝐿0 of the pair (𝐿0 , 𝐿0 ) should have the function 𝐻 𝐿𝑓 0 ,𝐿0 as its Hamiltonian function, defined using any choice of a 𝐶 1 -small Morse function 𝑓 as described above. Furthermore, we allow only for functions 𝑓 that are sufficiently 𝐶 1 -small such that O(𝐻 𝐿𝑓 0 ,𝐿0 ) = Crit( 𝑓 ). Note that for every 𝑦 ∈ O(𝐻 𝐿𝑓 0 ,𝐿0 ) we have 𝐴(𝑦) = 𝑓 (𝑦). The second restriction is that the Hamiltonian functions 𝐻 𝐿𝑖 ,𝐿 𝑗 in the Floer data D𝐿𝑖 ,𝐿 𝑗 , 𝑖 ≠ 𝑗, are all 0. It is possible to impose these additional restriction and still maintain regularity since we have assumed that the Lagrangians 𝐿0 , 𝐿1 , . . . , 𝐿 𝑘 intersect pairwise transversely. With these choices we have for every 𝑖 ≠ 𝑗: O(𝐻 𝐿𝑖 ,𝐿 𝑗 ) = 𝐿 𝑖 ∩ 𝐿 𝑗 ,

𝐴(𝑧) = 0,

for all 𝑧 ∈ O(𝐻 𝐿𝑖 ,𝐿 𝑗 ).

00 ⊂ 𝐸0 . We denote the space of all such regular choices of perturbation data by 𝐸reg reg 00 00 We remark that the Morse function 𝑓 is not fixed over 𝐸reg and each choice 𝑝 ∈ 𝐸reg 00 comes with its own function 𝑓 . Finally, note that N is still in the closure of 𝐸reg . We now appeal to the theory developed in Section 3. Consider the Fukaya category F𝑢𝑘( C; 𝑝) (see Section 3.2) as well as the Fukaya category of cobordisms F𝑢𝑘 cob ( e C1/2 ; 𝜄 𝛾 (𝑝, ℎ)) (see Section 3.4 and (3.25) in Section 3.6). Recall that we have an “inclusion” functor

I𝛾;𝑝,ℎ : F𝑢𝑘( C; 𝑝) −→ F𝑢𝑘 cob e C1/2 ; 𝜄 𝛾 (𝑝, ℎ) .




Denote by W the Yoneda module corresponding to the object 𝑊 ∈ Ob F𝑢𝑘 cob ( e C1/2 ; 𝜄 𝛾 (𝑝, ℎ))




and consider its pull-back by the functor I𝛾;𝑝,ℎ : ∗ M𝑊;𝛾,𝑝,ℎ := I𝛾;𝑝,ℎ W.

Recall from Sections 3.3, 3.4 and 3.6 that the 𝐴∞ -categories F𝑢𝑘( C; 𝑝),

F𝑢𝑘 cob e C1/2 ; 𝜄 𝛾 (𝑝, ℎ)




as well as the 𝐴∞ -functor I𝛾;𝑝,ℎ are all weakly filtered. Moreover, by Section 3.7 the module M𝑊;𝛾,𝑝,ℎ if weakly filtered too. By Propositions 3.4, and points (3.6), (3.6) on page 69 the discrepancy of this module is bounded from above by (𝑝, ℎ) = (𝜖1 (𝑝, ℎ), 𝜖2 (𝑝, ℎ), . . . , 𝜖 𝑑 (𝑝, ℎ), . . . ) which satisfies lim 𝜖 𝑑 (𝑝, ℎ) → 0 for every 𝑑, as 𝑝 → 𝑝0 ∈ N and ℎ → 0 (the latter in the 𝐶 1 -topology). Throughout the proof we will repeatedly deal with quantities having the same asymptotics as 𝜖 𝑑 (𝑝, ℎ). In order to simplify the text we introduce the following notation. Let N0 ⊂ N and let (𝑝, ℎ) ↦−→ 𝐶(𝑝, ℎ) be a real valued function defined for 𝑝 0 whose closure contains N , and ℎ ∈ H0 . in a subset of 𝐸reg 0 prof We will write 𝐶(𝑝, ℎ) ∈ 𝑂(N0 ) to indicate that for every 𝑝 0 ∈ N0 we have lim 𝐶(𝑝, ℎ) = 0 as 𝑝 → 𝑝0 and ℎ → 0 (the latter in the 𝐶 1 -topology).

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By Proposition 3.5 (and (3.32)) we have 𝜙𝑘

(5.9)

𝜙 𝑘−1

𝑆 𝑠 ℎ M𝑊;𝛾,𝑝,ℎ = C𝑜𝑛𝑒(L𝑘 −−→ C𝑜𝑛𝑒(L𝑘−1 −−−−→ C𝑜𝑛𝑒(· · · 𝜙2

𝜙1

· · · C𝑜𝑛𝑒(L2 −−→ C𝑜𝑛𝑒(L1 −−→ L0 ))···))), where 𝑠 ℎ → 0 as ℎ → 0. (Recall from § 2.3.4 that 𝑆 𝑠 ℎ M𝑉;𝛾,𝑝,ℎ stands for the module M𝑉;𝛾,𝑝,ℎ with action-shift by 𝑠 ℎ .) The modules L𝑖 in (5.9) are the Yoneda modules of the 𝐿 𝑖 ’s. The notation 𝜙 𝑖 stands for 𝜙 𝑖 = (𝜙 𝑖 , 0, δ (𝒊) ), with 𝜙 𝑖 being a homomorphism of modules that shifts action by ≤ 0 and has discrepancy ≤ δ (𝒊) (𝑝, ℎ) where for every (𝑗)

𝑑 we have 𝛿 𝑑 (𝑝, ℎ) ∈ 𝑂(N). For simplicity of notation set δ(𝑝, ℎ) := max δ (1) (𝑝, ℎ), . . . , δ (𝒌) (𝑝, ℎ) ,





so that the discrepancy of all the 𝜙 𝑖 ’s is ≤ δ(𝑝, ℎ) and we still have 𝛿 𝑑 (𝑝, ℎ) ∈ 𝑂(N) for all 𝑑. Consider the filtered chain complex 𝒞𝑝,ℎ := 𝑆 𝑠 ℎ M𝑊;𝛾,𝑝,ℎ (𝐿0 ) endowed with the differential coming from the 𝜇1 -operation of M𝑊;𝛾,𝑝,ℎ . By definition 𝒞𝑝,ℎ = 𝑆 𝑠 ℎ CF(𝛾 × 𝐿0 , 𝑊; D𝛾×𝐿0 ,𝑊 ), where D𝛾×𝐿0 ,𝑊 is the Floer datum prescribed by 𝜄 𝛾 (𝑝, ℎ). By (5.9) the Floer complex of (𝐿0 , 𝐿0 ) is a subcomplex of 𝒞𝑝,ℎ , or more precisely, we have an action preserving inclusion of chain complexes: (5.10)

CF(𝐿0 , 𝐿0 ; 𝑝) ⊂ 𝒞𝑝,ℎ ,

where D𝐿0 ,𝐿0 is specified by 𝑝 and is subject to the additional restrictions imposed earlier in the proof. To simplify the notation, we will denote from now on for a pair of Lagrangians (𝐿0 , 𝐿00) by CF(𝐿0 , 𝐿00; 𝑝) the Floer complex CF(𝐿0 , 𝐿00; D𝐿0 ,𝐿00 ), where D𝐿0 ,𝐿00 is the Floer datum specified by 𝑝. Recall that we also have the curve 𝛾0 ⊂ ℝ2 with 𝛾0 ∩ 𝜋(𝑊) = ∅. Choose a Floer datum D0 for (𝛾0 × 𝐿0 , 𝑊) with a sufficiently 𝐶 2 -small Hamiltonian function so that CF(𝛾0 × 𝐿0 , 𝑊; D0) = 0. Now 𝛾 × 𝐿0 can be Hamiltonian isotoped to 𝛾0 × 𝐿0 via an isotopy horizontal at infinity with Hofer length ≤ S(𝑊)+ 21 𝜖. By standard Floer theory (see e.g. [FOOO09a, Section 5.3.2]) the identity map on 𝒞𝑝,ℎ is null homotopic via a chain homotopy which shifts action by ≤ S(𝑊) + 21 𝜖. Translated to the formalism of (2.44) in Section 2.7 this means that 𝐵 ℎ (id𝒞𝑝,ℎ ) ≤ S(𝑊) + 12 𝜖, hence by (2.46) we have (5.11)

𝛽(𝒞𝑝,ℎ ) ≤ S(𝑊) + 21 𝜖,

where 𝛽(𝒞𝑝,ℎ ) is the boundary depth of the (acyclic) chain complex 𝒞𝑝,ℎ as defined in Section 2.7. We now appeal to Theorem 2.14 applied to the weakly filtered iterated cone (5.9). We apply this theorem with 𝑋 = 𝐿0 and 𝜌 𝑖 = 0. We obtain a new weakly filtered module Msuch that M(𝐿0 ) has a differential 𝜇1M as described in that theorem together with a filtered chain isomorphism 𝜎1 : 𝒞𝑝,ℎ → M(𝐿0 ). An inspection of the sizes of shifts and discrepancies of the various maps involved in Theorem 2.14 shows that there exists a constant 𝑠 𝜎 (𝑝, ℎ) ∈ 𝑂(N) such that 𝜎1 shifts filtration by ≤ 𝑠 𝜎 (𝑝, ℎ).

ASTÉRISQUE 426

91

5.1. PROOF OF THEOREM 5.1

Additionally, Theorem 2.14 implies that CF(𝐿0 , 𝐿0 ; 𝑝) is also a filtered subcomplex of M(𝐿0 ) and that pr0 ◦ 𝜎1 maps CF(𝐿0 , 𝐿0 ; 𝑝) ⊂ C𝑝,ℎ to CF(𝐿0 , 𝐿0 ; 𝑝) ⊂ M(𝐿0 ) via the identity map: (pr0 ◦ 𝜎1 ) CF(𝐿0 ,𝐿0 ;𝑝) = id. Here pr0 : M(𝐿0 ) → CF(𝐿0 , 𝐿0 ; 𝑝) is the projection onto the 0-th factor of M(𝐿0 ). Consider now the homology unit 𝑒 𝐿0 ∈ CF(𝐿0 , 𝐿0 ; 𝑝) as constructed in (3.10). By standard Floer theory 𝑒 𝐿0 = 𝑞 (recall that 𝑞 is the unique maximum of 𝑓 : 𝐿0 → ℝ). Let 𝑐 ∈ CF(𝐿0 , 𝐿0 ; 𝑝) and 𝛾 ∈ O(𝐻 𝐿0 ,𝐿0 ) a generator, where 𝐻 𝐿0 ,𝐿0 is the Hamiltonian function of the Floer datum specified by 𝑝 for (𝐿0 , 𝐿0 ). We denote by h𝑐, 𝛾i ∈ Λ the coefficient of 𝛾 when writing 𝑐 as a linear combination of elements of O(𝐻 𝐿0 ,𝐿0 ) with coefficients in Λ. We will need the following.

Lemma 5.3. — For every chain 𝑐 ∈ CF(𝐿0 , 𝐿0 ; 𝑝) we have h𝜇1 (𝑐), 𝑞i = 0. We postpone the proof of the lemma and continue with the proof of Theorem 5.1. Put 𝐶 𝑓 := max𝑥∈𝐿0 | 𝑓 (𝑥)|, 𝐶 (1) (𝑝, ℎ) := 𝐶 𝑓 + 𝑠 𝜎 (𝑝, ℎ). By (5.11) there exists (5.12)

𝑏 0 ∈ 𝒞𝑝,ℎ with 𝐴(𝑏 0; 𝒞𝑝,ℎ ) ≤ 𝐴(𝑒 𝐿0 ; 𝒞𝑝,ℎ ) + S(𝑊) + 21 𝜖 ≤ 𝐶 𝑓 + S(𝑊) + 12 𝜖 , 𝒞𝑝,ℎ

such that 𝑒 𝐿0 = 𝜇1 (𝑏 0). Recall from point (2.14) of Theorem 2.14 that pr0 ◦ 𝜎1 𝒞𝑝,ℎ

and apply pr0 ◦ 𝜎1 to the equality 𝑒 𝐿0 = 𝜇1 (5.13)

CF(𝐿0 ,𝐿0 ;𝑝)

= id. Set 𝑏 := 𝜎1 (𝑏 0)

(𝑏 0). We obtain

𝐴 𝑏; M(𝐿0 ) ≤ 𝐶 (1) (𝑝, ℎ) + S(𝑊) + 12 𝜖,

𝑒 𝐿0 = pr0 ◦ 𝜇1M(𝑏),




where 𝐶 (1) (𝑝, ℎ) := 𝐶 𝑓 + 𝑠 𝜎 (𝑝, ℎ). Obviously 𝐶 (1) (𝑝, ℎ) ∈ 𝑂(N). (Note that 𝑓 → 0 as 𝑝 → 𝑝0 ∈ N.) Using the splitting (2.30) write 𝑏 = 𝑏 0 + · · · + 𝑏 𝑘 , with 𝑏 𝑖 ∈ CF(𝐿0 , 𝐿 𝑖 ; 𝑝) and 𝐴 𝑏 𝑖 ; CF(𝐿0 , 𝐿 𝑖 ; 𝑝) ≤ 𝐶 (2) (𝑝, ℎ) + S(𝑊) + 12 𝜖,




where 𝐶 (2) (𝑝, ℎ) is a new constant such that lim 𝐶 (2) (𝑝, ℎ) ∈ 𝑂(N). Continuing to apply Theorem 2.14 we have (5.14)

𝑘 Õ

𝑞 = pr0 ◦ 𝜇1M(𝑏) =

CF(𝐿0 ,𝐿0 ;𝑝)

𝑎0,𝑗 (𝑏 𝑗 ) = 𝜇1

𝑗=0

(𝑏 0 ) +

𝑘 Õ

𝑎 0,𝑗 (𝑏 𝑗 ),

𝑗=1

where the operators 𝑎 𝑖,𝑗 are the entries of the matrix representation of 𝜇1M with respect to the splitting (2.30). By Lemma 5.3, h𝜇1 (𝑏0 ), 𝑞i = 0, hence there exists 1 ≤ 𝑗0 ≤ 𝑘 such that (5.15)



𝑎0,𝑗0 (𝑏 𝑗0 ), 𝑞 ≠ 0,

𝜈 h𝑎0,𝑗0 (𝑏 𝑗0 ), 𝑞i ≤ 𝜈(1) = 0.






Here 𝜈 is the standard valuation of Λ (see (3.11)). By Theorem 2.14 there exist chains 𝑐 𝑖0 ,𝑖00 ∈ CF(𝐿 𝑖0 , 𝐿 𝑖00 ; 𝑝), for all 𝑖 0 < 𝑖 00, with 𝐴(𝑐 𝑖0 ,𝑖00 ) ≤ 𝐶 (3) (𝑝, ℎ), where 𝐶 (3) (𝑝, ℎ) ∈ 𝑂(N) and such that 𝑎0,𝑗0 (𝑏 𝑗0 ) =

Õ

F𝑢𝑘( C;𝑝)

𝜇𝑑

(𝑏 𝑗0 , 𝑐 𝑖 𝑑 ,𝑖 𝑑−1 , . . . , 𝑐 𝑖2 ,𝑖1 ),

2≤𝑑, 𝑖

where 𝑖 = (𝑖 1 , . . . , 𝑖 𝑑 ) runs over all partitions 0 = 𝑖1 < 𝑖2 · · · < 𝑖 𝑑−1 < 𝑖 𝑑 = 𝑗0 . SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2021

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CHAPTER 5. PROOF OF THE MAIN GEOMETRIC STATEMENTS

It follows that there exists a partition 𝑖 0 = (𝑖10 , . . . , 𝑖 0𝑑 ) with 𝑑 ≥ 2, for which

D

F𝑢𝑘( C;𝑝)

𝜇𝑑

E

(𝑏 𝑗0 , 𝑐 𝑖 0 ,𝑖 0 , . . . , 𝑐 𝑖 0 ,𝑖 0 ), 𝑞 ≠ 0, 𝑑 𝑑−1

2 1



F𝑢𝑘( C;𝑝)

𝜈 𝜇𝑑

(𝑏 𝑗0 , 𝑐 𝑖 0 ,𝑖 0 , . . . , 𝑐 𝑖 0 ,𝑖 0 ), 𝑞 𝑑 𝑑−1



≤ 0.

2 1

Writing 𝑏 𝑗0 as a linear combination (over Λ) of elements from 𝐿0 ∩ 𝐿 𝑗0 and similarly for the 𝑐 𝑖 𝑟0 ,𝑖 0 ’s, we deduce that there exist 𝑥 ∈ 𝐿0 ∩ 𝐿 𝑗0 , 𝑃(𝑇) ∈ Λ, and 𝑧 𝑟 ∈ 𝐿 𝑖 𝑟0 ∩ 𝐿 𝑖 0 , 𝑟−1 𝑟−1 𝑄 𝑟 ∈ Λ for 𝑟 = 2, . . . , 𝑑, such that 𝐴(𝑃(𝑇)𝑥) ≤ 𝐶 (2) (𝑝, ℎ) + S(𝑊) + 21 𝜖, 𝐴(𝑄 𝑟 (𝑇)𝑧 𝑟 ) ≤ 𝐶 (3) (𝑝, ℎ),

D

for all 2 ≤ 𝑟 ≤ 𝑑,

F𝑢𝑘( C;𝑝) 𝜇𝑑 (𝑃(𝑇)𝑥, 𝑄 𝑑 (𝑇)𝑧 𝑑 , . . . , 𝑄 2 (𝑇)𝑧 2 ),



F𝑢𝑘( C;𝑝)

𝜈 𝜇𝑑

E

𝑞 ≠ 0,

(𝑃(𝑇)𝑥, 𝑄 𝑑 (𝑇)𝑧 𝑑 , . . . , 𝑄 2 (𝑇)𝑧2 ), 𝑞



≤ 0.

Note that 𝑑, as well as the points 𝑥, 𝑧 𝑑 , . . . , 𝑧 2 , 𝑞, all depend on (𝑝, ℎ), but for the moment we suppress this from the notation. Since 𝐴(𝑃(𝑇)𝑥) = −𝜈(𝑃(𝑇)) and 𝐴(𝑄 𝑟 (𝑇)𝑧 𝑟 ) = −𝜈(𝑄 𝑟 (𝑇)) we obtain (5.16)



F𝑢𝑘( C;𝑝)

𝜈 𝜇𝑑

(𝑥, 𝑧 𝑑 , . . . , 𝑧 2 ), 𝑞



≤ S(𝑊) + 12 𝜖 + 𝐶 (4) (𝑝, ℎ),

where 𝐶 (4) (𝑝, ℎ) ∈ 𝑂(N). 00 for Denote by D(𝑝) = (𝐾(𝑝), 𝐽(𝑝)) the perturbation datum prescribed by 𝑝 ∈ 𝐸reg the tuple of Lagrangians (𝐿0 , 𝐿 𝑗0 , 𝐿 𝑖 𝑑−1 , . . . , 𝐿 𝑖2 , 𝐿0 ). It follows from (5.16) that there exists a non-constant Floer polygon 𝑢 ∈ M(𝑥, 𝑧 𝑑 , . . . , 𝑧 2 , 𝑞; D(𝑝)) with 𝜔(𝑢) ≤ S(𝑊) + 21 𝜖 + 𝐶 (4) (𝑝, ℎ). Let 𝑝0 ∈ N be any choice of perturbation data which assigns to the tuple of Lagrangians (𝐿0 , 𝐿 𝑗0 , 𝐿 𝑖 𝑑−1 , . . . , 𝐿 𝑖2 , 𝐿0 ) the perturbation data D(𝑝0 ) = (𝐾 = 0, 𝐽(𝑝0 )), where 𝐽(𝑝 0 ) is a family of almost complex structures that coincide with 𝐽 𝐵 on 𝐵 (see (5.8)). Fix a generic 𝐶 1 -small Morse function 𝑓 as on page 88. We now choose a sequence 00 with (𝑝 , ℎ ) → (𝑝 , 0) as 𝑛 → ∞, and with the following additional {(𝑝 𝑛 , ℎ 𝑛 )} in 𝐸reg 𝑛 𝑛 0 property. The Hamiltonian function 𝐻 𝐿0 ,𝐿0 (𝑛) prescribed by 𝑝 𝑛 for the Floer datum D𝐿0 ,𝐿0 (𝑝 𝑛 ) of (𝐿0 , 𝐿0 ) is 𝐻 𝐿1 0 ,𝐿0 , i.e. constructed as on page 88 but with the function 𝑛1 𝑓 𝑛

𝑓

instead of 𝑓 . Consequently, the point 𝑞 (the maximum of 𝑛1 𝑓 ) does not depend on 𝑛. Passing to a subsequence of {(𝑝 𝑛 , ℎ 𝑛 )} if necessary we may assume that both 𝑑 as well as the points 𝑥, 𝑧 𝑑 , . . . , 𝑧 2 above do not depend on 𝑛 either. (Note that by Theorem 2.14, 𝑑 ≤ 𝑘, so there are only finitely many possible values for 𝑑.) In summary, we obtain a sequence 𝑢𝑛 ∈ M(𝑥, 𝑧 𝑑 , . . . , 𝑧 2 , 𝑞; D(𝑝 𝑛 )) with 𝜔(𝑢𝑛 ) ≤ S(𝑊) + 12 𝜖 + 𝐶 (4) (𝑝 𝑛 , ℎ 𝑛 ).

ASTÉRISQUE 426

5.1. PROOF OF THEOREM 5.1

93

By a compactness result [OZ], [OZ11] (see also [FO97], [Oh96b], [Oh96a]) there exists a subsequence of {𝑢𝑛 } which converges to a union of Floer polygons 𝑣0 , 𝑣1 , . . . , 𝑣 𝑙 , 𝑙 ≥ 0, together with a (possibly broken) negative gradient trajectory 𝜂 of 𝑓 . 9 The Floer polygons 𝑣 𝑖 map the boundary components of their domains of definition to some of the Lagrangians in the collection 𝐿0 , 𝐿 𝑗0 , 𝐿 𝑖 𝑑−1 , . . . , 𝐿 𝑖2 , 𝐿0 . Moreover, 𝑣0 maps one of its boundary components to 𝐿0 . The maps 𝑣 𝑖 satisfy the Floer equation corresponding to the perturbation data prescribed by 𝑝0 . Consequently they are all genuine pseudo-holomorphic (i.e. without Hamiltonian perturbations) with respect to the (domain-dependent) almost complex structures prescribed by 𝑝0 . In particular, one has 𝜔(𝑣 𝑖 ) ≥ 0 for every 𝑖. Í As 𝜔(𝑢𝑛 ) ≤ S(𝑊) + 12 𝜖 + 𝐶 (4) (𝑝 𝑛 , ℎ 𝑛 ) for every 𝑛, we have 𝑙𝑖=0 𝜔(𝑣 𝑖 ) ≤ S(𝑊) + 12 𝜖, hence (5.17)

𝜔(𝑣0 ) ≤ S(𝑊) + 21 𝜖.

The other part of the limit of {𝑢𝑛 }, namely the negative gradient trajectory 𝜂 of 𝑓 , emanates from an 𝐿0 -boundary point of one of the polygons, say 𝑣0 , and ends at the point 𝑞. Consider now 𝑣0 and 𝜂. Note that 𝜂 must be the constant trajectory at the point 𝑞 since it goes into 𝑞 which is a maximum of 𝑓 . It follows that the polygon 𝑣0 passes (along its boundary) through the point 𝑞. We now appeal to the special form of 𝐽(𝑝0 ) over the ball 𝐵. Recall that 𝑣0 is 𝐽(𝑝0 )-holomorphic. Thus restricting 𝑣0 to the subdomain (of its definition) which is mapped to Int (𝐵) we obtain a proper 𝐽 𝐵 -holomorphic curve 𝑣00 parametrized by a noncompact Riemann surface with one boundary component. Moreover that boundary component is mapped to 𝐵∩𝐿0 , and 𝑞 ∈ 𝐵 which is the center of the ball is in the image of that boundary component. Passing to the standard ball 𝐵(𝑟) via the symplectic embedding 𝑒 mentioned in (5.8) we obtain from 𝑣00 a proper 𝐽std -holomorphic curve 𝑣000 inside 𝐵(𝑟) which passes through 0 and its boundary is mapped to 𝐵ℝ (𝑟) ⊂ 𝐵2𝑛 (𝑟). Applying a reflection along ℝ 𝑛 × 0 to 𝑣 000, and gluing the result to 𝑣000 we obtain a proper 𝐽std -holomorphic curve (without boundary) e 𝑣000 in Int 𝐵2𝑛 (𝑟) which passes through 0. By the Lelong inequality we have 𝜋𝑟 2 ≤ 𝜔std ( e 𝑣000). Putting everything together we obtain 𝛿(𝐿0 , 𝑆) − 𝜖 ≤ 𝜋𝑟 2 ≤ 𝜔( e 𝑣000) = 2𝜔(𝑣000) ≤ 2𝜔(𝑣0 ) ≤ 2S(𝑊) + 𝜖. Since this inequality holds for all 𝜖 > 0 the desired inequality (5.3) follows. Proof of Lemma 5.3. — Recall that the Hamiltonian function in the Floer data D𝐿0 ,𝐿0 of (𝐿0 , 𝐿0 ) is 𝐻 𝐿𝑓 0 ,𝐿0 and we have O(𝐻 𝐿𝑓 0 ,𝐿0 ) = Crit( 𝑓 ). Let 𝑢 : ℝ × [0, 1] → 𝑀 be a Floer strip connecting 𝑥 − to 𝑥+ , where 𝑥± ∈ Crit( 𝑓 ). Identifying (𝐷 \ {−1, +1}, 𝜕𝐷 \ {−1, +1}) with (ℝ × [0, 1], ℝ × {0} ∪ ℝ × {1}) we obtain from 𝑢 a map 𝑢 0 : (𝐷 \ {−1, +1}, 𝜕𝐷 \ {−1, +1}) → (𝑀, 𝐿0 ) that extends continuously 9. “Broken” means that the trajectory might pass through several critical points of 𝑓 .

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to a map 𝑢¯ 0 : (𝐷, 𝜕𝐷) → (𝑀, 𝐿0 ). Since 𝐿0 is weakly exact we have 𝜔(𝑢¯ 0) = 0, hence 𝜔(𝑢) = 0. By (3.13) it follows that 𝑓 (𝑥− ) = 𝑓 (𝑥+ ) + 𝐸(𝑢), where 𝐸(𝑢) is the energy of 𝑢 (see (3.2)). As 𝐸(𝑢) ≥ 0 we have 𝑓 (𝑥− ) ≥ 𝑓 (𝑥+ ) with equality iff 𝐸(𝑢) = 0. Suppose by contradiction that h𝜇1 (𝑥), 𝑞i ≠ 0 for some 𝑥 ∈ Crit( 𝑓 ). Let 𝑢 be a Floer strip that contributes to 𝜇1 (𝑥) and connects 𝑥 to 𝑞. By the above, we have 𝑓 (𝑥) ≥ 𝑓 (𝑞). Since 𝑞 is the unique maximum of 𝑓 it follows that 𝑥 = 𝑞. Moreover, 𝐸(𝑢) = 0. The latter implies that 𝜕𝑠 𝑢 ≡ 0. But this can happen only if 𝑢 is the constant strip at 𝑞 which contradicts the fact that 𝑢 contributes to 𝜇1 (𝑥).  To complete the proof of inequality (5.3) it remains only to treat the case when the Lagrangians 𝐿0 , 𝐿1 , . . . , 𝐿 𝑘 do not intersect pairwise transversely. Fix 𝜖 > 0. We apply 𝑘 Hamiltonian isotopies, one to each Lagrangian 𝐿 𝑖 , 1 ≤ 𝑖 ≤ 𝑘, such that the following holds: 1) The images 𝐿01 , . . . , 𝐿0𝑘 of 𝐿1 , . . . , 𝐿 𝑘 after these isotopies are such that 𝐿0 , 𝐿01 , . . . , 𝐿0𝑘 intersect pairwise transversely. 2) The Hofer length of each of these isotopies is ≤ 𝜖/𝑘. 3) 𝛿(𝐿0 ; 𝑆) − 𝜖 ≤ 𝛿(𝐿0 ; 𝑆0), where 𝑆0 = 𝐿01 ∪ · · · ∪ 𝐿0𝑘 . (𝐿1 , . . . , 𝐿 𝑘 ) be a weakly exact cobordism. We now glue to each of Let 𝑉 : 𝐿0 the negative ends 𝐿 𝑖 of 𝑉 the Lagrangian suspension associated to the preceding Hamiltonian isotopy used to move 𝐿 𝑖 to 𝐿0𝑖 . The result is a new cobordism 𝑉 0 : 𝐿0 (𝐿01 , . . . , 𝐿0𝑘 ) whose shadow satisfies S(𝑉 0) ≤ S(𝑉) + 𝜖. Since the ends of 𝑉 0 intersect pairwise transversely it follows from what we have already proved that S(𝑉 0) ≥ 12 𝛿(𝐿0 ; 𝑆0). Therefore: 1 2 𝛿(𝐿0 ; 𝑆)

− 12 𝜖 ≤ 12 𝛿(𝐿0 ; 𝑆0) ≤ S(𝑉 0) ≤ S(𝑉) + 𝜖.

As this holds for all 𝜖 > 0 the result readily follows. This completes the proof of inequality (5.3). We now turn to the proofs of the other two statements of Theorem 5.1. 5.1.1. Proof of statement (5.1). — As in the previous part of the proof, we first assume that the Lagrangians 𝐿1 , . . . , 𝐿 𝑘 intersect pairwise transversely. Fix 𝜖 > 0 small enough such that 0

S(𝑉) + 𝜖 < 21 𝛿Σ (𝑁 ∪ 𝑆) − 12 𝜖.

(5.18) Fix also 𝑟 > 0 with 0

(5.19)

0

𝛿Σ (𝑁 ∪ 𝑆) − 𝜖 ≤ 𝜋𝑟 2 < 𝛿Σ (𝑁 ∪ 𝑆).

Write Σ0 = 𝑁 ∩𝑆 = {𝑥 1 , . . . , 𝑥 𝑚 } for the double points of 𝑁 ∪𝑆, and let 𝑒 𝑥 𝑖 : 𝐵(𝑟) → 𝑀, 𝑖 = 1, . . . , 𝑚, be a collection of symplectic embeddings with the properties as in the 0 definition of 𝛿Σ on page 85 (we take 𝕃 = 𝑁 ∪ 𝑆 and 𝑄 = ∅ in that definition). Ð𝑚 Denote 𝐵 := 𝑖=1 image (𝑒 𝑥 𝑖 ) and let 𝐽 𝐵 be the complex structure on 𝐵 whose value on image (𝑒 𝑥 𝑖 ) is the push forward (𝑒 𝑥 𝑖 )∗ (𝐽std ) of the standard complex structure 𝐽std of 𝐵(𝑟) via the map 𝑒 𝑥 𝑖 .

ASTÉRISQUE 426

5.1. PROOF OF THEOREM 5.1

95

Figure 6. The curves 𝛾 and 𝛾0 and the cobordism 𝑉.

We consider now two curves 𝛾, 𝛾0 of the the same shape as in the earlier part of the proof (see Figure 6) and such that (similarly to (5.7)) there exists a Hamiltonian isotopy, horizontal at infinity, 𝜑𝑡 : ℝ2 → ℝ2 , 𝑡 ∈ [0, 1], with 𝜑0 = id, 𝜑1 (𝛾) = 𝛾0 and with (5.20) length{𝜑𝑡 } ≤ S(𝑉) + 12 𝜖. Next we set up the Fukaya categories involved in the proof. Let C be the collection of Lagrangians 𝐿1 , . . . , 𝐿 𝑘 , 𝐿. We will work with the Fukaya F𝑢𝑘( C; 𝑝) defined with choices of perturbation data 𝑝 with the following restrictions. The Floer data of (𝑁 , 𝐿 𝑖 ), prescribed by 𝑝, are of the type D𝑁 ,𝐿𝑖 = (𝐻 𝑁 ,𝐿𝑖 = 0, 𝐽(𝑝)), where 𝐽(𝑝) = {𝐽𝑡 (𝑝)} is a family of almost complex structures such that 𝐽𝑡 (𝑝) 𝐵 = 𝐽 𝐵 for all 𝑡. The Floer data D𝐿𝑖 ,𝐿 𝑗 𝑖 ≠ 𝑗 have the 0 Hamiltonian function. Finally, the perturbation data D𝑁 ,𝐿𝑖 𝑑 ,...,𝐿𝑖1 , 𝑑 ≥ 2, all have vanishing Hamiltonian form, i.e. they are of the type (𝐾 = 0, 𝐽). Due to the assumption that 𝑁 , 𝐿1 , . . . , 𝐿 𝑘 intersect pairwise transversely, regular choices of perturbation data with the above properties do exist. We denote the space of such 00 . (It is important to note that the restriction that 𝐽 (𝑝) 𝐵 regular choices by 𝐸reg 𝑡 𝐵 = 𝐽 for every 𝑡 does not pose any regularity problem since every Floer strip or polygon relevant for the definition of F𝑢𝑘( C; 𝑝) cannot have its image lying entirely inside 𝐵, and outside of 𝐵 we have not posed any restrictions on the choice of almost complex structures.) We set up the Fukaya categories F𝑢𝑘 cob ( e C1/2 , 𝜄 𝛾 (𝑝, ℎ)) and F𝑢𝑘 cob ( e C1/2 , 𝜄 𝛾0 (𝑝, ℎ)) and the associated inclusion functors in the same way as in the previous part of the proof. Let V, V0 be the Yoneda modules corresponding to 𝑉, one time viewed as an object 𝑉 ∈ Ob(F𝑢𝑘 cob ( e C1/2 ; 𝜄 𝛾 (𝑝, ℎ))) and one time as 𝑉 ∈ Ob(F𝑢𝑘 cob ( e C1/2 ; 𝜄 𝛾0 (𝑝, ℎ))). Consider the pull-back modules ∗ M𝑉;𝛾,𝑝,ℎ := I𝛾;𝑝,ℎ V,

∗ M𝑉;𝛾0 ,𝑝,ℎ := I𝛾;𝑝,ℎ V0 .

By Proposition 3.5 (and (3.32)) we have 𝜙𝑘

𝜙 𝑘−1

(5.21) 𝑆 𝑠 ℎ M𝑉;𝛾,𝑝,ℎ = C𝑜𝑛𝑒 L𝑘 −−→ C𝑜𝑛𝑒 L𝑘−1 −−−−→ 𝜙2



C𝑜𝑛𝑒 · · · C𝑜𝑛𝑒 L2 −−→ L1 ··· ,

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where 𝑠 ℎ → 0 as ℎ → 0, and similarly to what we have had on page 90, 𝜙¯ 𝑖 = (𝜙 𝑖 , 0, δ (𝒊) ), with 𝜙 𝑖 a homomorphism of modules that shifts action by ≤ 0 and has discrepancy ≤ δ(𝑝, ℎ), where for every 𝑑 we have 𝛿 𝑑 (𝑝, ℎ) ∈ 𝑂(N). By similar 0 arguments, 𝑆 𝑠 ℎ M𝑉;𝛾0 ,𝑝,ℎ = L, where L is the Yoneda module of 𝐿, and 𝑠 0ℎ → 0 as ℎ → 0. Consider now the chain complexes C𝑝,ℎ := M𝑉;𝛾,𝑝,ℎ (𝑁),

0 C𝑝,ℎ := M𝑉;𝛾0 ,𝑝,ℎ (𝑁)

endowed with the differential coming from the 𝐴∞ -modules M𝑉;𝛾,𝑝,ℎ , M𝑉;𝛾0 ,𝑝,ℎ . 10 By definition C𝑝,ℎ = CF(𝛾 × 𝑁 , 𝑉; D𝛾×𝑁 ,𝑉 ), where D𝛾×𝑁 ,𝑉 is the Floer datum prescribed by 𝜄 𝛾 (𝑝, ℎ). Similarly 0 C𝑝,ℎ = CF(𝛾0 × 𝑁 , 𝑉; D𝛾0 ×𝑁 ,𝑉 ),

where D𝛾0 ×𝑁 ,𝑉 is the Floer datum prescribed by 𝜄 𝛾0 (𝑝, ℎ). Consider now the Hamiltoe𝑡 := 𝜑𝑡 × id : ℝ2 × 𝑀 → ℝ2 × 𝑀, 𝑡 ∈ [0, 1], where 𝜑𝑡 is the Hamilnian isotopy 𝜑 e𝑡 } is horizontal at infinity and by (5.20) tonian isotopy from page 95. Note that { 𝜑 1 e1 (𝛾 × 𝑁) = 𝛾0 × 𝑁, by standard Floer theory has Hofer length ≤ S(𝑉) + 2 𝜖. Since 𝜑 (see e.g. [FOOO09a, Chapter 5]) this isotopy induces two chain maps 0 𝜙 : C𝑝,ℎ −→ C𝑝,ℎ ,

0 𝜓 : C𝑝,ℎ −→ C𝑝,ℎ ,

which are both filtered and such that 𝜓 ◦ 𝜙 is chain homotopic to id by a chain homotopy that shifts action by ≤ S(𝑉) + 𝜖. More specifically 𝜓 ◦ 𝜙 = id +𝐾𝑑 C𝑝,ℎ + 𝑑 C𝑝,ℎ 𝐾, where 𝑑 C𝑝,ℎ is the differential of C𝑝,ℎ and 𝐾 is a Λ-linear map that shifts action by ≤ S(𝑉) + 𝜖. Using the formalism of (2.44) this means that (5.22)

𝐵 ℎ (𝜓 ◦ 𝜙 − id) ≤ S(𝑉) + 𝜖.

We now appeal to Theorem 2.14, by which we obtain the following: ⊲ A chain complex M(𝑁) whose underlying Λ-module coincides with C𝑝,ℎ and whose differential 𝜇1M(𝑁) is described by (2.31) from Theorem 2.14. An isomorphism of chain complexes 𝜎1 : M(𝑁) → C𝑝,ℎ such that both 𝜎1 and its inverse 𝜎1−1 : C𝑝,ℎ → M(𝑁) shift action by ≤ 𝐶 (1) (𝑝, ℎ), where 𝐶 (1) (𝑝, ℎ) ∈ 𝑂(N). ⊲

We now estimate the action drop 𝛿 𝜇1M(𝑁) (as defined in (2.42)) of the differential of the chain complex M(𝑁). By Theorem 2.14 the differential 𝜇1M(𝑁) comprises various 𝜇𝑑 -operations, 1 ≤ 𝑑 ≤ 𝑘, associated to tuples of Lagrangians of the type (𝑁 , 𝐿 𝑖 𝑑 , 𝐿 𝑖 𝑑−1 , . . . , 𝐿 𝑖2 , 𝐿 𝑖1 ), where 𝑖 = 𝑖 1 < · · · < 𝑖 𝑑 ≤ 𝑗, 1 ≤ 𝑖 ≤ 𝑗 ≤ 𝑘. Recall also that 00 were chosen with vanishing Hamiltonian perturbation the perturbation data 𝑝 ∈ 𝐸reg for tuple of Lagrangians as above. Therefore, the above mentioned 𝜇𝑑 -operations are defined by counting (unperturbed) pseudo-holomorphic polygons 𝑢 with corners mapped to intersection points between consecutive pairs of Lagrangians in tuples as above. 𝜇1M(𝑁)

10. Note that C𝑝,ℎ defined here is different than the C𝑝,ℎ from page 90.

ASTÉRISQUE 426

97

5.1. PROOF OF THEOREM 5.1

Each polygon 𝑢 contributing to these 𝜇𝑑 -operations has an intersection point in 𝑁 ∩ 𝐿 𝑗 as one of its inputs and an intersection point in 𝑁 ∩ 𝐿 𝑖 as its output. Moreover, these polygons are 𝐽 𝐵 -holomorphic over 𝐵. We thus obtain 𝜔(𝑢) ≥ 𝜔(image (𝑢) ∩ 𝐵) ≥ 14 𝜋𝑟 2 + 41 𝜋𝑟 2 = 12 𝜋𝑟 2 , where the first inequality hold because 𝑢 is unperturbed-pseudo-holomorphic over its entire domain, while the second inequality follows from a Lelong-inequality type of argument (see e.g. [BC07], [BC06]). Combining the preceding inequalities with (5.19) we deduce that every Floer polygon 𝑢 that participate in the calculation of the differ0 ential 𝜇1M(𝑁) must satisfy 𝜔(𝑢) ≥ 12 𝛿Σ (𝑁 ∪ 𝑆) − 12 𝜖. It follows that 0

𝛿𝜇1M(𝑁) ≥ 12 𝛿Σ (𝑁 ∪ 𝑆) − 12 𝜖.

(5.23)

In view of the map 𝜎1 and its inverse 𝜎1−1 , mentioned earlier in the proof, we deduce the following estimate for the action drop of the differential of C𝑝,ℎ : 0

𝛿 𝑑 C𝑝,ℎ ≥ 21 𝛿Σ (𝑁 ∪ 𝑆) − 12 𝜖 − 2𝐶 (1) (𝑝, ℎ). 00 close enough to N and the profile funcAs 𝐶 (1) (𝑝, ℎ) ∈ 𝑂(N), by choosing 𝑝 ∈ 𝐸reg tion ℎ small enough, we may assume in view of (5.18) that 1 Σ0 2 𝛿 (𝑁

∪ 𝑆) − 21 𝜖 − 2𝐶 (1) (𝑝, ℎ) > S(𝑉) + 𝜖.

Combining the above together with (5.22) we obtain 𝛿 𝑑 C𝑝,ℎ > S(𝑉) + 𝜖 ≥ 𝐵 ℎ (𝜓 ◦ 𝜙 − id). By Lemma 2.15 (applied with 𝐶 = C𝑝,ℎ , 𝑓 = 𝜓 ◦ 𝜙, 𝑔 = id) we deduce that 𝜓 ◦ 𝜙 is 0 0 is injective too, hence dim C injective. It follows that 𝜙 : C𝑝,ℎ → C𝑝,ℎ Λ 𝑝,ℎ ≤ dimΛ C𝑝,ℎ . But 𝑘 Ê Ê Ê 0 C𝑝,ℎ =

Λ · 𝑥,

C𝑝,ℎ =

𝑖=1 𝑥∈𝑁∩𝐿 𝑖

Λ · 𝑥,

𝑥∈𝑁∩𝐿

which implies the desired inequality (5.4). This completes the proof of statement (5.1) under the additional assumption that 𝐿1 , . . . , 𝐿 𝑘 intersect pairwise transversely. It remains to treat the case when the Lagrangians 𝐿1 , . . . , 𝐿 𝑘 do not necessarily intersect pairwise transversely. Let 𝑉 be a cobordism as in the statement of the theorem. Fix 𝑟 > 0 and 𝜖 > 0 with 0

(5.24)

S(𝑉) + 𝜖 < 𝜋𝑟 2 < 𝛿Σ (𝑁 ∪ 𝑆).

Let 𝑒 𝑥 : 𝐵(𝑟) → 𝑀, 𝑥 ∈ Σ0 = 𝑁 ∩ 𝑆, be a collection of symplectic embeddings as in Ð𝑘 0 the definition of 𝛿Σ (𝑁 ∪ 𝑆) on page 85. Since Σ0 = 𝑖=1 (𝑁 ∩ 𝐿 𝑖 ) and the latter union is disjoint every 𝑥 ∈ Σ0 belongs to precisely one of the Lagrangians 𝐿1 , . . . , 𝐿 𝑘 . Now let 𝑦 ∈ Σ0 and assume that 𝑦 ∈ 𝑁 ∩ 𝐿 𝑖 . Let 𝑗 ≠ 𝑖. It is easy to see from the assumptions 0 imposed on the embeddings 𝑒 𝑥 in the definition of 𝛿Σ (𝑁 ∪ 𝑆) that 𝐿 𝑗 ∩ 𝑒 𝑦 𝐵(𝑟) = ∅.




In particular Ð 𝐿 𝑗 ∩ 𝐿 𝑖 lies outside of 𝑒 𝑦 (𝐵(𝑟)). It follows that of 𝐵 := 𝑥∈Σ0 image 𝑒 𝑥 (𝐵(𝑟)).

Ð

𝑖 0 0 small enough and 𝑟 > 0 so that 00

(5.25)

S(𝑉) + 𝜖 < 41 𝛿Σ (𝑆; 𝑁) − 14 𝜖,

00

00

𝛿Σ (𝑆; 𝑁) − 𝜖 ≤ 𝜋𝑟 2 < 𝛿Σ (𝑆; 𝑁).

Next, fix symplectic embeddings 𝑒 𝑥 : 𝐵(𝑟) → 𝑀, 𝑥 ∈ Σ00, as in the definition of 00 𝛿Σ (𝑆; 𝑁) on page 85. Fix also curves 𝛾, 𝛾0 as in the proof of statement (5.1). We set up the Fukaya categories F𝑢𝑘( C; 𝑝), F𝑢𝑘 cob ( e C1/2 , 𝜄 𝛾 (𝑝, ℎ)), F𝑢𝑘 cob ( e C1/2 , 𝜄 𝛾0 (𝑝, ℎ)) and 0 the inclusion functors I𝛾;𝑝,ℎ , I𝛾 ;𝑝,ℎ , in the same way as in the proof of statement (5.1). 0 , and the two chain maps 𝜙 : C We then define the chain complexes C𝑝,ℎ , C𝑝,ℎ 𝑝,ℎ → 0 0 C𝑝,ℎ , 𝜓 : C𝑝,ℎ → C𝑝,ℎ , with (5.26)

𝜓 ◦ 𝜙 = id +𝐾 ◦ 𝑑 C𝑝,ℎ + 𝑑 C𝑝,ℎ ◦ 𝐾,

where 𝐾 shifts action by ≤ S(𝑉) + 𝜖. As before, we now use Theorem 2.14 and obtain a chain complex M(𝑁) whose underlying Λ-module coincides with C𝑝,ℎ and equals

(5.27)

M(𝑁) =

𝑘 Ê 𝑖=1

ASTÉRISQUE 426

CF(𝑁 , 𝐿 𝑖 ; D𝑁 ,𝐿𝑖 ).

99

5.1. PROOF OF THEOREM 5.1

M(𝑛)

By Theorem 2.14 the differential 𝜇1 can be written with respect to the splitting (5.27) as an upper triangular matrix of operators (𝑎 𝑖,𝑗 ) with diagonal elements CF(𝑁 ,𝐿 𝑖 ; D𝑁 ,𝐿 𝑖 )

𝑎 𝑖,𝑖 = 𝜇1 Write

𝜇1M(𝑁)

⊲ 𝑑0 =

.

= 𝑑0 + 𝑑1 , where: CF(𝑁 ,𝐿 𝑖 ; D𝑁 ,𝐿 𝑖 ) 𝑖=1 𝜇1

É𝑘

with respect to (5.27) and

⊲ 𝑑1 : M(𝑁) → M(𝑁) is the operator represented by the part of the matrix (𝑎 𝑖,𝑗 ) that lies strictly above the diagonal. The operator 𝑑1 consists of sums of 𝜇𝑑 -operations, 𝑑 ≥ 2, where among the inputs of each such operation there is at least one point from 𝐿 𝑖 ∩𝐿 𝑗 , 𝑖 < 𝑗. A similar argument to the one used on page 96 in estimating 𝛿𝜇1M(𝑁) in the proof of statement (5.1) shows that 𝛿 𝑑1 ≥ 14 𝜋𝑟 2 . Here, 𝛿 𝑑1 is the action drop of 𝑑1 (see Section 2.7, page 44). Combining with (5.25) we get 00

𝛿 𝑑1 ≥ 41 𝛿Σ (𝑆; 𝑁) − 41 𝜖.

(5.28)

Put 𝑓 0 := 𝜓 ◦ 𝜙 : C𝑝,ℎ → C𝑝,ℎ . By (5.26) we have 𝐵 ℎ ( 𝑓 0 − id) ≤ S(𝑉) + 𝜖. Recall from Theorem 2.14 the isomorphism of chain complexes 𝜎1 : C𝑝,ℎ → M(𝑁) such that both 𝜎1 and its inverse 𝜎1−1 shift action by ≤ 𝐶 (1) (𝑝, ℎ), where 𝐶 (1) (𝑝, ℎ) ∈ 𝑂(N). Consider 𝑓 := 𝜎1 ◦ 𝑓 0 ◦ 𝜎1−1 : M(𝑁) −→ M(𝑁). We would like to apply Proposition 2.18 to 𝐶 = M(𝑁), 𝑑0 , 𝑑1 as defined above and the map 𝑓 . We have 𝑓 − id = (𝜎1 ◦ 𝐾 ◦ 𝜎1−1 ) ◦ 𝑑 C𝑝,ℎ + 𝑑 C𝑝,ℎ ◦ (𝜎1 ◦ 𝐾 ◦ 𝜎1−1 ), hence 𝐵 ℎ ( 𝑓 − id) ≤ S(𝑉) + 𝜖 + 2𝐶 (1) (𝑝, ℎ). As 𝐶 (1) (𝑝, ℎ) ∈ 𝑂(N), by taking 𝑝 close enough to Nand the profile function ℎ small enough, we may assume in view of (5.25) that 00 S(𝑉) + 𝜖 + 2𝐶 (1) (𝑝, ℎ) < 14 𝛿Σ (𝑆; 𝑁) − 41 𝜖. Together with (5.28) we now obtain 𝐵 ℎ ( 𝑓 − id) < 𝛿 𝑑1 .

(5.29)

In order to apply Proposition 2.18 it remains to check that (5.30)

dimΛ 𝐻∗ M(𝑁), 𝑑0 ≥ dimΛ 𝐻∗ M(𝑁), 𝜇1M(𝑁) .







This follows from standard results in homological algebra since 𝐻∗ M(𝑁), 𝑑0 =




𝑘 Ê

HF(𝑁 , 𝐿 𝑖 ),

𝐻∗ M(𝑁), 𝜇1M(𝑁)  𝐻∗ ( C𝑝,ℎ , 𝑑 C𝑃,ℎ )




𝑖=1

and C𝑝,ℎ is an iterated cone of the type C𝑝,ℎ = C𝑜𝑛𝑒 CF(𝑁 , 𝐿 𝑘 ) → C𝑜𝑛𝑒 CF(𝑁 , 𝐿 𝑘−1 ) → C𝑜𝑛𝑒 · · · C𝑜𝑛𝑒 CF(𝑁 , 𝐿2 ) → CF(𝑁 , 𝐿1 ) ··· .






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We are now in position to apply Proposition 2.18, by which we obtain dimΛ image ( 𝑓 ) ≥ dimΛ 𝐻∗ (M(𝑁), 𝑑0 ) =




𝑘 Õ

dimΛ HF(𝑁 , 𝐿 𝑖 ).

𝑖=1 𝑘 On the other hand dimΛ (image ( 𝑓 )) ≤ dimΛ M(𝑁) = 𝑖=1 #(𝑁 ∩ 𝐿 𝑖 ). Putting the last two inequalities together yields (5.5) and concludes the proof of statement (5.1). 

Í

5.1.3. Remark. — The following argument, due to Misha Khanevsky, leads to a more direct proof of an inequality as in the first part of Theorem 5.1 but gives a weaker estimate. We reproduce the argument here with Khanevsky’s permission. A result of Usher (Theorem 4.9 in [Ush14]) claims that, given two Lagrangians 𝑉 and 𝑉 0 that intersect transversely and non-trivially, there exists 𝛿 > 0 depending on 𝑉 and 𝑉 0, such that the energy (in the sense of Hofer geometry) required to disjoin 𝑉 from 𝑉 0 is greater than 𝛿. This result was proven for compact or (tame at infinity) symplectic manifolds but can be adjusted without any difficulty to the case of Lagrangians withÐ cylindrical ends in ℂ × 𝑀. Assume that 𝐿 ⊄ 𝑖 𝐿 𝑖 and let 𝑇 be a small Lagrangian torus, disjoint from all 𝐿 𝑖 ’s, and such that 𝑇 intersects 𝐿 transversely and non-trivially. Let 𝛾0 be a curve as in Figure 6 and let 𝑉 0 = 𝛾0 × 𝑇. Thus 𝑉 0 and 𝑉 intersect non-trivially and transversely (see also Figure 4). The isotopy Ψ taking the curve 𝛾 to the curve 𝛾0 in Figure 6 disjoins 𝑉 0 from 𝑉 and thus its energy 𝐸(Ψ) has to exceed 𝛿. At the same time, Ψ can be picked in such a way that 𝐸(Ψ) is as close as needed to S(𝑉) and thus we deduce the inequality S(𝑉) ≥ 𝛿 which finishes Khanevsky’s argument. However, notice that the dependence on 𝑇 of the constant 𝛿 here means that it is generally smaller than 𝛿(𝐿; 𝑆) from the statement of Theorem 5.1. Note also that this argument does not imply the points (5.1) and (5.1) of the statement and it also can not be adjusted to estimate the algebraic measurements that we will see later in Corollary 6.13.

5.2. Proof of Theorem 5.2 The proof of inequality (5.4) given in Section 5.1 carries over to the monotone case without any modifications. We now explain how to adjust the proof of (5.3) given in Section 5.1 in order to prove (5.6). We may assume throughout the proof that S(𝑉) < 𝐴𝐿 , for otherwise inequality (5.6) is trivially satisfied. We need to prove that S(𝑉) ≥ 21 𝛿(𝐿; 𝑆). We fix 𝜖 > 0 as in the proof of (5.3) but we require additionally that (5.31)

S(𝑊) + 𝜖 < 𝐴𝐿 .

The proof now goes along the same lines as the proof of (5.3), detailed in Section 5.1, up to the point where we had to use Lemma 5.3 (see page 91). That lemma does not hold in the monotone case, and we will now use the following lemma instead:

ASTÉRISQUE 426

101

5.3. THE QUASI-EXACT AND QUASI-MONOTONE CASES

Lemma 5.4. — Let 𝑐 ∈ Crit( 𝑓 ), viewed as an element of O(𝐻 𝐿𝑓 0 ,𝐿0 ).

CF(𝐿0 ,𝐿0 ;𝑝) 

CF(𝐿0 ,𝐿0 ;𝑝) 
If 𝜇1

(𝑐), 𝑞 ≠ 0 then 𝜈 𝜇1

(𝑐), 𝑞

≥ 𝐴 𝐿0 .

We postpone the proof for a while and continue with the proof of Theorem 5.2. As in the proof of Theorem 5.2 we decompose the element 𝑏 from (5.13) as 𝑏 = 𝑏0 + · · · + 𝑏 𝑘 with 𝑏 𝑖 ∈ CF(𝐿0 , 𝐿 𝑖 ; 𝑝). We cannot deduce that h𝜇1 (𝑏 0 ), 𝑞i = 0, as earlier. However by Lemma 5.4 and (5.31) we still obtain 𝜈 h𝜇1 (𝑏 0 ), 𝑞i ≥ 𝐴𝐿0 − S(𝑊) − 𝐶 (2) (𝑝, ℎ) − 12 𝜖 > 12 𝜖 − 𝐶 (2) (𝑝, ℎ),




where 𝐶 (2) (𝑝, ℎ) ∈ 𝑂(N). By taking 𝑝 close enough to 𝑝0 ∈ N and ℎ small enough we may assume that 𝜈(h𝜇1 (𝑏 0 ), 𝑞i) > 0. In view of (5.14) we can now deduce, as before, that there exists 1 ≤ 𝑗0 ≤ 𝑘 such that (5.15) holds. From this point on, the proof continues exactly as carried out in the weakly exact case in Section 5.1. It remains to prove the preceding lemma. Proof of Lemma 5.4. — Let 𝑢 ∈ M(𝑐, 𝑞; D𝐿0 ,𝐿0 ) be a Floer strip that goes from 𝑐 to 𝑞 and contributes to 𝜇1CF(𝐿0 ,𝐿0 ;𝑝) (𝑥). We need to show that 𝜔(𝑢) ≥ 𝐴𝐿0 . Indeed, as in the proof of Lemma 5.3 on page 91, after identifying

ℝ × [0, 1], ℝ × {0} ∪ ℝ × {1}




with

𝐷 \ {−1, +1}, 𝜕𝐷 \ {−1, +1}




the map 𝑢 extends continuously to a map 𝑢¯ : (𝐷, 𝜕𝐷) → (𝑀, 𝐿0 ). The dimension of the component of 𝑢 in the space M∗ (𝑐, 𝑞; D𝐿0 ,𝐿0 ) of non-parametrized Floer trajectories connecting 𝑐 to 𝑞 is given by dim M𝑢∗ (𝑐, 𝑞; D𝐿0 ,𝐿0 ) = |𝑐| − |𝑞| − 1 + 𝜇 [ 𝑢¯ ] = |𝑐| − 𝑛 − 1 + 𝜇 [ 𝑢¯ ] ,







¯ ∈ 𝐻2𝐷 (𝑀, 𝐿0 ) is the homology class induced where 𝜇 is the Maslov index and [ 𝑢] ∗ by 𝑢¯ . Since dim M𝑢 (𝑐, 𝑞; D𝐿0 ,𝐿0 ) ≥ 0 we must have ¯ ≥ 𝑛 + 1 − |𝑐| > 0. 𝜇 [ 𝑢]




By monotonicity of 𝐿0 we have 𝜔([ 𝑢¯ ]) ≥ 𝐴𝐿0 , hence 𝜔(𝑢) ≥ 𝐴𝐿0 . This concludes the proof of Lemma 5.4.  The proof of Theorem 5.2 is now complete.

5.3. The quasi-exact and quasi-monotone cases For the applications in Chapter 6, versions of Theorems 5.1 and 5.2 will be important for the quasi-exact and quasi-monotone cases. The definitions of these classes of Lagrangian cobordisms appear in Chapter 4 (see Definitions 4.2 and 4.4). We have the following generalization of Theorems 5.1 and 5.2 to the quasi-exact and quasi-monotone cases.

Theorem 5.5. — Let 𝐿, 𝐿1 , . . . , 𝐿 𝑘 be weakly exact Lagrangians (resp. monotone Lagran(𝐿1 , . . . , 𝐿 𝑘 ) a quasi-exact (resp. quasi-monotone) gians in L𝑎𝑔 mon,d (𝑀)) and 𝑉 : 𝐿 Lagrangian cobordism. Let 𝐾𝑉 ⊂ ℝ2 be a compact subset, homeomorphic to a closed 2-disk,

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CHAPTER 5. PROOF OF THE MAIN GEOMETRIC STATEMENTS

which is quasi-exact (resp. quasi-monotone) admissible for 𝑉. Then all the statements of Theorem 5.1 (resp. Theorem 5.2) continue to hold with S(𝑉) replaced by Area(𝐾𝑉 ). 5.3.1. Remarks 1) For 𝐾𝑉 as in the theorem we have Area(𝐾𝑉 ) ≥ S(𝑉). 2) If 𝑉 is a connected monotone cobordism then all its ends have the same Maslov-2 disk count: d 𝐿 = d 𝐿𝑖 = d 𝑉 . However, the latter is not clear if we only assume that 𝑉 is quasi-monotone. The issue is that we do not know whether there exists an almost complex structure 𝐽𝑉 for which both (𝐽𝑉 , 𝐾𝑉 ) is quasi-monotone admissible and in addition 𝐽𝑉 is regular for all 𝐽𝑉 -holomorphic disks of Maslov-2. A typical argument would be to perturb 𝐽𝑉 inside 𝐾𝑉 to achieve regularity and then try to argue by Gromov compactness that for a small enough perturbation 𝐽𝑉𝜖 all pseudoholomorphic disks have Maslov index ≥ 2. The problem with this approach is that as 𝜖 → 0 there might be 𝐽𝑉𝜖 -holomorphic disks 𝑢𝜖 with 𝜇(𝑢𝜖 ) ≤ 0 and with 𝜔(𝑢𝜖 ) → ∞, hence we cannot apply Gromov compactness to 𝑢𝜖 as 𝜖 → 0. For this reason, in Theorem 5.5 for the quasi-monotone case, we have assumed explicitly that all the Lagrangians 𝐿, 𝐿 𝑖 have the same Maslov-2 disk count d. Of course, in the monotone case, Theorem 5.2, this is not needed as it follows from the assumption that 𝑉 is monotone and connected. 5.3.2. Proof of Theorem 5.5. — In view of the theory developed in Chapter 4 (especially Propositions 4.3 and 4.6), the proof is essentially the same as the proofs of Theorems 5.1 and 5.2 as presented above. The main change is that the projection of the “non-cylindrical” part of 𝑉 should now be replaced by 𝐾𝑉 . Other than that, instead of working with the modules V, W, M𝑉;𝛾,𝑝,ℎ , M𝑊;𝛾,𝑝,ℎ one uses their quasi𝑞 𝑞 exact or quasi-monotone versions V𝑞 , W𝑞 , M𝑉;𝛾,𝑝,ℎ , M𝑊;𝛾,𝑝,ℎ , where 𝑞 stands for either 𝑞 = qe or 𝑞 = qm. 

ASTÉRISQUE 426

CHAPTER 6 METRICS ON SPACES OF LAGRANGIANS AND EXAMPLES

This chapter gives some context to the phenomena reflected in Theorem 5.1 and discusses a number of applications and ramifications. The first goal is to introduce metrics on the space of Lagrangian submanifolds that come from shadow measurements. Roughly speaking, our metrics will be defined by infimizing the shadow over all (multiply ended) Lagrangian cobordisms with two of their ends coinciding with two given Lagrangian submanifolds. As usual, the difficult part is in showing that this procedure leads to a non-degenerate measurement, and the main ingredient in establishing the non-degeneracy of our metrics will be Theorem 5.1 and its various versions. Of course, in order to obtain non-degeneracy we need to restrict the class of Lagrangians in 𝑀 and the class of cobordisms in ℝ2 ×𝑀 in our considerations. The two classes of Lagrangians (in 𝑀) that we will focus on, are weakly-exact Lagrangians and monotone ones. Naturally, we would like to use cobordisms of the same class (weakly-exact or monotone) in defining the metrics. However, here a new problem arises. In order to retain the triangle inequality for our metrics we need to infimize shadows over a class of Lagrangian cobordisms that is closed under composition (or gluing) of two cobordisms along a pair of matching ends. As it turns out, neither the class of weakly-exact cobordisms nor the class of monotone ones seems to enjoy this property (unless one imposes additional topological restrictions). It is at this point that we need to appeal to the more general class of quasi-exact and quasi-monotone cobordisms. The next section elaborates on this issue and how to solve it.

6.1. Setting up the right class of cobordisms Let: ∗ = we (weakly exact) or ∗ = (mon, d) (monotone with Maslov-2 disk count equal to d). Denote by L𝑎𝑔 ∗ (𝑀) the collection of Lagrangian submanifolds 𝐿 ⊂ 𝑀 of class ∗. Let 𝑄 be a class of Lagrangian cobordisms with ends in L𝑎𝑔 ∗ (𝑀).

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We will denote by L𝑎𝑔 𝑄,∗ (ℝ2 × 𝑀) the collection of Lagrangian cobordisms of class 𝑄 with ends in L𝑎𝑔 ∗ (𝑀). We say that 𝑄 (or L𝑎𝑔 𝑄,∗ (ℝ2 × 𝑀)) is closed under composition if for every two (𝐿1 , . . . , 𝐿𝑟 ) and 𝑊 : 𝐿 𝑗 (𝐾 1 , . . . , 𝐾 𝑠 ) in L𝑎𝑔 𝑄,∗ (ℝ2 × 𝑀) cobordisms 𝑉 : 𝐿 their composition along 𝐿 𝑗 , 𝑊 ◦𝑉 : 𝐿

(𝐿1 , . . . , 𝐿 𝑗−1 , 𝐾1 , . . . , 𝐾 𝑠 , 𝐿 𝑗+1 , . . . , 𝐿𝑟 ) ,

is also in L𝑎𝑔 𝑄,∗ (ℝ2 × 𝑀). Here and in what follows we always assume that the matching end 𝐿 𝑗 is connected. It is easy to see by an application of the Van Kampen theorem that if we consider 𝑊, 𝑉 monotone and both inclusions 𝑊 → 𝑀 and 𝑉 → 𝑀 are trivial in 𝜋1 , then 𝑊 ◦ 𝑉 is again monotone with 𝜋1 (𝑊 ◦ 𝑉) → 𝜋1 (𝑀) trivial (these are the assumptions in [BC14]). Moreover, if 𝑊, 𝑉 are weakly exact, connected cobordisms with a single positive end and a single negative end both connected, then 𝑊 ◦ 𝑉 is weakly exact. Indeed, by results in [BS19] a weakly exact simple cobordism 𝑉 has the property that the map induced on 𝜋1 by the inclusion of an end of 𝑉 into 𝑉 is epimorphic. Using this fact, the Van Kampen theorem again implies the claim. However, these conditions are quite restrictive and, without them, the class of weakly exact cobordisms generally seems not to be closed under composition, and the same for monotone cobordisms. At the same time we will see soon that the classes of quasi-exact cobordisms (with weakly exact ends) and the class of quasi-monotone cobordisms (with ends in L𝑎𝑔 mon,d (𝑀)) are closed under composition. This is the reason we will appeal to these classes of cobordisms. However, for our applications we will actually need to somewhat restrict the classes of quasi-exact and quasi-monotone cobordisms as follows.

Definition 6.1 (Tightly quasi-exact and quasi-monotone cobordisms). — Let 𝑉 ⊂ ℝ2 × 𝑀 be a quasi-exact (resp. quasi-monotone) cobordism with ends in L𝑎𝑔 we (𝑀) (resp. L𝑎𝑔 mon,d (𝑀)). In the “(mon, d)” case assume in addition that not all the ends of 𝑉 are void. We say that 𝑉 is tightly quasi-exact (resp. tightly quasi-monotone) if for every compact subset 𝐾𝑉 ⊂ ℝ2 , homeomorphic to a closed 2-disk, for which 𝑉 is cylindrical over ℝ2 \ Int (𝐾𝑉 ) (see Definition 4.1), there exists 𝐽𝑉 such that (𝐽𝑉 , 𝐾𝑉 ) is quasi-exact (resp. quasi-monotone) admissible (see Definitions 4.2 and 4.4). We will elaborate more on the reasons for introducing the classes of tightly quasiexact/quasi-monotone cobordisms in Remark 6.2.1 below. 6.1.1. Remarks 1) If 𝑉 is tightly quasi-exact, then for every 𝜖 > 0 there exists a quasi-exact admissible (𝐽𝑉 , 𝐾𝑉 ) with S(𝑉) ≤ Area(𝐾𝑉 ) ≤ S(𝑉) + 𝜖. The same holds also in the tightly quasi-monotone case. 2) Every weakly exact cobordism is tightly quasi-exact. As we will see below compositions of weakly exact cobordism along one pair of matching ends is tightly quasi-exact. A similar remark applies to quasi-monotone cobordisms.

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105

3) In principle it seems that the class of tightly quasi-exact (resp. quasi-monotone) cobordisms is smaller than the quasi-exact (resp. quasi-monotone) ones. However, we are not aware of any concrete examples of quasi-exact (resp. monotone) cobordisms that are not tightly quasi-exact (resp. quasi-monotone).

Proposition 6.2. — Each of the following classes 𝑄 of cobordisms is closed under composition: (i) Exact cobordisms with exact ends. (ii) Quasi-exact cobordisms with weakly exact ends. (iii) Quasi-monotone cobordisms with ends in L𝑎𝑔 mon,d (𝑀). (iv) Tightly quasi-exact cobordisms with weakly exact ends. (v) Tightly quasi-monotone cobordisms with ends in L𝑎𝑔 mon,d (𝑀). For the proof of this proposition we need the following variant of Lemma 4.5, which can be proved by similar way.

Lemma 6.3. — Let (𝑉 , 𝐽𝑉 , 𝐾𝑉 ) be quasi-exact with weakly exact ends (resp. quasi-monotone with ends in L𝑎𝑔 mon,d (𝑀)). Let 𝜎 : ℝ2 → ℝ2 be a diffeomorphism which coincides with the identity in a neighborhood of 𝐾𝑉 . Assume that 𝜎 sends the horizontal rays of 𝜋(𝑉) to other horizontal half-lines so that 𝑉 0 = (𝜎 × id)(𝑉) is also a cobordism. Under these assumptions, (𝑉 0 , 𝐽𝑉 , 𝐾𝑉 ) is also quasi-exact (resp. quasi-monotone). The proof is similar to the proof of Lemma 4.5. We now prove the previous proposition. Proof of Proposition 6.2. — Let 𝑉 : 𝐿 (𝐿1 , . . . , 𝐿𝑟 ) and 𝑊 : 𝐿 𝑗 (𝐾 1 , . . . , 𝐾 𝑠 ) be two cobordisms of class 𝑄. ⊲ Case “𝑄 = exact cobordisms with exact ends” Denote by e 𝜆 = 𝜆 ⊕ 𝜆ℝ2 the primitive of 𝜔 e = 𝜔 ⊕ 𝜔ℝ2 with respect to which we define exactness (here 𝜆 is the given primitive of 𝜔 and 𝑑𝜆ℝ2 = 𝜔ℝ2 ). Let 𝐹𝑉 : 𝑉 → ℝ and 𝐹𝑊 : 𝑊 → ℝ be primitives of e 𝜆 𝑉 and e 𝜆 𝑊 respectively. Let 𝛼 ≈ (−1, 1) be the 2 projection to ℝ of the neck of 𝑊 ◦ 𝑉 resulting from the gluing of 𝑊 and 𝑉 along their 𝐿 𝑗 -ends. By adding a suitable constant we can arrange that 𝐹𝑉 and 𝐹𝑊 agree along 𝛼 × 𝐿 𝑗 , hence e 𝜆 𝑊◦𝑉 is exact. (Note that 𝐿 𝑗 is connected, as by assumption composition of cobordisms is performed only along a pair of connected matching ends. See the beginning of Section 6.1.) ⊲ Case “𝑄 = quasi-exact” We first use Lemma 6.3 to rearrange conveniently the ends of 𝑉 to get a quasiexact cobordism 𝑉 0 whose ends are all positive except for a single negative end that coincides with 𝐿 𝑗 . We then glue 𝑉 to 𝑉 0 along the end 𝐿 𝑗 . More explicitly, we translate 𝑉 0 (together with the almost complex structure 𝐽𝑉 ) along the plane so that 𝐾𝑉 ⊂ 𝜋−1 ((1, ∞) × ℝ) and 𝑉 0 ∩ 𝜋−1 ([0, 1] × ℝ) = [0, 1] × {0} × 𝐿 𝑗 . Similarly, we translate 𝑊 (together with 𝐽𝑊 ) along the plane so that 𝐾𝑊 ⊂ 𝜋−1 ((−∞, −1) × ℝ) and

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𝑊 ∩ 𝜋−1 ([−1, 0] × ℝ) = [−1, 0] × {0} × 𝐿 𝑗 . We then define 𝑊 ◦ 𝑉 0 as the union 𝑊 ◦ 𝑉 0 = 𝑉 0 ∩ [0, ∞) × ℝ × 𝑀





∪ 𝑊 ∩ (−∞, 0] × ℝ × 𝑀 .





In the region (−1, 1) × ℝ × 𝑀 both almost complex structures 𝐽𝑉 and 𝐽𝑊 are fiberwise split so we can interpolate between corresponding fiber structures thus getting a new almost complex structure e 𝐽 that is split in the exterior of 𝐾𝑉 ∪ 𝐾𝑊 and coincides with 𝐽𝑉 on [1, ∞) × ℝ and with 𝐽𝑊 on (−∞, −1] × ℝ. The cobordism (𝑊 ◦ 𝑉 0 , e 𝐽 ) is quasi-exact by an immediate application of the open mapping theorem combined with the fact that 𝐿 𝑗 is weakly exact and that (𝑉 0 , 𝐽𝑉 ) and (𝑊 , 𝐽𝑊 ) are quasi-exact. Finally, we use Lemma 6.3 again to move the remaining 𝐿 𝑖 ends of 𝑊 ◦ 𝑉 0 to the left, thus getting that the cobordism 𝑊 ◦ 𝑉 in the statement is quasi-exact. The case of tight quasi-exact cobordisms follows immediately from the previous argument. Indeed, if 𝐾𝑊◦𝑉 ⊂ ℝ2 is a compact subset homeomorphic to a closed 2disk with 𝑊 ◦ 𝑉 being cylindrical over ℝ2 \ Int (𝐾𝑊◦𝑉 ), then Int (𝐾𝑊◦𝑉 ) must contain the (bounded!) plane curve forming the neck of the gluing of 𝑊 with 𝑉 along 𝐿 𝑗 . We now take two disjoint subsets 𝐾𝑉 , 𝐾𝑊 ⊂ 𝐾𝑊◦𝑉 each homeomorphic to a closed 2-disk and such that 𝑉 and 𝑊 are cylindrical over ℝ2 \ Int (𝐾𝑉 ) and over ℝ2 \ Int (𝐾𝑊 ) respectively. By the tightness assumption there exists 𝐽𝑉 and 𝐽𝑊 for which (𝑉 , 𝐽𝑉 , 𝐾𝑉 ) and (𝑊 , 𝐽𝑊 , 𝐾𝑊 ) are both quasi-exact. Let 𝑆 ⊂ ℝ2 be a thin strip (homeomorphic to [−1, 1] × [−𝜖, 𝜖]) that contains in its interior the neck of the gluing of W and V along 𝐿 𝑗 and such that 𝑆 ⊂ Int (𝐾𝑊◦𝑉 ). By positioning the strip 𝑆 appropriately we may assume that 𝐾𝑉 ∪ 𝐾𝑊 ∪ 𝑆 is homeomorphic to a closed 2-disk. By the previous argument (for “𝑄= quasi-exact”), 𝐽𝑉 𝐾𝑉 ×𝑀 and 𝐽𝑊 𝐾𝑊 ×𝑀 extend to an almost complex structure 𝐽𝑊◦𝑉 on ℝ2 × 𝑀 which makes (𝐽𝑊◦𝑉 , 𝐾𝑉 ∪ 𝐾𝑊 ∪ 𝑆) quasi-exact admissible for 𝑊 ◦ 𝑉. Since 𝐾𝑉 ∪ 𝐾𝑊 ∪ 𝑆 ⊂ 𝐾𝑊◦𝑉 , the pair (𝐽𝑊◦𝑉 , 𝐾𝑊◦𝑉 ) is also quasi-exact admissible for 𝑊 ◦ 𝑉. Finally, the proof of the statements in the quasi-monotone and tightly quasimonotone cases is the same as for the classes of quasi-exact and tightly quasi-exact cobordisms.  6.1.2. Remark. — Let 𝑉 : 𝐿 (𝐿1 , . . . , 𝐿𝑟 ) and 𝑊 : 𝐿 𝑗 quasi-exact cobordisms with weakly exact ends, and let 𝑊 ◦𝑉 : 𝐿

(𝐾 1 , . . . , 𝐾 𝑠 ) be two

(𝐿1 , . . . , 𝐿 𝑗−1 , 𝐾1 , . . . , 𝐾 𝑠 , 𝐿 𝑗+1 , . . . , 𝐿𝑟 )

be their composition along 𝐿 𝑗 , which by Proposition 6.2 is again quasi-exact. Let 𝐾𝑉 , 𝐾𝑊 ⊂ ℝ2 be compact subsets which are quasi-exact admissible for 𝑉 and 𝑊 respectively. The proof of Proposition 6.2 shows that for every 𝜖 > 0 there exists a 𝜖 compact subsets 𝐾𝑊◦𝑉 ⊂ ℝ2 which is quasi-exact admissible for 𝑊 ◦ 𝑉 and such that 𝜖 Area(𝐾𝑊◦𝑉 ) ≤ Area(𝐾𝑉 ) + Area(𝐾𝑊 ) + 𝜖.

6.2. Shadow metrics on spaces of Lagrangian submanifolds Let (𝑀, 𝜔) be a symplectic manifold. Fix a class L𝑎𝑔 ∗ (𝑀) of Lagrangian submanifolds of 𝑀, where ∗ can be either “we” (i.e. weakly exact) or “(mon, d)” (i.e. monotone

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with a fixed Maslov-2 disk count d, see Section 3.5). In case (𝑀, 𝜔 = d𝜆) is an exact symplectic manifold we allow also ∗ = ex, i.e. exact Lagrangians. In case ∗ = we, let 𝑄 be the class of Lagrangian cobordisms which are tightly quasi-exact, and in case ∗ = (mon, d) let 𝑄 be the class of tightly quasi-monotone Lagrangian cobordisms with ends in L𝑎𝑔 mon,d (𝑀). Finally, if ∗ = ex we can take 𝑄 to be either the class of exact Lagrangian cobordisms with exact ends or the class of quasi-exact cobordisms with exact ends. For the definition of exact cobordisms we fix a primitive 𝜆ℝ2 of 𝜔ℝ2 and take e 𝜆 = 𝜆 ⊕ 𝜆ℝ2 as the primitive of 𝜔 e for the purpose of defining exact cobordisms. Fix a family F ⊂ L𝑎𝑔 ∗ (𝑀) of Lagrangian submanifolds of 𝑀. Let 𝐿 and 𝐿0 be two other Lagrangians in L𝑎𝑔 ∗ (𝑀). Theorem 5.1 and its various generalizations (Theorems 5.2 and 5.5) suggest the definition of the following two sequences of numbers. The definition of these numbers has a geometric underpinning in that it is based on the existence of certain cobordisms. First, for each 𝑎 > 0, define the 𝑎-cone-length of 𝐿0 relative to 𝐿 (with respect to F) as (6.1)

𝑙 𝑎F(𝐿0 , 𝐿) := min 𝑘 ∈ ℕ ; ∃𝑉 : 𝐿0



(𝐿1 , . . . , 𝐿 𝑠−1 , 𝐿, 𝐿 𝑠 , . . . , 𝐿 𝑘 ), 𝐿 𝑖 ∈ F, S(𝑉) ≤ 𝑎 .



Here, the minimum is taken only over cobordisms 𝑉 ∈ L𝑎𝑔 𝑄 (ℝ2 × 𝑀), i.e. in the class 𝑄. We stress that we allow 𝑉 to be disconnected, and that 𝑉 ∈ L𝑎𝑔 𝑄 (ℝ2 × 𝑀) means that every path connected component of 𝑉 is of class 𝑄. We use the convention that the number 𝑙 𝑎F(𝐿0 , 𝐿) equals 0 if 𝐿 and 𝐿0 are related by a simple cobordism 𝑉 : 𝐿0 𝐿 of shadow ≤ 𝑎 (a cobordism with just two possibly non-void ends, one positive and one negative, is called simple). We set 𝑙 𝑎F(𝐿0 , 𝐿) = ∞ if no cobordism 𝑉 as above exists. We will omit Ffrom the notation when there is no risk of confusion. It is clear that 𝑙 𝑎F(𝐿0 , 𝐿) is non-increasing in 𝑎 and symmetric with respect to 𝐿, 𝐿0. Next, define 𝑙 F(𝐿0 , 𝐿) := lim𝑎→∞ 𝑙 𝑎F(𝐿0 , 𝐿) to be the absolute cone length of 𝐿0 relative to 𝐿 and 𝑙0F(𝐿0 , 𝐿) := lim𝑎→0 𝑙 𝑎F(𝐿0 , 𝐿). In view of Theorem 5.1 it is natural to also estimate the minimal shadow required for splittings as in the definition of 𝑙 𝑎F and thus define a second family of natural measurements as follows. For every 𝑘 ∈ ℕ define: (6.2) 𝑑 𝑘F(𝐿0 , 𝐿) := inf S(𝑉) ; 𝑉 : 𝐿0



(𝐿1 , . . . , 𝐿 𝑠−1 , 𝐿, 𝐿 𝑠 , . . . , 𝐿𝑟 ), 𝐿 𝑖 ∈ F, 𝑟 ≤ 𝑘 .



Again, the infimum is taken only over cobordisms 𝑉 of class 𝑄 and we allow 𝑉 to be disconnected. This is significant as, for instance, if Fcontains a representative in each Hamiltonian isotopy class of the Lagrangians in L𝑎𝑔 ∗ (𝑀), then 𝑑2F(𝐿0 , 𝐿) is finite for all 𝐿, 𝐿0 ∈ L𝑎𝑔 ∗ (𝑀) (one can take 𝑉 as an appropriate union 𝑉0 ∪ 𝑉1 of two disjoint 𝐿01 , 𝑉1 : ∅ (𝐿, 𝐿1 ) with 𝐿1 and 𝐿01 respectively Lagrangian suspensions 𝑉0 : 𝐿0 Hamiltonian isotopic to 𝐿 and to 𝐿0). We take 𝑑 𝑘F(𝐿0 , 𝐿) = ∞ if no cobordisms 𝑉 as in (6.2) exist. Again, 𝑑 𝑘F is symmetric in (𝐿, 𝐿0), and 𝑑 𝑘F(𝐿0 , 𝐿) is non-increasing in 𝑘. Note that 𝑑0F is the “shadow” metric on elementary cobordism equivalence classes

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as defined in [CS19]. Thus for a Hamiltonian diffeomorphism 𝜙, we have 𝑑0F 𝜙(𝐿), 𝐿 ≤ k𝜙k 𝐻 ,




where k k 𝐻 denotes the Hofer norm of 𝜙. The following inequality is immediate in view of Proposition 6.2:

.

F 0 F 0 00 00 F 𝑑 𝑘+𝑘 0 (𝐿, 𝐿 ) ≤ 𝑑 𝑘 (𝐿, 𝐿 ) + 𝑑 𝑘 0 (𝐿 , 𝐿 ).

(6.3)

Obviously, we also have 𝑑 F F

𝑙 𝑎 (𝐿0 ,𝐿)

(𝐿0 , 𝐿) ≤ 𝑎 and 𝑙 FF

𝑑 𝑘 (𝐿0 ,𝐿)

(𝐿0 , 𝐿) ≤ 𝑘.

Finally, we define also the following measurement: 𝑑 F(𝐿, 𝐿0) = lim 𝑑 𝑘F(𝐿, 𝐿0) = inf 𝑑 𝑘F(𝐿, 𝐿0).

(6.4)

𝑘→∞

𝑘≥0

Or more explicitly (6.5)

𝑑 F(𝐿, 𝐿0) = inf S(𝑉) ; 𝑉 : 𝐿0

(𝐿1 , . . . , 𝐿 𝑠−1 , 𝐿, 𝐿 𝑠 , . . . , 𝐿𝑟 ),



𝐿 𝑖 ∈ F, 𝑉 ∈ L𝑎𝑔 𝑄 (ℝ2 × 𝑀) .



From the above it follows that 𝑑 F( , ) is a pseudo-metric called the shadow pseudo-metric associated to F. By definition, 𝑑 F(𝐿, 𝐿0) is infinite only if there are no cobordisms relating 𝐿 to 𝐿0 and with all the other ends in F.

..

Theorem 5.5 implies:

Corollary 6.4. — If 𝑑 F(𝐿0 , 𝐿) = 0, then 𝐿 ⊂ 𝐿0 ∪

Ð

𝐾∈F 𝐾

and 𝐿0 ⊂ 𝐿 ∪

Ð

𝐾∈F 𝐾.

Proof. — If the first inclusion in this statement does not hold, then (6.6)

𝛿(𝐿; 𝐿0 ∪

Ø

𝐾) > 0.

𝐾∈F

If 𝑄 is the class of exact cobordisms with exact ends, then the first point of Theorem 5.1 implies that 𝑑 F(𝐿0 , 𝐿) cannot vanish. If 𝑄 is either “tightly quasi-exact Lagrangians with weakly exact ends” or “tightly quasi-monotone Lagrangians with ends in L𝑎𝑔 mon,d (𝑀)”, then again, by Theorem 5.5 together with point 1) of Remark 6.1.1 it follows that 𝑑 F(𝐿0 , 𝐿) can not vanish. The argument for the second inclusion is the same.  It is easy to see (Remark 6.3.2 below) that the pseudo-metric 𝑑 F given by (6.4) is in general degenerate. However, we have

Corollary 6.5. — Let F and F0 be two families of Lagrangians in L𝑎𝑔 ∗ (𝑀) such Ð Ð 0

that the intersection ( 𝐾∈F 𝐾) ∩ ( 𝐾0 ∈F0 𝐾 ) is totally disconnected (e.g. discrete). Then the pseudo-metric on L𝑎𝑔 ∗ (𝑀) defined by 0 0 b 𝑑 F,F := max{𝑑 F, 𝑑 F }

is non-degenerate.

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0

Proof. — If b 𝑑 F,F (𝐿, 𝐿0) = 0 we deduce from Corollary 6.4 that 𝐿 ⊂ 𝐿0 ∪ 𝐾∈F 𝐾 and Ð 𝐿 ⊂ 𝐿0 ∪ 𝐾0 ∈F0 𝐾 0. Assume that there is a point 𝑥 ∈ 𝐿 such that 𝑥 ∉ 𝐿0. Then there is Ð an open disk 𝐷 ⊂ 𝐿 with 𝐷 ∩ 𝐿0 = ∅. It follows that one has 𝐷 ⊂ 𝐾∈F 𝐾 as well as Ð Ð Ð 𝐷 ⊂ 𝐾0 ∈F0 𝐾 0 which is not possible because the set ( 𝐾∈F 𝐾) ∩ ( 𝐾0 ∈F0 𝐾 0) is totally disconnected. We conclude that 𝐿 ⊂ 𝐿0. The roles of 𝐿 and 𝐿0 being symmetric, we deduce that 𝐿 = 𝐿0. 

Ð

Notice that if 𝐿0 = 𝜙(𝐿) with 𝜙 a Hamiltonian diffeomorphism, then 0 b 𝑑 F,F (𝐿, 𝐿0) ≤ k𝜙k 𝐻 .

Given a family F that is finite (but this can also work in more general instances) it is easy to produce an additional family F0 that satisfies the assumption of Corollary 6.5. This can be achieved, for instance, by transporting each element of F by an appropriate Hamiltonian isotopy. 0 We will not analyze here in detail the properties of the metrics b 𝑑 F,F but there are two simple observations that we include.

Corollary 6.6. — For every Hamiltonian diffeomorphism 𝜙 of 𝑀 we have F,F0 0 b 𝑑 (𝐿, 𝐿0) − b 𝑑 F,F (𝜙(𝐿), 𝜙(𝐿0)) ≤ 2k𝜙k 𝐻 , (6.7) 𝜙(F),𝜙(F0 ) 0 b 𝑑 (𝐿, 𝐿0) − b 𝑑 F,F (𝐿, 𝐿0) ≤ 2k𝜙k 𝐻 . 0

Therefore, Ham(𝑀, 𝜔) acts by quasi-isometries on the metric space (L𝑎𝑔 ∗ (𝑀), b 𝑑 F,F ). Moreover, the identity is a quasi-isometry between the two metric spaces L𝑎𝑔 ∗ (𝑀), b 𝑑 𝜙(F),𝜙(F ) , L𝑎𝑔 ∗ (𝑀), b 𝑑 F,F . 0

0







Proof. — A cobordism 𝑉 : 𝐿 (𝐹1 , . . . , 𝐿0 , . . . , 𝐹 𝑘 ) can be extended, by gluing appropriate Lagrangian suspensions to the ends 𝐿 and 𝐿0, to a cobordism 𝑉 0 : 𝜙(𝐿)

(𝐹1 , . . . , 𝜙(𝐿0), . . . , 𝐹 𝑘 )

of shadow S(𝑉 0) ≤ S(𝑉) + 2k𝜙k 𝐻 . The first inequality in the statement then follows rapidly, by applying the same argument to 𝜙 −1 . Similarly, to deduce the second inequality, consider 𝑉 : 𝐿 (𝐹1 , . . . , 𝐿0 , . . . , 𝐹 𝑘 ). By applying 𝜙 to 𝑉 we get 𝜙(𝑉) : 𝜙(𝐿)

𝜙(𝐹1 ), . . . , 𝜙(𝐿0), . . . , 𝜙(𝐹 𝑘 ) .




Extend both ends 𝜙(𝐿) and 𝜙(𝐿0) by Lagrangian suspensions thus getting 𝑉 00 (𝜙(𝐹1 ), . . . , 𝐿0 , . . . , 𝜙(𝐹 𝑘 )) of shadow bounded by S(𝑉) + 2k𝜙k 𝐻 and the desired inequality follows easily.  6.2.1. Remark. — Consider the case ∗ = we. As indicated at the beginnig of Chapter 6 we could not have defined the pseudo-metric 𝑑 F by infimizing in (6.5) only over weakly exact cobordisms. The reason is that compositions of weakly exact cobordisms might not be weakly exact, hence the triangle inequality (6.3) might not hold. It is for this reason that we needed to enlarge the class of cobordisms to quasi-exact. Next we explain why infimizing shadows over quasi-exact cobordisms still does not give the correct definition and why we need to appeal to tightly quasi-exact ones.

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The reason is that Theorem 5.5 (as opposed to Theorem 5.1) gives us only a lower bound for Area(𝐾𝑉 ) rather than for S(𝑉). (Here 𝐾𝑉 ⊂ ℝ2 is a compact subset which is quasi-exact admissible for 𝑉.) However, if 𝑉 is tightly quasi-exact then by point 1) of Remark 6.1.1 we have inf Area(𝐾𝑉 ) ; 𝐾𝑉 ⊂ ℝ2 is quasi-exact admissible for 𝑉 = S(𝑉).





Therefore the definition in (6.5) has the desired properites. Of course, one could attempt to define another pseudo-metric similar to 𝑑 F, by (6.8) 𝑑 F,qe (𝐿, 𝐿0) = inf Area(𝐾𝑉 ) ; 𝑉 : 𝐿0

(𝐿1 , . . . , 𝐿 𝑠−1 , 𝐿, 𝐿 𝑠 , . . . , 𝐿𝑟 ),



𝐿 𝑖 ∈ F, 𝑉 ∈ L𝑎𝑔 qe (ℝ2 × 𝑀), 𝐾𝑉 is quasi-exact admissible for 𝑉 ,



where the infimum is taken over all quasi-exact cobordisms as in (6.8). In view of Theorem 5.5 and Remark 6.1.2, this yields a pseudo-metric with similar properties to 𝑑 F. Similar remarks apply to the case ∗ = (mon, d) and to quasi-monotone versus tightly quasi-monotone cobordisms. Note that in the case ∗ = ex (exact Lagrangians) one can safely take 𝑄 to be the class of exact cobordisms, since compositions of exact cobordisms is exact and moreover Theorem 5.1 applies to exact cobordisms.  0

The construction of the metrics b 𝑑 F,F admits several variations. For instance, let U = {𝑈 𝑖 } 𝑖∈𝐼 be a family of open sets 𝑈 𝑖 ⊂ 𝑀 and let F𝑖 = {𝐿 ∈ L𝑎𝑔 ∗ (𝑀) ; 𝐿 ∩𝑈 𝑖 = ∅}. For each index 𝑖 ∈ 𝐼 we then have a shadow pseudo-metric 𝑑 F𝑖 . Define a new pseudometric:  𝐷 U = sup 𝑑 F𝑖 ; 𝑖 ∈ 𝐼 . For the next corollary we will make use of the following. For 𝐿 ∈ L𝑎𝑔 ∗ (𝑀) let: Δ(𝐿; U) = inf 𝑠 ; ∀𝑖 ∈ 𝐼, ∃𝜙 Hamiltonian diffeomorphism with 𝜙(𝐿) ∩ 𝑈 𝑖 = ∅, k𝜙k 𝐻 ≤ 𝑠 .



Corollary 6.7. — With the notation above we have Ð (i) If U is a covering of 𝑀 in the sense that

𝑖

𝑈 𝑖 = 𝑀, then 𝐷 U is non-degenerate.

(ii) For all 𝐿, 𝐿0 ∈ L𝑎𝑔 ∗ (𝑀) such that Δ(𝐿; U) and Δ(𝐿0; U) are finite, we have 𝐷 U(𝐿, 𝐿0) ≤ Δ(𝐿; U) + Δ(𝐿0; U). Proof. — The first point follows immediately from Corollary 6.4. For the second point fix some 𝑠 > Δ(𝐿; U), 𝑠 0 > Δ(𝐿0; U) and pick one family F𝑖 . There is a cobordism 𝑉 :𝐿 (𝐿01 , 𝐿0 , 𝐿1 ) such that 𝑉 is a disjoint union of two Lagrangian suspensions 𝑉0 : 𝐿 𝐿1 and 𝑉1 : ∅ (𝐿01 , 𝐿0) such that 𝐿1 , 𝐿01 ∈ F𝑖 and S(𝑉0 ) ≤ 𝑠, S(𝑉1 ) ≤ 𝑠 0. This means that 𝑑 F𝑖 (𝐿, 𝐿0) ≤ 𝑠 + 𝑠 0 which implies the claim. 

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0

There are other variants of the definition of the metric b 𝑑 F,F that have interesting features. For instance, by considering in (6.2) only cobordisms (𝐿1 , . . . , 𝐿 𝑘 , 𝐿), in other words cobordisms for which 𝐿0 is the posi𝑉 : 𝐿0 tive end and 𝐿 is the top negative end, one gets a measurement 𝑡 𝑘F(𝐿0 , 𝐿). It has similar properties to 𝑑 𝑘F, except that it is not symmetric. We define 𝑡 F(𝐿0 , 𝐿) as in (6.4) and we symmetrize by putting 𝑟 F(𝐿0 , 𝐿) =

1 2

𝑡 F(𝐿0 , 𝐿) + 𝑡 F(𝐿, 𝐿0) ,




thus obtaining a new pseudo-metric. This pseudo-metric satifies the conclusion of Corollary 6.4 and can be used in the rest of the preceding constructions, leading to 0 metrics b 𝑟 F,F that satisfy the conclusions of Corollaries 6.5, 6.6 and 6.7, where in 6.7 the pseudo-metric 𝐷 U is replaced with 𝑅 U = sup{𝑟 F𝑖 ; 𝑖 ∈ 𝐼}. An additional interesting feature of the pseudo-metrics 𝑅 U is the following:

Corollary 6.8. — With the notation above fix 𝐿 ∈ L𝑎𝑔 ∗ (𝑀) and assume that U is a covering of 𝑀. There exists a constant 𝛿 > 0 depending on 𝐿 and Usuch that, if 𝐿0 ∈ L𝑎𝑔 ∗ (𝑀) is disjoint from 𝐿, then 𝑅 U(𝐿, 𝐿0) ≥ 𝛿.

Proof. — The crucial remark is that, by inspecting the proof of the first part of (𝐿1 , . . . , 𝐿 𝑘 , 𝐿0) in L𝑎𝑔 𝑄 (ℝ2 × 𝑀) and such Theorem 5.1, we see that given 𝑉 : 𝐿 Ð 1 that 𝐿 ∩ 𝐿0 = ∅, then S(𝑉) ≥ 2 𝛿(𝐿; 𝑆) where 𝑆 = 𝑖 𝐿 𝑖 but 𝑆 – and thus 𝛿(𝐿, 𝑆) – does not depend on 𝐿0. As Uis a covering of 𝑀 there exists some index 𝑖 ∈ 𝐼 and an open set 𝑈 ⊂ 𝑈 𝑖 such that 𝑈 is the image of an embedding 𝑒 : 𝐵(𝑟) → 𝑀 with 𝑒 −1 (𝐿) = 𝐵ℝ (𝑟). Obviously, 𝑈 is disjoint from all the elements of F𝑖 and thus 𝑅 U(𝐿, 𝐿0) ≥ 41 𝜋𝑟 2 .  To ilustrate Corollary 6.8, consider 𝑀 = 𝕋 2 = 𝑆1 × 𝑆1 with 𝐿 = {𝑥∗ } × 𝑆1 and 𝐿 𝑘 = {𝑥 𝑘 } × 𝑆 1 , where 𝑥 𝑘 is a sequence in 𝑆1 with 𝑥 𝑘 → 𝑥∗ as 𝑘 → ∞ and 𝑥 𝑘 ≠ 𝑥 ∗ for all 𝑘. Clearly 𝐿 𝑘 converges to 𝐿 in the Hausdorff distance. By Corollary 6.8 all the Lagrangians 𝐿 𝑘 remain at a bounded distance from 𝐿 in the 𝑅 U-metric. 6.2.2. Remarks 1) It is well-known that there are other natural metrics defined on L𝑎𝑔 ∗ (𝑀). The most famous is Hofer’s Lagrangian metric, used since the work of Chekanov [Che00], which infimizes the Hofer energy needed to carry one Lagrangian to the other. Another interesting more algebraic metric, smaller than the Hofer metric, is the spectral metric due to Viterbo. Both these metrics are infinite as soon as the two Lagrangians compared are not Hamiltonian isotopic. A metric smaller than the Hofer norm, and based on simple Lagrangian cobordism has been introduced in [CS19]: it measures the distance between 𝐿 and 𝐿0 by infimizing the shadow of cobordisms having only 𝐿 and 𝐿0 as ends. This metric is finite on each simple cobordism class 𝑑 ∅,∅ . This metric is again often and, with the notation above, it coincides with 𝑑 ∅ = b infinite. For instance, in the exact case, as soon as 𝐿 and 𝐿0 have non-isomorphic homologies, the simple shadow distance between 𝐿 and 𝐿0 is infinite. Indeed, if 𝐿 and 𝐿0 are related by an exact simple cobordism, then 𝐿 and 𝐿0 have isomorphic singular homologies [BC13] (more rigidity is actually true, see [Sua17]). For other

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results on the simple shadow metric see [Bis], [Bis19a], [Bis19b]. It is already known that without appropriate constraints on the class of admissible Lagrangians and cobordisms, such as those imposed here, even the simple cobordism metric 𝑑∅ is degenerate [CS19]. 2) A notion of cone-length is familiar in homotopy theory as a measure of complexity for topological spaces [Cor94]. 6.3. Some examples and calculations 6.3.1. Curves on tori and related examples. — We fix a family of Lagrangians F, to be specified later, and omit it from the notation of the measurements 𝑙 𝑎F, 𝑑 𝑘F,𝑙 F, 𝑑 F. If 𝜙 is a Hamiltonian diffeomorphism, then clearly 𝑙 𝑎 (𝜙(𝐿), 𝐿) = 0 as soon as 𝑎 ≥ k𝜙k 𝐻 and so 𝑙(𝜙(𝐿), 𝐿) = 0. However, we will see below classes of examples with 0 < 𝑙 𝑎 (𝜙(𝐿), 𝐿) < ∞. Intuitively, an inequality of the type 1 ≤ 𝑙 𝑎 (𝜙(𝐿), 𝐿) < ∞ seems to indicate that 𝜙 distorts 𝐿 (at least for our choices of classes F). The examples below that satisfy 1 ≤ 𝑙 𝑎 (𝜙(𝐿), 𝐿) < ∞ also satisfy 𝑑 𝑙𝑎 (𝜙(𝐿),𝐿) (𝜙(𝐿), 𝐿) < 𝑑0 (𝜙(𝐿), 𝐿) ≤ k𝜙k 𝐻 . In other words, in these examples the “optimal” (in the sense of minimizing the shadow) approximation of 𝜙(𝐿) through elements of the set {𝐿} ∪ F requires more elements than just 𝐿. Moreover, the relevant 𝑑 𝑘 ’s are small enough so that inequality (5.4) of Theorem 5.1 applies and indeed, as predicted by the theorem, in these examples the number of intersection points 𝜙(𝐿) ∩ 𝑁, where 𝑁 is an appropriate other Lagrangian 𝑁 ∈ L𝑎𝑔 ∗ (𝑀), is much higher than the usual lower bound, given by the rank of the Floer homology group HF(𝑁 , 𝐿). Consider the 2-dimensional torus 𝑀 = 𝑇 2 endowed with an area form. We identify 𝑇 2 with the square [−1, 1] × [−1, 1] with the usual identifications of the edges. We consider five Lagrangians on 𝑇 2 , described on the square [−1, 1] × [−1, 1] by 𝐿 = [−1, 1] × {0},

𝑆1 = {− 21 − 𝜖} × [−1, 1], 𝑆2 = {− 12 + 𝜖} × [−1, 1], 𝑆3 = { 21 − 𝜖} × [−1, 1], 𝑆4 = { 12 + 𝜖} × [−1, 1].

Here 0 < 𝜖 ≤ 81 . We will construct a new Lagrangian obtained through surgery between 𝐿 and the 𝑆 𝑖 ’s. We use the surgery conventions from [BC13] and define – see Figure 7: 𝐿0 = 𝑆3 # (𝑆2 # (𝐿 # 𝑆1 )) # 𝑆4 .



(6.9)



In the surgeries above we use handles of equal size in the sense that the area enclosed by each handle is equal to a fixed 𝛿 > 0 with 𝛿 very small. We will also make use of the two rectangles 𝐾 1 = [− 21 − 2𝜖, − 12 + 2𝜖] × [−𝜖, 𝜖],

𝐾2 = [ 12 − 2𝜖,

1 2

+ 2𝜖] × [−𝜖, 𝜖]

and we put 𝐾 = 𝐾 1 ∪ 𝐾 2 (see again Figure 7).

Lemma 6.9. — Let F = {𝑆1 , 𝑆2 , 𝑆3 , 𝑆4 } and assume that 𝛿 < 21 𝜖2 . We have (i) 𝑑0 (𝐿0 , 𝐿) = 4𝜖, 𝑑4 (𝐿0 , 𝐿) ≤ 2𝛿, 𝑙(𝐿0 , 𝐿) = 0, 𝑙2𝛿 (𝐿0 , 𝐿) = 4.

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Figure 7. The Lagrangians 𝐿, 𝐿0 = 𝑆3 # [(𝑆2 # (𝐿 # 𝑆1 )) # 𝑆4 ], and 𝑁 in 𝑇 2 .

(ii) For any weakly-exact Lagrangian 𝑁 ⊂ 𝑇 2 with 𝑁 ∩ 𝐾 = ∅, we have # (𝑁 ∩ 𝐿0) ≥ 𝑟 𝑘 HF(𝑁 , 𝐿) +




(6.10)

4 Õ

𝑟 𝑘 HF(𝑁 , 𝑆 𝑖 ) .




𝑖=1

𝑁0

𝑁0

(iii) If is a weakly exact Lagrangian ⊂ 𝑇 2 , then either 𝑁 0 ∩ 𝐿 ≠ ∅ or, for any 0 Hamiltonian diffeomorphism 𝜙 with 𝜙(𝐿) = 𝐿 we have 𝜙(𝑁 0) ∩ 𝐾 ≠ ∅. Floer homology is considered here with coefficients in ℤ/2. Notice that HF(𝑁 , 𝐿0)  HF(𝑁 , 𝐿) so the inequality (6.10) indicates an “excess” of intersection points. An example of a Lagrangian 𝑁 as at point ii is simply 𝑁 = [−1, 1] × {−2𝜖}. Proof. — By inspecting again Figure 7 and possibly extending the representation of the torus by adding vertically two additional fundamental domains to the square [−1, 1] × [−1, 1] one can see that there is a Hamiltonian isotopy 𝜙 : 𝑇 2 → 𝑇 2 so that 𝐿0 = 𝜙(𝐿) (this is because the upper and lower “bends” in the picture encompass equal areas). The expression in (6.9) show that there is a cobordism 𝑉 : 𝐿0 → (𝑆3 , 𝑆2 , 𝐿, 𝑆1 , 𝑆4 ) given as the trace of the respective surgeries (as given in [BC13]) and because the 𝑆 𝑖 ’s are disjoint and all the handles are of area 𝛿 we have S(𝑉) ≤ 2𝛿. The reason for the factor 2 is that the handles associated to the surgeries on the “left” and those on the “right” can not be assumed to have a superposing projection; the constant is 2 and not 4 because the two handles on the left (and similarly for the two handles on the right) can be assumed to have overlapping projections. It is a simple exercise to show that 𝛿(𝐿0; 𝐿) = 8𝜖.

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From the first part of Theorem 5.1 we deduce 𝑑0 (𝐿0 , 𝐿) ≥ 4𝜖. It is also easy to see that one can find a Hamiltonian 𝐻 : 𝑇 2 → ℝ with variation equal to 4𝜖 and so that 𝜙1𝐻 (𝐿) = 𝐿0. Therefore, 𝑑0 (𝐿0 , 𝐿) = 4𝜖. On the other hand, recall the assumption 𝛿 < 𝜖2 /2. Therefore we have 𝑑4 (𝐿0 , 𝐿) ≤ 2𝛿. We now estimate cone-length. Clearly, the absolute number is 𝑙(𝐿0 , 𝐿) = 0. From the existence of the cobordism 𝑉 we deduce 𝑙2𝛿 (𝐿0 , 𝐿) ≤ 4. We want to show 𝑙2𝛿 (𝐿0 , 𝐿) = 4. Assume that 𝑙2𝛿 (𝐿0 , 𝐿) ≤ 3. Therefore there exists a cobordism 𝑉 0 : 𝐿0 → (𝐿1 , 𝐿2 , 𝐿3 , 𝐿4 ) where one of the 𝐿 𝑖 ’s equals 𝐿 and the other three are picked among the 𝑆 𝑖 ’s (or are void) and the shadow of 𝑉 0 is at most 2𝛿. Without loss of generality, assume that 𝑆1 is not among the 𝐿 𝑖 ’s. We now consider the number 𝛿(𝐿0; 𝐿 ∪ 𝑆2 ∪ 𝑆3 ∪ 𝑆4 ). By using a disk centered along the part of 𝑆1 contained in 𝐿0 we see that 𝛿(𝐿0; 𝐿 ∪ 𝑆1 ∪ 𝑆2 ) ≥ 8𝜖. By the first part of Theorem 5.1 it follows S(𝑉 0) ≥ 4𝜖 which contradicts 𝛿 ≤ 2𝜖2 . The two other points of the Lemma also follow from Theorem 5.1 (they possibly admit also more elementary, direct proofs). Point (b) of the Theorem implies that for any weakly-exact Lagrangian 𝑁 ⊂ 𝑇 2 so that 𝑁 ∩ 𝐾 = ∅, we have (6.10). Indeed, we may find disks around the (unique) intersection point of each of the 𝑆 𝑖 ’s with 𝐿 that are of area 4𝜖2 , have the real part along 𝐿 and the imaginary part along 𝑆 𝑖 , are contained in 𝐾, and any two of these disks have disjoint interiors. As 𝑁 avoids 𝐾, this means ÐÐ 𝛿Σ (𝕃 ; 𝑁) ≥ 4𝜖2 for 𝕃 = 𝐿 𝑖 𝑆 𝑖 and Σ the intersection points of the 𝑆 𝑖 ’s with 𝐿. The last point of the Lemma follows in a similar way. Assuming also that 𝑁 0 ∩ 𝐿 = ∅ we also have 𝜙(𝑁 0) ∩ 𝐿0 = ∅. If we also have 𝜙(𝑁 0) ∩ 𝐾 = ∅, then 𝜙(𝑁 0) satisfies inequality (6.10) (with 𝜙(𝑁 0) in the place of 𝑁). From the fact that 𝑁 0 is weakly exact we deduce that the singular homology class of 𝑁 0 is the same as that of 𝐿 and thus HF(𝑁 0 , 𝑆 𝑖 ) does not vanish. But this leads to a contradiction with 𝜙(𝑁 0) ∩ 𝐿0 = ∅.  It is easy to construct examples similar to the one above in higher dimensions. For instance, one can consider 𝑀 = (𝑇 2 × 𝑇 2 , 𝜔 ⊕ 𝜔) and take 𝐿¯ = 𝐿 × 𝐿, 𝑆¯ 𝑖 = 𝑆 𝑖 × 𝑆 𝑖 etc. We will see some less trivial extensions in the next subsection. 6.3.2. Remarks. — The examples above also point out two deficiencies of the pseudometric 𝑑 F. 1) 𝑑 F is generally degenerate. For example, 𝑑3F(𝑆1 , 𝑆2 ) = 0, hence 𝑑 F(𝑆1 , 𝑆2 ) = 0. Ý (𝑆1 , 𝑆2 , 𝑆2 ) be the cobordism 𝑉 = 𝛾0 × 𝑆1 𝛾1 × 𝑆2 where Indeed, let 𝑉 : 𝑆1 𝛾0 = ℝ + 𝑖 ⊂ ℂ and 𝛾1 is a curve in ℂ that has two horizontal negative ends, one at height 2 and the other at height 3 and is disjoint from 𝛾0 . The same construction shows that for any family Fwith more than one element the resulting pseudo-metric is degenerate. In the above examples the cobordisms 𝑉 are disconnected and they also have vanishing shadow. However, there are also examples of connected cobordisms 𝑊𝜖 with constant ends and positive shadow such that lim𝜖→0 S(𝑊𝜖 ) = 0. For instance, with the notation above, consider a curve 𝛾 ⊂ ℝ2 which has a “⊃” shape with its lower end going to −∞ along the horizontal line 𝑦 = −1 and its upper end going to −∞ along the horizontal line 𝑦 = 1. Let 𝛾0 be the 𝑥-axis, 𝑦 = 0. Consider now the surgery

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115

𝑊𝜖 := (𝛾 × 𝑆1 ) #𝜖 (𝛾0 × 𝐿) ⊂ ℝ2 × 𝑇 2 . (Note that, in contrast to the construction of e.g. 𝐿0 above, the surgery here is performed in the space ℝ2 ×𝑇 2 .) Clearly 𝑊𝜖 is a (connected) (𝑆1 , 𝐿, 𝑆1 ) and lim𝜖→0 S(𝑊𝜖 ) = 0. weakly exact Lagrangian cobordism 𝑊𝜖 : 𝐿 0 2) In general, even if both 𝐿 and 𝐿 belong to the triangulated completion of the family F, it can be difficult to know whether 𝑑 F(𝐿, 𝐿0) is finite because there might not be any practical way to construct cobordisms with ends 𝐿, 𝐿0 and elements of F. 6.3.3. Matching cycles in simple Lefschetz fibrations. — We revisit here the phenomena described above in a different context and we also present examples of symplectic diffeomorphisms 𝜙 : 𝑀 → 𝑀 with 𝑙(𝜙 𝑘 (𝐿), 𝐿) = 𝑘 (in these examples 𝜙 is a Dehn twist). The manifold 𝑀 is now taken to be the total space of a Lefschetz fibration 𝜋 : 𝑀 −→ ℂ over ℂ with general fiber the cotangent bundle of a sphere 𝐾 (in particular 𝑀 is not compact). We will assume that the Lefschetz fibration has exactly three singularities 𝑥 1 , 𝑥2 , 𝑥3 , whose projection on ℂ is arranged as in Figure 8 below. We also assume that there are two matching cycles relating the three singularities that we denote by 𝑆, from 𝑥1 to 𝑥 2 , and 𝐿, from 𝑥2 to 𝑥3 – as in the same figure.

Figure 8. The matching cycles 𝑆, 𝑆1 , 𝑆2 and 𝐿 and the Lagrangians 𝐿1 , 𝐿2 , 𝐿02 constructed by surgery (and small perturbation) from them.

Notice that 𝐿 and 𝑆 intersect (transversely) in a single point. Moreover, recall that with the notation in [BC13], [BC17] we have that 𝑆 # 𝐿 is Hamiltonian isotopic to the Dehn twist 𝜏𝑆 (𝐿), and, similarly, 𝐿 # 𝑆 is Hamiltonian isotopic to 𝜏𝑆−1 (𝐿). An important point to emphasize here is that the Dehn twist 𝜏𝑆 (𝐿) is only well defined up to Hamiltonian isotopy. On the other hand, the models for 𝜏𝑆 (𝐿) (and 𝜏𝑆−1 (𝐿)) given by surgery, as before, are precisely determined as soon as the local data of the surgery is fixed (the surgery handle and the precise Darboux chart around the intersection point). We will also need two other matching cycles 𝑆1 and 𝑆2 with a projection as in Figure 8 a. They are both Hamiltonian isotopic to 𝑆. The two spheres 𝑆1 and 𝑆2 intersect transversely at the points 𝑥1 and 𝑥 2 and each of them intersects transversely 𝐿 at the point 𝑥2 . We now consider the following three Lagrangians: 𝐿1 which is obtained from 𝑆1 #𝐿 after a small Hamiltonian isotopy such that its projection is as in Figure 8 b,

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𝐿2 given as a small deformation of 𝑆2 # 𝐿1 and 𝐿02 , a small deformation of 𝐿1 # 𝑆2 such that their projections are as in the same figure, part c and d, respectively. Notice that 𝐿1 is a model for 𝜏𝑆 (𝐿) and that 𝐿2 and 𝐿02 are models for 𝜏𝑆2 (𝐿) and 𝐿 = 𝜏𝑆−1 𝜏𝑆 (𝐿), respectively. In particular, there is a Hamiltonian isotopy 𝜙 such that 𝐿02 = 𝜙(𝐿). Fix the family F = {𝑆1 , 𝑆2 }. The first remark is that by taking the surgery handles sufficiently small we have 𝑑2 (𝐿02 , 𝐿) < 𝑑0 (𝐿02 , 𝐿) < ∞. Further, let 𝐾 0 be a Hamiltonian perturbation of the vanishing sphere 𝐾 in the general fiber. Let 𝑁 be the trail of 𝐾 0 along a curve as in Figure 8 d. We now claim that 𝑙(𝐿2 , 𝐿) = 2. Indeed, by construction we have a cobordism 𝐿2 (𝑆2 , 𝑆1 , 𝐿), hence 𝑙(𝐿2 , 𝐿) ≤ 2. Now, it is not hard to see that HF(𝑁 , 𝐿2 ) = HF(𝑁 , 𝑆1 ) ⊕ HF(𝑁 , 𝑆2 ) = HF(𝐾, 𝐾) ⊕ HF(𝐾, 𝐾) (one can use Seidel’s exact triangle associated to a Dehn twist for this computation, or alternatively Theorem A from [BC17] with 𝑉 = 𝐿2 ). Since HF(𝑁 , 𝐿) = 0 and HF(𝑁 , 𝐿2 ) ≠ 0 it follows that 𝐿2 is not Hamiltonian isotopic to 𝐿, hence 𝑙(𝐿2 , 𝐿) ≥ 1. (𝐿, 𝐹) Moreover, 𝑙(𝐿2 , 𝐿) ≠ 1 for otherwise we would have either a cobordism 𝐿2 or 𝐿2 (𝐹, 𝐿) with 𝐹 ∈ {𝑆1 , 𝑆2 }. As HF(𝑁 , 𝐿) = 0, the latter would imply that HF(𝑁 , 𝐿2 )  HF(𝑁 , 𝐹)  HF(𝑁 , 𝑆)  HF(𝐾, 𝐾), a contradiction. This proves that 𝑙(𝐿2 , 𝐿) = 2. On the other hand, HF(𝑁 , 𝐿02 ) = 0. However, by taking the surgery handles in the constructions above sufficiently small we see that #(𝑁 ∩ 𝐿02 ) ≥ 2 rk(HF(𝐾, 𝐾)), as predicted by Theorem 5.1. Notice also that if the surgery handle is not small enough, or, alternatively, 𝑁 avoids 𝐿02 by passing closer to 𝑥1 , then 𝑁 is disjoint from 𝐿02 . The last remark in this setting is the following. By taking more copies of the sphere 𝑆, (for instance four, as on the left of Figure 9), we can construct, in a way similar to the above, models 𝐿 𝑘 for 𝜏𝑆𝑘 (𝐿). In Figure 9, on the right, we represent 𝜏𝑆4 (𝐿) in this way. As before, it is easy to compute HF(𝑁 , 𝐿 𝑘 ) =

𝑘 Ê

HF(𝐾, 𝐾).

𝑖=1

This shows that 𝑙(𝐿 𝑘 , 𝐿) = 𝑘 (this is a reflection of the well-known fact that the class of 𝜏𝑆 is not a torsion element in 𝜋0 Symp(𝑀), see [KS02], [Sei00]).

Figure 9. A model for 𝜏𝑆4 (𝐿).

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6.3.4. Trace of surgery. — The numbers 𝑑 𝑘 are hard to compute as it is difficult in general to identify cobordisms with fixed ends and with minimal shadow. However, we will see here how to use inequality (5.5) to show the “optimality” of decompositions given by the trace of certain surgeries at one point. We focus on just one example. As in § 6.3.1 we take 𝑀 = 𝑇 = 𝑆1 × 𝑆1 and we fix 𝑆1 an 𝐿 as in that subsection. We now consider 𝐿00 = 𝐿 # 𝑆1 and, again as in § 6.3.1, we assume that the area of the handle used in the surgery giving 𝐿00 is equal to 𝛿. We fix F = {𝑆1 , 𝑆2 , 𝑆3 , 𝑆4 } as in Lemma 6.9. Notice that the shadow of the trace of the surgery 𝑉 : 𝐿00 = 𝐿 # 𝑆1 → (𝐿, 𝑆1 ) is equal to 𝛿.

Corollary 6.10. — For 𝛿 small enough we have 𝑑1 (𝐿00 , 𝐿) = 𝛿. In other words, there is no decomposition of 𝐿00 in terms of the family 𝐿∪Fthrough a cobordism with two negative ends and of shadow smaller than 𝛿. Proof. — Suppose that there is a cobordism 𝑉 0 : 𝐿00 → (𝐿1 , 𝐿2 ) such that one of the 𝐿 𝑖 ’s equals 𝐿, the other equals one of the 𝑆 𝑖 ’s and S(𝑉 0) = 𝛿0 < 𝛿. We first notice that 𝑆1 needs to appear among the 𝐿 𝑖 ’s. Indeed, suppose, for instance that (𝐿1 , 𝐿2 ) = (𝐿, 𝑆2 ). In this case, consider a disk based on the part of 𝐿00 that coincides with 𝑆1 and is disjoint from 𝑆2 as well as from 𝐿 and whose real part is along 𝐿00. The area of such a disk can be assumed to be as close as needed to 2(4𝜖 − 𝛿), where 𝜖 is defined in § 6.3.1. By now applying the first part of Theorem 5.1 we deduce that 𝛿 > S(𝑉 0) ≥ 4𝜖 − 𝛿 which is a contradiction if 𝛿 is small enough. In conclusion, we deduce that the two negative ends of 𝑉 0 coincide with 𝐿 and 𝑆1 . Consider now the Lagrangian 𝑁 as in Figure 10 and denote by 𝑜 the intersection of 𝐿 and 𝑆1 .

Figure 10. The triangle 𝑐𝑜𝑎 is of area 𝐴 with 𝛿 > 𝐴 > 𝛿0 .

The properties of 𝑁 are the following: 𝑁 is Hamiltonian isotopic to 𝑆1 ; it intersects 𝑆1 transversely at precisely two points 𝑎 and 𝑏 and it intersects 𝐿 transversely at one point 𝑐; 𝑁 intersects 𝐿00 transversely at the point 𝑏; the small triangle of vertices 𝑐, 𝑜, 𝑎 is of area 𝐴 with 𝛿0 < 𝐴 < 𝛿. We use the Lagrangian 𝑁 as follows. First, notice that by assuming 𝛿 small enough, and writing 𝕃 = 𝐿 ∪ 𝑆1 , we can find the relevant

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disks centered at 𝑜 so as to estimate 𝛿Σ𝕃 (𝕃 ; 𝑁) ≥ 4𝐴. By applying (5.5) in Theorem 5.1 we deduce 1 = #(𝑁 ∩ 𝐿00) ≥ dim HF(𝑁 , 𝐿) + dim HF(𝑁 , 𝑆1 ) = 3 which is a contradiction and thus proves that 𝑉 0 does not exist.  6.4. Algebraic metrics on L𝑎𝑔 ∗ (𝑀) The main purpose of this subsection is to notice that it is possible to define pseudometrics similar to those in Section 6.2 but that only exploit the algebraic structures involved and that do not appeal to cobordism. We emphasize that, as before, our pseudo-metrics may take infinite values. The proof of the first part of Theorem 5.1 0 implies not only the non-degeneracy of b 𝑑 F,F but also that of its algebraic counterpart. The main advantage of these pseudo-metrics is that when F generates 𝐷F𝑢𝑘 ∗ (𝑀) some of these algebraic pseudo-metrics are finite by definition, independently of the existence of cobordisms – see Remark 6.4.6. Additionally, the construction of 0 both metrics b 𝑑 F,F as well as that of their algebraic counterparts fits a more general, abstract pattern, potentially useful in other contexts, that we outline here. 6.4.1. Weighted triangulated categories. — Let X be a triangulated category and let X0 be a family of objects of Xthat generate Xthrough triangular completion. The purpose of this subsection is to describe a procedure leading to a (pseudo) metric on X0 . The pseudo-metrics 𝑑 F in Section 6.2 are of this type but, as we shall see further below, other choices are possible. There is a category denoted by 𝑇 𝑆 X that was introduced in [BC13], [BC14]. This category is monoidal and its objects are finite ordered families (𝐾 1 , . . . , 𝐾 𝑟 ) with 𝐾 𝑖 in O𝑏(X) with the operation given by concatenation. Up to a certain natural equivalence relation, the morphisms in 𝑇 𝑆 X are direct sums of basic morphisms 𝜙¯ from a family formed of a single object of X to a general family, 𝜙¯ : 𝐾 → (𝐾 1 , . . . , 𝐾 𝑠 ). Such a morphism 𝜙¯ is a triple (𝜙, 𝑎, 𝜂), where 𝑎 ∈ Ob(X), 𝜂 is a decomposition of 𝑎 through iterated distinguished triangles, namely: (6.11)

𝑎 = C𝑜𝑛𝑒 𝐾 𝑠 → C𝑜𝑛𝑒 𝐾 𝑠−1 → · · · → C𝑜𝑛𝑒 𝐾 2 → 𝐾 1 · · ·








and 𝜙 : 𝐾 → 𝑎 is an isomorphism. The tuple (𝐾 1 , . . . , 𝐾 𝑠 ) is called the linearization of the cone decomposition (6.11). In essence, the morphisms in 𝑇 𝑆 X parametrize all the cone-decompositions of the objects in X. Composition in 𝑇 𝑆 X comes down to refinement of cone-decompositions. Denote by 𝑇 𝑆 X0 the full subcategory of 𝑇 𝑆 Xthat has objects (𝐾 1 , . . . , 𝐾 𝑟 ) with 𝐾 𝑖 ∈ X0 , 1 ≤ 𝑖 ≤ 𝑟. Assume that we are given a weight 𝑤 : Mor𝑇 𝑆 X0 → [0, ∞] such that (6.12)

¯ + 𝑤(𝜓), ¯ ≤ 𝑤( 𝜙) ¯ 𝑤( 𝜙¯ ◦ 𝜓)

𝑤(id𝑋 ) = 0,

for all 𝑋 ,

where id𝑋 is the identity morphism viewed as defined on the family formed by the single object 𝑋 and with values in the same family. We will refer to this 𝑤 as a weight on X. Fix also a family F ⊂ X0 . In this setting, we define (compare to (6.2)): (6.13)

ASTÉRISQUE 426

¯ ; 𝜙¯ : 𝐾 0 → (𝐹1 , . . . , 𝐾, . . . , 𝐹𝑟 ), 𝐹𝑖 ∈ F, ∀𝑖 . 𝑠 F(𝐾 0 , 𝐾) = inf 𝑤( 𝜙)





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119

We set 𝑠 F to be ∞ if there are no morphisms as in (6.13). If 𝑤 is finite and if F generates X, then 𝑠 F is finite. Clearly 𝑠 F satisfies the triangle inequality but it is not symmetric in general. Defining 𝑠¯ F(𝐾 0 , 𝐾) :=

1 2

𝑠 F(𝐾 0 , 𝐾) + 𝑠 F(𝐾, 𝐾 0) ,




we obtain a pseudo-metric on the set of objects of X. We will refer to the pseudometrics obtained by this procedure as weighted fragmentation pseudo-metrics. The case of interest in this paper is X = 𝐷F𝑢𝑘 ∗ (𝑀) with X0 consisting of all the Yoneda modules associated to the Lagrangians in L𝑎𝑔 ∗ (𝑀). In our notation, the category F𝑢𝑘 ∗ (𝑀) is defined as described at the beginning of Chapter 3, without reference to filtrations. The shadow pseudo-metric 𝑑 F from Section 6.2 is a first example of a (class) of weighted fragmentation pseudo-metrics associated to a weight 𝑤 defined as follows. Recall from [BC14], [CC16] that there is a monoidal cobordism category C𝑜𝑏 𝑄 (𝑀) whose objects are families (𝐿1 , . . . , 𝐿 𝑠 ) with 𝐿 𝑖 ∈ L𝑎𝑔 ∗ (𝑀) and with morphisms (for(𝐿1 , . . . , 𝐿 𝑠 ) mal sums) of cobordisms in the class 𝑄 that are the type 𝑉 : 𝐿 (modulo an appropriate equivalence relation; the monoidal operation is concatenation). There is a monoidal functor, denoted in [BC14] by e Fbut that, to avoid confusion e in notation, we will denote here by Φ : (6.14)

e : Cob 𝑄 (𝑀) −→ 𝑇 𝑆 𝐷F𝑢𝑘 ∗ (𝑀) . Φ


On objects, this functor associates to a Lagrangian 𝐿 its Yoneda module L and its properties have been used extensively earlier in the paper, starting from Section 3.7. In the setting, X = 𝐷F𝑢𝑘 ∗ (𝑀), for a morphism 𝜙¯ ∈ Mor𝑇 𝑆 X0 we define the shadow weight of 𝜙¯ by (6.15)

¯ = inf S(𝑉) ; Φ(𝑉) e 𝑤 S( 𝜙) = 𝜙¯





and it is easy to see from the various definitions involved that 𝑑 F coincides with the weighted fragmentation pseudo-metric 𝑠¯ F associated to 𝑤 S. Additionally, recall from Corollary 6.5 that, by using an appropriate perturbation F0, we obtain an actual 0 0 metric b 𝑑 F,F = max{𝑑 F, 𝑑 F }. 6.4.2. Remark. — The category 𝑇 𝑆 X was inspired by the work on Lagrangian cobordism and might seem artificial in itself. However, we will remark here that (in a slightly modified form) it is the natural categorification of the Grothendieck group 𝐾(X). This group is defined as the quotient of the free abelian group generated by the objects in X modulo the relations 𝐵 = 𝐴 + 𝐶 whenever 𝐴 → 𝐵 → 𝐶 is a distinguished triangle in X. We will work here in a simplified setting and take the identity for the shift functor. As a consequence 𝐾(X) is a ℤ/2 vector space. Alternatively, 𝐾(X) can also be defined as the free monoid of finite ordered families (𝐾 1 , . . . , 𝐾 𝑟 ) where 𝐾 𝑖 ∈ O𝑏(X), with the operation being given by concatenation of families, modulo the relations 𝐾 1 + 𝐾 2 + · · · + 𝐾 𝑠 = 0 whenever there exists a cone decomposition of 0 with linearization (𝐾 1 , . . . , 𝐾 𝑠 ). When X is small, there is a b𝑆 X, closely associated to 𝑇 𝑆 X, that categorifies 𝐾(X) in the usual sense category, 𝑇

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(meaning that it is a monoidal category with the property that the monoid formed by b𝑆 X, families the isomorphism classes of its objects is 𝐾(X)). The basic idea is that, in 𝑇 that are linearizations of acyclic cones are declared isomorphic to 0. More formally, b𝑆 X is defined if the category X is small and is constructed in three steps: first we 𝑇 add to the morphisms in 𝑇 𝑆 X the morphism 0 → ∅ (and the relevant compositions) thus getting 𝑇 𝑆 X+ ; secondly, we localize 𝑇 𝑆 X+ at the family of morphisms A = 𝜙 ∈ Mor𝑇 𝑆 X+ ; 𝜙 : 0 → (𝐾 1 , . . . , 𝐾 𝑠 )





(here 0 is viewed as a family formed by the single element 0; this is equivalent to adding inverses to all the morphisms having 0 as domain and adding relations so that associativity of composition is still satisfied); finally, we complete in the monoidal sense by allowing formal sums for all the new and old morphisms. 6.4.3. Energy of retracts of weakly filtered modules. — Assume that M is a weakly filtered module over the weakly filtered 𝐴∞ -category A with discrepancy ≤ 𝑚 as in § 2.3.1 and that 𝜓 : M → M is a weakly filtered module homomorphism with discrepancy ≤  ℎ which is null-homotopic. Following the terminology in Section 2.7, we consider the homotopical boundary level of 𝜓, 𝐵 ℎ (𝜓;  ℎ ) := 𝛽 ℎ (𝜓;  ℎ ) + 𝐴(𝜓). Let 𝑓 : M0 → M1 be a morphism of weakly filtered modules and define: 𝜌( 𝑓 ) = inf max 𝐵 ℎ (𝑔 ◦ 𝑓 − id;  ℎ ), 𝐴(𝑔) + 𝐴( 𝑓 ), 0



(6.16)




𝑔

where the infimum is taken over all weakly filtered module morphisms ℎ

with 𝑔 ◦ 𝑓 ∈ hom𝜖 (M0 , M0 ) , 𝑔 ◦ 𝑓 ' idM0 .

𝑔 : M1 −→ M0 ,

In case no such 𝑔 exists we put 𝜌( 𝑓 ) = ∞. The measurement 𝜌 estimates the minimal energy required to find a left homotopy inverse for 𝑓 . 6.4.4. Remark. — Similar notions are familiar in Floer theory, generally to compare two quasi-isomorphic chain complexes, and in that case the infimum above is taken also over all morphisms 𝑓 and one also takes into account a homotopy 𝑓 ◦ 𝑔 ' idM1 . For instance, see [UZ16]. The next two lemmas give simple properties of 𝜌 that will be useful below. 𝑓

𝑓0

Lemma 6.11. — Given M0 → M1 , M1 → M2 , one has (6.17)

𝜌( 𝑓 0 ◦ 𝑓 ) ≤ 𝜌( 𝑓 ) + 𝜌( 𝑓 0). 𝑔

𝑔0

Proof. — Indeed, assume M1 → M0 , M2 → M1 are weakly filtered module maps and 𝜂 : 𝑔 ◦ 𝑓 ' idM0 , 𝜂 : 𝑔 0 ◦ 𝑓 0 ' idM1 are the respective homotopies. Assume that 𝑓 , 𝑔, 𝜂, 𝑓 0 , 𝑔 0 , 𝜂0 shift filtrations by ≤ 𝑠, 𝑟, 𝑘, 𝑠 0 , 𝑟 0 , 𝑘 0, respectively. These numbers can be taken larger but as close as desired to the respective action levels. Notice that 𝑓 0 ◦ 𝑓

ASTÉRISQUE 426

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121

shifts filtrations by ≤ 𝑠 + 𝑠 0, 𝑔 ◦ 𝑔 0 shifts filtrations by ≤ 𝑟 + 𝑟 0 and, moreover, the homotopy 𝜂¯ = 𝑔 ◦ 𝜂0 ◦ 𝑓 + 𝜂 : 𝑔 ◦ 𝑔 0 ◦ 𝑓 0 ◦ 𝑓 ' idM0 shifts filtrations by ≤ max{𝑟 + 𝑠 + 𝑘 0 , 𝑘}. This implies the claim.



To state the second property, assume that the weakly filtered module M1 can be written as a weakly filtered iterated cone M1 = C𝑜𝑛𝑒(𝐾 𝑠 → C𝑜𝑛𝑒 𝐾 𝑠−1 → · · · · · · → C𝑜𝑛𝑒 N → C𝑜𝑛𝑒 𝐾 𝑖−1 → · · · C𝑜𝑛𝑒(𝐾 2 → 𝐾 1 · · ·








···





and that there is another weakly filtered module N0 together with weakly filtered maps 𝑢 : N → N0 and 𝑣 : N0 → N and a weakly filtered homotopy 𝜉 : 𝑣 ◦ 𝑢 ' idN.

Lemma 6.12. — There is another weakly filtered module M10 that can be written as a filtered iterated cone of the same form as the decomposition for M1 except with N0 replacing N and there is an associated map 𝑢 0 : M1 → M10 so that 𝜌(𝑢 0) ≤ max{𝐴(𝑢) + 𝐴(𝑣), 𝐴(𝜉), 0}. As a corollary we deduce that given M1 , N as well as N0 and a weakly filtered map 𝑢 : N → N0 with 𝜌(𝑢) < ∞, then, for any 𝜖 > 0, there exists a weakly filtered module M10 and a map 𝑢 0 : M1 → M10 as in the lemma such that 𝜌(𝑢 0) ≤ 𝜌(𝑢) + 𝜖.

(6.18)

Proof of Lemma 6.12. — By recurrence, the proof is easily reduced to showing the statement for two particular types of decompositions: 𝜙

M1 = C𝑜𝑛𝑒 N −→ 𝐾 1




𝜙

and

M1 = C𝑜𝑛𝑒 𝐾 2 −→ N .




We will only treat here the first case the second being entirely similar. Without loss of generality, we may assume that 𝜙 does not shift action filtrations. Assume that the map 𝑣 : N0 → N shifts filtrations by ≤ 𝑟, the map 𝑢 shifts filtrations by ≤ 𝑠 and 𝜉 shifts filtration by ≤ 𝑘. Following the definitions of weakly filtered cones in Section 2.4 we construct M10 as follows. Let 𝑣¯ : 𝑆−𝑟 N0 → N be given by the map 𝑣 after shifting the filtration of its domain up by 𝑟. Define 𝜙0 := 𝜙 ◦ 𝑣¯ : 𝑆 −𝑟 N0 −→ 𝐾 1 and put M10 = C𝑜𝑛𝑒(𝜙0). With the notation in (2.7), this cone is defined by taking the action shift of 𝜙0 to be 0. There are module morphisms 𝑣 0 : M10 → M1 defined as ¯ 𝜙 ◦ 𝜉 + id𝐾1 ) where 𝑢¯ : N → 𝑆 −𝑟 N0 is the 𝑣 0 = (𝑣¯ , id𝐾1 ) and 𝑢 0 : M1 → M10 , 𝑢 0 = (𝑢, map 𝑢 with its target with a shifted filtration (these equations have to be interpreted component by component, as in the definition of the structure maps of cones of 𝐴∞ modules). There is also a homotopy 𝜉0 : M1 → M1 , 𝜉0 : 𝑣 0 ◦ 𝑢 0 ' id given by the formula 𝜉0 = (𝜉, 0). Notice that 𝑣 0 does not shift filtrations; 𝑢 0 shifts action filtrations by ≤ max{𝑟 + 𝑠, 𝑘}; 𝜉0 shifts filtration by ≤ 𝑘. As we can take 𝑟, 𝑠, 𝑘 larger but as close as desired to, respectively, 𝐴(𝑣), 𝐴(𝑢), 𝐴(𝜉) this implies the claim. 

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6.4.5. Algebraic weights on 𝑇 𝑆 𝐷F𝑢𝑘 ∗ (𝑀). — We now use the measurement 𝜌 introduced in § 6.4.3 to define an algebraic weight 𝑤, in the sense of § 6.4.1. We will assume here that X = 𝐷F𝑢𝑘 ∗ (𝑀) and that X0 consists of the Yoneda modules associated to the Lagrangians in L𝑎𝑔 ∗ (𝑀). We will appeal to the constructions from Section 3.3. Recall from Proposition 3.1 0 we associate a weakly filtered that to a system of coherent perturbation data 𝑝 ∈ 𝐸reg ∗ 𝐴∞ -category F𝑢𝑘( C; 𝑝), where C = L𝑎𝑔 (𝑀). We also recall that we denote by N the family of coherent perturbation data D = (𝐾, 𝐽) with 𝐾 ≡ 0. Proposition 3.1 also shows that for 𝑝0 ∈ N the discrepancies of the categories F𝑢𝑘( C; 𝑝) tend to zero when 𝑝 → 𝑝0 . We will denote by F𝑢𝑘( C; 𝑝)Δ the category of all (finite) iterated weakly filtered cones that one can construct – as in Section 2.4 – out of the objects of F𝑢𝑘( C; 𝑝). There is clearly a functor F𝑢𝑘( C; 𝑝)Δ −→ 𝐷F𝑢𝑘 ∗ (𝑀) that forgets filtrations on objects and associates to each morphism its homology class (again, at the same time forgetting the filtration). We denote by [𝑋] the image of an object 𝑋 through this functor and similarly for morphisms. The distinguished triangles in 𝐷F𝑢𝑘 ∗ (𝑀) are associated through this functor to the cone attachments in F𝑢𝑘( C; 𝑝)Δ and there is a similar correspondence for the iterated cones. Let 𝜙¯ : L → (L1 , . . . , L𝑘 ), 𝜙¯ = (𝜙, 𝑎, 𝜂) be a morphism in 𝑇 𝑆 X0 (see § 6.4.1) and (6.19)

¯ := inf 𝜌(𝛼) ; 𝛼 ∈ MorF𝑢𝑘( C;𝑝)Δ , 𝛼 : L → M, 𝑤 𝑝 (𝜙)



such that M admits an iterated cone ¯ =𝜂 . decomposition 𝜂¯ with [𝛼] = 𝜙, [M] = 𝑎, [𝜂] ¯ infimizes 𝜌 among all the filtered models of the morphism 𝜙¯ In summary, 𝑤 𝑝 ( 𝜙) inside F𝑢𝑘( C; 𝑝). The weight 𝑤 𝑝 satisfies (6.12), hence can be used as in § 6.4.1 to define a pseudo-metric 𝑠¯ 𝑝F. It is useful to define also similar notions for points 𝑝 0 ∈ N. For this purpose, we set
¯ = lim sup(𝑤 𝑝 𝜙) ¯ . 𝑤 𝑝0 ( 𝜙) 𝑝→𝑝 0

It is easy to see that 𝑤 𝑝0 continues to satisfy (6.12) and therefore there is a corresponding weighted fragmentation pseudo-metric 𝑠¯ 𝑝F0 . It follows from the proof of the first part of Theorem 5.1 that

Corollary 6.13. — Let 𝜙¯ : L → (L1 , . . . , L𝑘 ) be a morphism in 𝑇 𝑆 𝐷F𝑢𝑘 ∗ (𝑀). (i) There exists 𝑝0 ∈ N such that, with the notation in Theorem 5.1, we have ¯ ≥ 1 𝛿(𝐿; 𝑆). 𝑤 𝑝0 ( 𝜙) 2

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123

e (𝐿1 , . . . , 𝐿 𝑘 ) with Φ(𝑉) = 𝜙¯ (ii) If there exists a Lagrangian cobordism 𝑉 : 𝐿 0 we have e is the functor from (6.14)), then for any 𝑝 ∈ 𝐸reg (where Φ ¯ S(𝑉) ≥ 𝑤 𝑝 ( 𝜙). For 𝜙¯ a morphism as in Corollary 6.13, define ¯ := sup 𝑤 𝑝0 ( 𝜙). ¯ 𝑤 alg ( 𝜙) 𝑝 0 ∈N

The weight 𝑤 alg still satisfies (6.12) and, as in § 6.4.1, there is an associated pseudoF metric, 𝑠¯alg on L𝑎𝑔 ∗ (𝑀). Point (i) of 6.13 implies that Corollary 6.4 remains valid F taking the place of 𝑑 F. Moreover, if F, F0 satisfy the assumption in Corolwith 𝑠¯alg lary 6.5, then the formula

 F F0 F,F0 b 𝑠alg = max 𝑠¯alg , 𝑠¯alg

(6.20)

0

F,F defines a metric on L𝑎𝑔 ∗ (𝑀). Point (ii) of Corollary 6.13 shows that b 𝑠alg is bounded 0 F,F b from above by the shadow metric 𝑑 from 6.5.

6.4.6. Remark. — Assume that F and F0 satisfy the hypothesis in Corollary 6.5 and that they both generate 𝐷F𝑢𝑘 ∗ (𝑀). In this case, the weights 𝑤 𝑝 are finite and thus 0 the pseudo-metrics 𝑠¯ 𝑝F as well as b 𝑠 𝑝F,F (which is defined by the obvious analogue of (6.20)) are also finite. On the other hand, for a fixed 𝑝 it is not clear that the pseudo0 F,F0 is non-degenerate but might be metric b 𝑠 𝑝F,F is non-degenerate. By contrast, b 𝑠alg infinite. Proof of Corollary 6.13. — Let 𝜙¯ = (𝜙, 𝑎, 𝜂) and consider a category F𝑢𝑘( C, 𝑝) and a map 𝛼 : L → M so that [𝛼] = 𝜙, [M] = 𝑎, and so that the cone-decomposition 𝜂 corresponds to the writing of M as a weakly filtered iterated cone: M = C𝑜𝑛𝑒 L𝑘 → C𝑜𝑛𝑒 L𝑘−1 · · · → C𝑜𝑛𝑒 L2 → L1





···





.

Let 𝛽 : M → L be another map and assume that 𝜁 : L → L is a homotopy so that 𝜁 : 𝛽 ◦ 𝛼 ' idL. Assume that 𝛼 shifts filtrations by ≤ 𝑠, 𝛽 shifts filtrations by ≤ 𝑟 and 𝜁 shifts filtrations by ≤ 𝑘. Consider 𝛽

M1 = Cone(M → L) and the inclusion 𝑖 : L → M1 , 𝑖 = (0, idL). As described in Section 2.4, when defining the cone M1 we use the value 𝑟 to write M1 = 𝑆−𝑟 M ⊕ L. We now notice that the map 𝜁¯ = (𝛼, 𝜁) : L → M1 is a homotopy 𝜁¯ : 𝑖 ' 0 and we see that 𝜁¯ shifts filtrations by ≤ max{𝑟 + 𝑠, 𝑘}. We deduce: (6.21)

𝐵 ℎ (𝑖) ≤ 𝜌(𝛼).

Using this remark we now return to the setting of the proof of Theorem 5.1. In particular, we pick the choice of perturbation data 𝑝 as in (5.10) and, for coherence of notation, we put 𝐿0 = 𝐿. Instead of the complex 𝒞𝑝,ℎ which has a geometric

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construction we use the complex M1 (𝐿0 ) constructed above. Inequality (5.12) is a consequence of (5.11). If we replace inequality (5.11) with (6.21) , we can still deduce an inequality similar to (5.12) but with 𝜌(𝛼) instead of S(𝑊). In other words, there is (6.22)

𝑏 0 ∈ M1 (𝐿0 )

with

𝐴 𝑏 0; M1 (𝐿0 ) ≤ 𝐴 𝑒 𝐿0 ; M1 (𝐿0 ) + 𝜌(𝛼) + 12 𝜖.







The reason is that we do not need to use in this argument the boundary depth of the chain complex M1 (𝐿0 ) (which in our algebraic setting might not even be acyclic) but only the boundary depth of the element 𝑒 𝐿0 which is controled by the boundary depth of the map 𝑖 : L = L0 → M1 which in turn is controled by 𝜌(𝛼). Given ¯ = inf[𝛼]=𝜙 𝜌(𝛼) we may assume that that 𝑤 𝑝 ( 𝜙) ¯ + 𝜖000 𝜌(𝛼) ≤ 𝑤 𝑝 ( 𝜙) and by continuing as in the proof of Theorem 5.1 we obtain, after making 𝑝 → 𝑝0 that there is a Floer polygon 𝑣0 (compare to (5.17)) such that ¯ + 1 𝜖 + 𝜖000 . 𝜔(𝑣0 ) ≤ 𝑤 𝑝0 ( 𝜙) 2

The argument ends by the same type of application of the Lelong inequality as in the proof of the Theorem 5.1. The proof of the second point of the corollary is again basically contained in the proof of Theorem 5.1. It uses the isotopy pictured in Figure 5 but applies the construction there directly to the cobordism 𝑉 in Figure 4 (and not to 𝑊). As in (5.9) we deduce the existence of a weakly filtered module 𝜙𝑘

𝜙 𝑘−1

𝜙2

(6.23) M𝑉;𝛾,𝑝,ℎ = C𝑜𝑛𝑒 L𝑘 −−→ C𝑜𝑛𝑒 L𝑘−1 −−−−→ C𝑜𝑛𝑒 · · · C𝑜𝑛𝑒 L2 −−→ L1 ))· · ·






(where we neglect a small shift that can be made to → 0). There is also a similar module M𝑉;𝛾0 ,𝑝,ℎ which is identified with the Yoneda module of 𝐿. Isotopy 𝛾0 → 𝛾 of Hofer length ≤ S(𝑉) + 21 𝜖 (see above (5.11)) induces module homomorphisms (see for instance [FOOO09a, Chapter 5], at least for modules over an 𝐴∞ -algebra, the case of 𝐴∞ -categories is similar; alternatively, a direct argument based on moving boundary conditions is also possible) 𝛼 : M𝑉;𝛾0 ,𝑝,ℎ −→ M𝑉;𝛾,𝑝,ℎ , 𝜂0

𝛽 : M𝑉;𝛾,𝑝,ℎ −→ M𝑉;𝛾0 ,𝑝,ℎ

as well as homotopies 𝜂 : 𝛽 ◦ 𝛼 ' id, : 𝛼 ◦ 𝛽 ' id that are all shifting actions by e we have Φ(𝑉) e not more than S(𝑉) + 12 𝜖. By the definition of the functor Φ = (𝜙, 𝑎, 𝜂) and [𝛼] = 𝜙, 𝑎 = [M𝑉;𝛾,𝑝,ℎ ] and, as we just indicated, we also have 𝜌(𝛼) ≤ S(𝑉) + 12 𝜖. ¯ ≤ S(𝑉) + 1 𝜖.  This means that by definition (6.19), 𝑤 𝑝 ( 𝜙) 2

ASTÉRISQUE 426

,

BIBLIOGRAPHY

[Arn80]

V. Arnol0d – “Lagrange and Legendre cobordisms. I”, Funktsional. Anal. i Prilozhen. 14 (1980), no. 3, p. 1–13, 96.

[Aur07]

D. Auroux – “Mirror symmetry and 𝑇-duality in the complement of an anticanonical divisor”, J. Gökova Geom. Topol. GGT 1 (2007), p. 51–91.

[BCa]

P. Biran & O. Cornea – In preparation.

[BCb]

P. Biran & O. Cornea – “Lagrangian cobordism and Fukaya categories”, ArXiv version (2018). Can be found at http://arxiv.org/pdf/1304. 6032.

[BCc]

P. Biran & O. Cornea – “A Lagrangian pictionary”, Preprint (2020). Can be found at https://arxiv.org/pdf/2003.07332.

[BC06]

J.-F. Barraud & O. Cornea – “Homotopic dynamics in symplectic topology”, in Morse theoretic methods in nonlinear analysis and in symplectic topology (Dordrecht) (P. Biran, O. Cornea & F. Lalonde, eds.), NATO Sci. Ser. II Math. Phys. Chem., vol. 217, Springer, 2006, p. 109–148.

[BC07]

, “Lagrangian intersections and the Serre spectral sequence”, Annals of Mathematics 166 (2007), p. 657–722.

[BC12]

P. Biran & O. Cornea – “Lagrangian topology and enumerative geometry”, Geom. Topol. 16 (2012), no. 2, p. 963–1052.

[BC13]

, “Lagrangian cobordism. I”, J. Amer. Math. Soc. 26 (2013), no. 2, p. 295–340.

[BC14]

P. Biran & O. Cornea – “Lagrangian cobordism and Fukaya categories”, Geom. Funct. Anal. 24 (2014), no. 6, p. 1731–1830.

126

BIBLIOGRAPHY

[BC17]

P. Biran & O. Cornea – “Cone-decompositions of Lagrangian cobordisms in Lefschetz fibrations”, Selecta Math. (N.S.) 23 (2017), no. 4, p. 2635–2704.

[BCS]

P. Biran, O. Cornea & E. Shelukhin – “Lagrangian shadows and triangulated categories”, Preprint (2018). Can be found at http://arxiv. org/pdf/1806.06630v1.

[Bis]

M. R. Bisgaard – “A distance expanding flow on exact Lagrangian cobordism classes”, Preprint (2016). Can be found at https://arxiv.org/pdf/ 1608.05821.

[Bis19a]

M. R. Bisgaard – “Invariants of Lagrangian cobordisms via spectral numbers”, J. Topol. Anal. 11 (2019), no. 1, p. 205–231.

[Bis19b]

, “Topology of (small) Lagrangian cobordisms”, Algebr. Geom. Topol. 19 (2019), no. 2, p. 701–742.

[BS19]

J.-F. Barraud & L. S. Suarez – “The fundamental group of a rigid Lagrangian cobordism”, Ann. Math. Qué. 43 (2019), no. 1, p. 125–144.

[CC16]

F. Charette & O. Cornea – “Categorification of Seidel’s representation”, Israel J. Math. 211 (2016), no. 1, p. 67–104.

[Che97]

Y. Chekanov – “Lagrangian embeddings and Lagrangian cobordism”, in Topics in singularity theory, Amer. Math. Soc. Transl. Ser. 2, vol. 180, Amer. Math. Soc., Providence, RI, 1997, p. 13–23.

[Che00]

Y. V. Chekanov – “Invariant Finsler metrics on the space of Lagrangian embeddings”, Math. Z. 234 (2000), no. 3, p. 605–619.

[Cor94]

O. Cornea – “Cone-length and Lusternik-Schnirelmann category”, Topology 33 (1994), p. 95–111.

[CR03]

O. Cornea & A. Ranicki – “Rigidity and gluing for Morse and Novikov complexes”, J. Eur. Math. Soc. 5 (2003), no. 4, p. 343–394.

[CS19]

O. Cornea & E. Shelukhin – “Lagrangian cobordism and metric invariants”, J. Differential Geom. 112 (2019), no. 1, p. 1–45.

[EP03]

M. Entov & L. Polterovich – “Calabi quasimorphism and quantum homology”, Int. Math. Res. Not. (2003), no. 30, p. 1635–1676.

[FO97]

K. Fukaya & Y.-G. Oh – “Zero-loop open strings in the cotangent bundle and Morse homotopy”, Asian J. Math. 1 (1997), no. 1, p. 96–180.

[FOOO09a] K. Fukaya, Y.-G. Oh, H. Ohta & K. Ono – Lagrangian intersection Floer theory: anomaly and obstruction. Part I, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI, 2009.

ASTÉRISQUE 426

BIBLIOGRAPHY

127

[FOOO09b]

, Lagrangian intersection Floer theory: anomaly and obstruction. Part II, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI, 2009.

[KS]

A. Kislev & E. Shelukhin – “Bounds on spectral norms and applications”, Preprint (2018). Can be found at arXiv:1810.09865 [math.SG].

[KS02]

M. Khovanov & P. Seidel – “Quivers, Floer cohomology, and braid group actions”, J. Amer. Math. Soc. 15 (2002), no. 1, p. 203–271.

[MS12]

D. McDuff & D. Salamon – 𝐽-holomorphic curves and symplectic topology, second ed., American Mathematical Society Colloquium Publications, vol. 52, American Mathematical Society, Providence, RI, 2012.

[Oh93]

Y.-G. Oh – “Floer cohomology of Lagrangian intersections and pseudoholomorphic disks. I.”, Comm. Pure Appl. Math. 46 (1993), no. 7, p. 949– 993.

[Oh95]

, “Addendum to: "Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I."”, Comm. Pure Appl. Math. 48 (1995), no. 11, p. 1299–1302.

[Oh96a]

, “Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings”, Internat. Math. Res. Notices 1996 (1996), no. 7, p. 305–346.

[Oh96b]

, “Relative floer and quantum cohomology and the symplectic topology of lagrangian submanifolds”, in Contact and symplectic geometry (C. B. Thomas, ed.), Publications of the Newton Institute, vol. 8, Cambridge Univ. Press, Cambridge, 1996, p. 201–267.

[Oh05a]

, “Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds”, in The breadth of symplectic and Poisson geometry. Festschrift in honor of Alan Weinstein (Boston, MA) (J. Marsden & T. Ratiu, eds.), Progr. Math., vol. 232, Birkhäuser Boston, 2005, p. 525– 570.

[Oh05b]

, “Spectral invariants, analysis of the Floer moduli space, and geometry of the Hamiltonian diffeomorphism group”, Duke Math. J. 130 (2005), no. 2, p. 199–295.

[Oh06]

, “Lectures on Floer theory and spectral invariants of Hamiltonian flows”, in Morse theoretic methods in nonlinear analysis and in symplectic topology (Dordrecht) (P. Biran, O. Cornea & F. Lalonde, eds.), NATO Sci. Ser. II Math. Phys. Chem., vol. 217, Springer, 2006, p. 321–416.

[OZ]

Y.-G. Oh & K. Zhu – “Thick-thin decomposition of Floer trajectories and adiabatic gluing”, Preprint (2011). Can be found at http://arxiv.org/ pdf/1103.3525.pdf.

SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2021

128

BIBLIOGRAPHY

[OZ11]

, “Floer trajectories with immersed nodes and scale-dependent gluing”, J. Symplectic Geom. 9 (2011), no. 4, p. 483–636.

[Sch00]

M. Schwarz – “On the action spectrum for closed symplectically aspherical manifolds”, Pacific J. Math. 193 (2000), no. 2, p. 419–461.

[Sei00]

P. Seidel – “Graded Lagrangian submanifolds”, Bull. Soc. Math. France 128 (2000), no. 1, p. 103–149.

[Sei08]

, Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008.

[Shea]

E. Shelukhin – “Symplectic cohomology and a conjecture of Viterbo”, Preprint (2019). Can be found at arXiv:1904.06798 [math.SG].

[Sheb]

, “Viterbo conjecture for Zoll symmetric spaces”, Preprint (2018). Can be found at arXiv:1811.05552 [math.SG].

[Sua17]

L. S. Suarez – “Exact Lagrangian cobordism and pseudo-isotopy”, Internat. J. Math. 28 (2017), no. 8, p. 1750059, 35.

[Ush11]

M. Usher – “Boundary depth in Floer theory and its applications to Hamiltonian dynamics and coisotropic submanifolds”, Israel J. Math. 184 (2011), p. 1–57.

[Ush13]

, “Hofer’s metrics and boundary depth”, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), no. 1, p. 57–128 (2013).

[Ush14]

, “Submanifolds and the Hofer norm”, J. Eur. Math. Soc. 16 (2014), p. 1571–1616.

[UZ16]

M. Usher & J. Zhang – “Persistent homology and Floer–Novikov theory”, Geom. Topol. 20 (2016), no. 6, p. 3333–3430.

[Vit]

C. Viterbo – “Symplectic homogenization”, Preprint (2014). Can be found at arXiv:0801.0206 [math.SG].

[Vit92]

C. Viterbo – “Symplectic topology as the geometry of generating functions”, Math. Ann. 292 (1992), no. 4, p. 685–710.

ASTÉRISQUE 426

ASTÉRISQUE

2021 423. K. ARDAKOV – Equivariant D-modules on rigid analytic spaces 2020 422. Séminaire BOURBAKI – Volume 2018/2019, exposés 1151–1165 421. J.H. BRUINIER, B. HOWARD, Stephen S. KUDLA, K. MADAPUSI PERA, M. RAPOPORT & T. YANG – Arithmetic divisors on orthogonal and unitary Shimura varieties 420. H. RINGSTRÖM – Linear systems of wave equations on cosmological backgrounds with convergent asymptotics 419. V. GORBOUNOV, O. GWILLIAM & B.R. WILLIAMS – Chiral differential operators via quantization of the holomorphic 𝜎-model 418. R. BEUZART-PLESSIS – A local trace formula for the Gan-Gross-Prasad conjecture for unitary groups : the Archimedean case 417. J.D. ADAMS, M.A.A. VAN LEEUWEN, P.E. TRAPA & D.A. VOGAN, Jr. – Unitary representations of real reductive groups 416. S. CROVISIER, R. KRIKORIAN, C. MATHEUS & S. SENTI (éditors) – Some aspects of the theory of dynamical systems : a tribute to Jean-Christophe Yoccoz (volume II) 415. S. CROVISIER, R. KRIKORIAN, C. MATHEUS & S. SENTI (éditors) – Some aspects of the theory of dynamical systems : a tribute to Jean-Christophe Yoccoz (volume I) 2019 414. Séminaire BOURBAKI – Volume 2017/2018, exposés 1136–1150 413. M. CRAINIC, R. LOJA FERNANDES & D. MARTINEZ – Renormalization in Quantum Field Theory (after R. Borcherds) 412. E. HERSCOVICH – Renormalization in Quantum Field Theory ( after R. Borcherds) 411. G. DAVID – Local regularity properties of almost- and quasiminimal sets with a sliding boundary condition 410. P. BERGER & J.-C. YOCCOZ – Strong regularity 409. F. CALEGARI & A. VENKATESH – A torsion Jacquet-Langlands correspondence 408. D. MAULIK & A. OKOUNKOV – Quantum Groups and Quantum Cohomology 407. Séminaire BOURBAKI – Volume 2016/2017, exposés 1120-1135 2018 406. Laurent FARGUES & J.-M. FONTAINE – Curves and vector bundles in 𝑝-adic Hodge theory 405. J.-F. BONY, S. FUJIIE, T. RAMOND & M. ZERZERI – Resonances for homoclinic trapped sets 404. O. MATTE & J.S .MØLLER – Feynman-Kac formulas for the ultra-violet renormalized Nelson models 403. M. BERTI, T. KAPPELER & R. MONTALTO – Large KAM tori for perturbations of the defocusing NLS equation 402. H. BAO & W. WANG – A new approach to Kazhdan-Lustig theory of type 𝐵 via quantum symmetric pairs 401. J. SZEFTEL – Parametrix for wave equations on a rough background III : space-time regularity of the phase 400. A. DUCROS – Families of Berkovich spaces 399. T. LIDMAN & C. MANOLESCU – The equivalence of two Seiberg-Witten Floer homologies 398. WEE TECK GAN, FAN GAO & M. H. WEISSMAN – 𝐿-groups and the Langlands program for covering groups — The Langlands-Weissman Program for Brylinski-Deligne extensions 397. S. RICHE & G. WILLIAMSON – Tilting modules and the 𝑝-canonical basis

2017 396. Y. SAKELLARIDIS & A. VENKATESH – Periods and Harmonic Analysis on Spherical Varieties 395. V. GUIRARDEL & G. LEVITT – JSJ decompositions of groups 394. JUNYI XIE – The Dynamical Mordell-Lang Conjecture for polynomial endomorphisms of the affine plane 393. G. BIEDERMANN, G. RAPTIS & M. STELZER – The realization space of an unstable coalgebra 392. G. DAVID, M. FILOCHE, D. JERISON & S. MAYBORODA – A free boundary problem for the localization of eigenfunctions 391. S. KELLY – Voevodsky motives and ldh-descent 390. SÉMINAIRE BOURBAKI – Volume 2015/2016, exposés 1104–1119 389. S. GRELLIER & P. GÉRARD – The cubic Szegö equation and Hankel operators 388. T. LÉVY – The master field on the plane 387. R. M. KAUFMANN & B. C. WARD – Feynman Categories 386. B. LEMAIRE & GUY HENNIART – Représentations des espaces tordus sur un groupe réductif connexe p-adique 2016 385. A. BRAVERMAN, M. FINKELBERG & H. NAKAJIMA – Instanton moduli spaces and 𝑊-algebras 384. T. BRADEN, A. LICATA, N. PROUDFOOT & B. WEBSTER – Quantizations of conical symplectic resolutions 383. S. GUILLERMOU, G. LEBEAU, A. PARUSIŃSKI, P. SCHAPIRA & J.-P. SCHNEIDERS – Subanalytic sheaves and Sobolev spaces 382. F. ANDREATTA, S. BIJAKOWSKI, A. IOVITA, P.L. KASSAEI, V. PILLONI, B. STROH, Y. TIAN & L. XIAO – Arithmétique 𝑝-adique des formes de Hilbert 381. L. BARBIERI-VIALE & B. KAHN – On the derived category of 1-motives 380. SÉMINAIRE BOURBAKI – Volume 2014/2015, exposés 1089–1103 379. O. BAUES & V. CORTÉS – Symplectic Lie groups 378. F. CASTEL – Geometric representations of the braid groups 377. S. HURDER & A. RECHTMAN – The dynamics of generic Kuperberg flows 376. K. FUKAYA, Y.-G. OH, H. OHTA & K. ONO – Lagrangian Floer theory and mirror symmetry on compact toric manifolds 2015 375. F. FAURE & M. TSUJII – Prequantum transfer operator for symplectic Anosov diffeomorphism 374. T. ALAZARD & J.-M. DELORT – Sobolev estimates for two dimensional gravity water waves 373. F. PAULIN, M. POLLICOTT & B. SCHAPIRA – Equilibrium states in negative curvature 372. R. FRIGERIO, J.-F. LAFONT & A. SISTO – Rigidity of high dimensional graph manifolds 371. K. KEDLAYA & R. LIU – Relative 𝑝-adic Hodge theory : Foundations 370. De la géométrie algébrique aux formes automorphes (II), J.-B. BOST, P. BOYER, A. GENESTIER, L. LAFFORGUE, S. LYSENKO, S. MOREL & B.C. NGO, éditeurs 369. De la géométrie algébrique aux formes automorphes (I), J.-B. BOST, P. BOYER, A. GENESTIER, L. LAFFORGUE, S. LYSENKO, S. MOREL & B.C. NGO, éditeurs 367-368. SÉMINAIRE BOURBAKI – Volume 2013/2014, exposés 1074–1088 2014 366. J. MARTÍN & M. MILMAN – Fractional Sobolev inequalities : symmetrization, isoperimetry and interpolation 365. B. KLEINER & J. LOTT – Local collapsing, orbifolds, and geometrization 363-364. L. ILLUSIE, Y. LASZLO & F. ORGOGOZO avec la collaboration de F. DÉGLISE, A. MOREAU, V. PILLONI, M. RAYNAUD, J. RIOU, B. STROH, M. TEMKIN et W. ZHENG – Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents. (Séminaire à l’École polytechnique 2006–2008) 362. M. JUNGE & M. PERRIN – Theory of H𝑝 -spaces for continuous filtrations in von Neumann algebras 361. SÉMINAIRE BOURBAKI – Volume 2012/2013, exposés 1059–1073 360. J.I. BURGOS GIL, P. PHILIPPON & M. SOMBRA – Arithmetic geometry of toric varieties. Metrics, measures and heights 359. M. BROUÉ, G. MALLE & J. MICHEL – Split spetses for primitive reflection groups

We introduce new metrics on spaces of Lagrangian submanifolds, not necessarily in a fixed Hamiltonian isotopy class. Our metrics arise from measurements involving Lagrangian cobordisms. We also show that splitting Lagrangians through cobordism has an energy cost and, from this cost being smaller than certain explicit bounds, we deduce some forms of rigidity of Lagrangian intersections. We also fit these constructions in the more general algebraic setting of triangulated categories, independent of Lagrangian cobordism. As a main technical tool, we develop aspects of the theory of (weakly) filtered 𝐴∞ -categories. Nous introduisons de nouvelles métriques sur des espaces des sous-variétés Lagrangiennes dont la classe d’isotopie Hamiltonienne n’est pas nécessairement fixée. Ces métriques proviennent de certaines quantités associées aux cobordismes Lagrangiens. Nous montrons également que la décomposition d’un Lagrangien à travers un cobordisme a un coût énergétique non-nul et, à partir d’une borne explicite de ce coût, nous déduisons des formes de rigidité des intersections Lagrangiennes. Nous donnons un sens à certaines de ces constructions dans le cadre algébrique plus général des catégories triangulées, indépendamment du cobordisme Lagrangien. Comme outil technique central, nous développons certains aspects de la théorie des catégories 𝐴∞ (faiblement) filtrées.