Table Of Content

Lagrangian Intersection Floer Theory Contents Volume I Preface xi Chapter 1. Introduction 1 1.1 What is Floer (co)homology 1 1.2 General theory of Lagrangian Floer cohomology 5 1.3 Applications to symplectic geometry 13 1.4 Relation to mirror symmetry 16 1.5 Chapter-wise outline of the main results 25 1.6 Acknowledgments 35 1.7 Conventions 36 Chapter 2. Review: Floer cohomology 39 2.1 Bordered stable maps and the Maslov index 39 2.1.1 The Maslov index: the relative first Chern number 39 2.1.2 The moduli space of bordered stable maps 43 2.2 The Novikov covering and the action functional 49 2.2.1 The Γ-equivalence 50 2.2.2 The action functional and the Maslov-Morse index 51 2.3 Review of Floer cohomology I: without anomaly 53 2.3.1 The L2-gradient equation of A 53 2.3.2 Floer’s definition: Z -coefficients. 57 2 2.3.3 Bott-Morse Floer cohomology 59 2.4 Review of Floer cohomology II: anomaly appearance 60 2.4.1 The Floer cochain module 61 2.4.2 The Floer moduli space 62 2.4.3 The Novikov ring ΛR(L) 66 2.4.4 Monotone Lagrangian submanifolds 69 2.4.5 Appearance of the primary obstruction 71 Chapter 3. The A∞ algebra associated to a Lagrangian submanifold 77 3.1 Outline of Chapter 3 77 3.2 Algebraic framework on filtered A∞ algebras 86 3.2.1 A∞ algebras and homomorphisms 86 3.2.2 Filtered A∞ algebras and homomorphisms 89 3.3 Algebraic framework on the homotopy unit 94 3.3.1 Definition of the homotopy unit 94 3.3.2 Unital (resp. homotopy unital) A∞ homomorphisms 97 3.4 A∞ deformation of the cup product 97 v vi CONTENTS 3.5 The filtered A∞ algebra associated to a Lagrangian submanifold 102 3.6 Bounding cochains and the A∞ Maurer-Cartan equation. 107 3.6.1 Bounding cochains and deformations 108 3.6.2 Obstruction for the existence of bounding cochain 111 3.6.3 Weak unobstructedness and existence of Floer cohomology 114 3.6.4 The superpotential and M(cid:2)(C) 117 3.7 A∞ bimodules and Floer cohomology 120 3.7.1 Algebraic framework 120 3.7.2 A∞ bimodule homomorphisms 123 3.7.3 Weak unobstructedness and deformations 125 3.7.4 The filtered A∞ bimodule C(L(1),L(0);Λ0,nov) 126 3.7.5 The Bott-Morse case 137 3.7.6 Examples 151 3.7.7 The multiplicative structure on Floer cohomology 155 3.8 Inserting marked points in the interior 156 3.8.1 The operator p 156 3.8.2 Applications to vanishing of the obstruction classes o (L) 159 k 3.8.3 Outline of the construction of the operator p 161 3.8.4 The operator q 165 3.8.5 Bulk deformation of filtered A∞ structures 168 3.8.6 Outline of the construction of the operator q 175 3.8.7 The operator r and the A∞ bimodule 178 3.8.8 Construction of the operator r 181 3.8.9 Generalization of the operator p 182 3.8.10 Proof of parts of Theorems B, C and G 188 Chapter 4. Homotopy equivalence of A∞ algebras 191 4.1 Outline of Chapters 4 and 5 191 4.2 Homotopy equivalence of A∞ algebras: the algebraic framework 197 4.2.1 Models of [0,1]×C 197 4.2.2 Homotopies between A∞ homomorphisms 205 4.2.3 The unital or homotopy-unital cases 208 4.3 Gauge equivalence of bounding cochains 211 4.3.1 Basic properties and the category HA∞ 211 4.3.2 M (C) and its homotopy invariance 215 weak 4.3.3 M (L) and its homotopy invariance 216 weak,def 4.4 Uniqueness of the model of [0,1]×C 217 4.4.1 Induction on the number filtration I 218 4.4.2 A structures and homomorphisms 219 K 4.4.3 Induction on the number filtration II 220 4.4.4 Unital case I: the unfiltered version 223 4.4.5 Coderivation and Hochschild cohomology 226 4.4.6 Induction on the energy filtration 230 4.4.7 Unital case II: the filtered version 232 4.5 Whitehead theorem in A∞ algebras 233 4.5.1 Extending A homomorphisms to A homomorphisms 234 K K+1 4.5.2 Proof of Theorem 4.2.45 I: the number filtration 236 4.5.3 Unital case: the unfiltered version 237 CONTENTS vii 4.5.4 Extending filtered A∞ homomorphism modulo Tλi to modulo Tλi+1 239 4.5.5 Proof of Theorem 4.2.45 II: the energy filtration 241 4.6 Homotopy equivalence of A∞ algebras: the geometric realization 242 4.6.1 Construction of A∞ homomorphisms 242 4.6.2 Homotopies between A∞ homomorphisms 249 4.6.3 Compositions 257 4.6.4 Homotopyequivalence andtheoperatorqI:changingthecycle in the interior 259 4.6.5 Homotopyequivalence andtheoperatorqII:invarianceofsym- plectic diffeomorphisms 1 261 4.6.6 Homotopy equivalence and the operator q III: invariance of symplectic diffeomorphisms 2 264 Chapter 5. Homotopy equivalence of A∞ bimodules 267 5.1 Novikov rings 267 5.1.1 Reduction to universal Novikov ring 267 5.1.2 Hamiltonian independence of the Novikov ring 270 5.1.3 Floer cohomologies over Λ(L(0),L(1);(cid:2) ) and Λ 272 0 nov 5.2 HomotopyequivalencesofA∞ bimodules: thealgebraicframework 275 5.2.1 Weakly filtered A∞ bimodule homomorphisms 275 5.2.2 Deformations of A∞ bimodule homomorphisms 276 5.2.3 Homotopies between A∞ bimodule homomorphisms 282 5.2.4 Gauge invariance and the category HA∞(C1,C0) 288 5.2.5 Obstructions to defining A∞ bimodule homomorphisms I 291 5.2.6 Whitehead theorem for A∞ bimodule homomorphisms 292 5.2.7 Obstructions to defining A∞ bimodule homomorphisms II 294 5.3 Homotopy equivalences of A∞ bimodules: the geometric realization 296 5.3.1 Construction of filtered A∞ bimodule homomorphisms 296 5.3.2 Moving Lagrangian submanifolds by Hamiltonian isotopies 306 5.3.3 Homotopies between bimodule homomorphisms 313 5.3.4 Compositions of Hamiltonian isotopies and of bimodule homomorphisms 319 5.3.5 An energy estimate. 321 5.3.6 The operators q,r and homotopy equivalence 326 5.3.7 Wrap-up of the proof of invariance of Floer cohomologies 327 5.4 Canonical models, formal super schemes and Kuranishi maps 330 5.4.1 Canonical models, Kuranishi maps and bounding cochains 330 5.4.2 The canonical models of filtered A∞ bimodules 336 5.4.3 Filtered A∞ bimodules and complex of coherent sheaves 337 5.4.4 Construction of the canonical model 339 5.4.5 Including the operator q 347 5.4.6 Wrap-up of the proofs of Theorems F, G, M, N and Corollaries O, P 349 Chapter 6. Spectral sequences 355 6.1 Statement of the results in Chapter 6 355 viii CONTENTS 6.1.1 The spectral sequence 355 6.1.2 Non-vanishing theorem and a Maslov class conjecture 357 6.1.3 Applications to Lagrangian intersections 360 6.2 A toy model: rational Lagrangian submanifolds 362 6.3 The algebraic construction of the spectral sequence 366 6.3.1 c.f.z. 367 6.3.2 d.g.c.f.z. (differential graded c.f.z.) 369 6.3.3 Construction and convergence 371 6.4 The spectral sequence associated to a Lagrangian submanifold 375 6.4.1 Construction 375 6.4.2 A condition for degeneration: proof of (D.3) 375 6.4.3 Non-vanishing theorem: proof of Theorem 6.1.9. 377 6.4.4 Application to the Maslov class conjecture: proofs of Theorems 6.1.15 and 6.1.17 381 6.4.5 Compatibility with the product structure 382 6.5 Applications to Lagrangian intersections 385 6.5.1 Proof of Theorem H 385 6.5.2 Proof of Theorem I 385 6.5.3 Torsion of the Floer cohomology and Hofer distance: Proof of Theorem J 388 6.5.4 Floer cohomologies of Lagrangian submanifolds that do not in- tersect cleanly 393 6.5.5 Unobstructedness modulo TE 395 Volume II Chapter 7. Transversality 397 7.1 Construction of the Kuranishi structure 398 7.1.1 Statement of the results in Section 7.1 398 7.1.2 Kuranishi charts on Mmain,reg(β): Fredholm theory 401 k+1 7.1.3 Kuranishi charts in the complement of Mmain,reg(β): gluing 404 k+1 7.1.4 Wrap-up of the proof of Propositions 7.1.1 and 7.1.2 418 7.1.5 The Kuranishi structure of Mmain(M(cid:3),L(cid:3),{J } : β;top(ρ)): k+1 ρ ρ A∞ map analog of Stasheff cells 425 7.2 Multisections and choice of a countable set of chains 435 7.2.1 Transversality at the diagonal 436 7.2.2 Inductive construction of compatible system of multisections in the Bott-Morse case 437 7.2.3 Perturbed moduli space running out of the Kuranishi neighbor- hood I 444 7.2.4 Statement of results 445 7.2.5 Proof of Proposition 7.2.35 449 7.2.6 Filtered A structures 458 n,k 7.2.7 Construction of filtered A structures 461 n,K 7.2.8 Perturbed moduli space running out of the Kuranishi neigbor- hood II 466 7.2.9 Construction of filtered A homomorphisms 468 n,K CONTENTS ix 7.2.10 Constructions of filtered An,K homotopies 483 7.2.11 Constructions of filtered A∞ homotopies I: a short cut 502 7.2.12 Constructions of filtered A∞ homotopies II: the algebraic framework on homotopy of homotopies 505 7.2.13 Constructions of filtered A∞ homotopies III: the geometric realization of homotopy of homotopies 534 7.2.14 Bifurcation vs cobordism method: an alternative proof 569 7.3 Construction of homotopy unit 574 7.3.1 Statement of the result and the idea of its proof 574 7.3.2 Proof of Theorem 7.3.1 576 7.3.3 Proof of (3.8.36) 587 7.4 Details of the construction of the operators p, q and r 589 7.4.1 Details of the construction of p 589 7.4.2 Construction of q I: the A version 595 n,K 7.4.3 Construction of q II: q is an L∞ homomorphism 596 (cid:3) 7.4.4 Construction of q III: the homotopy invariance of Der(B(C[1]), (cid:3) B(C[1])) 601 7.4.5 Construction of q IV: wrap-up of the proof of Theorem 3.8.32 621 7.4.6 Proof of Theorem Y 625 7.4.7 AlgebraicformulationofrI:DerB(C ,C ;D)anditshomotopy 1 0 invariance 631 7.4.8 Algebraic formulation of r II: via bifurcation argument 637 7.4.9 Algebraic formulation of r III: via cobordism argument 640 7.4.10 Algebraic formulation of p I: the cyclic bar complex is an L∞ module 644 7.4.11 Algebraic formulation of p II: p induces an L∞ module homomorphism 647 7.5 Compatibility with rational homotopy theory 650 7.5.1 Statement of results 650 7.5.2 Virtual fundamental chain in de Rham theory 652 7.5.3 The Kuranishi structure of Mmain(β ) 654 k+1 0 7.5.4 Construction of the A homomorphism I 655 K 7.5.5 Construction of the A homomorphism II 663 K 7.5.6 The A∞ map to a topological monoid and N(cid:3) 669 k+1 Chapter 8. Orientation 675 8.1 Orientation of the moduli space of unmarked discs 675 8.1.1 The case of holomorphic discs 675 8.1.2 The example of non-orientable family index 684 8.1.3 The case of connecting orbits in Floer theory 686 8.1.4 Change of relatively spin structure and orientation 690 8.2 Convention and preliminaries 691 8.3 Orientationofthemodulispaceofmarkeddiscsandofthesingular strata of the moduli space 698 8.4 Orientation of M (β;P ,... ,P ). 703 (cid:4)+1 1 (cid:4) 8.4.1 Definition of the orientation of M (β;P ,... ,P ) 703 (cid:4)+1 1 (cid:4) 8.4.2 Cyclic symmetry and orientation 705 8.5 The filtered A∞ algebra case 708 8.6 Orientation of the moduli space of constant maps 713 x CONTENTS 8.7 Orientation of the moduli space of connecting orbits 716 8.8 The Bott-Morse case 719 8.9 Orientations of the top-moduli spaces and the twp-moduli spaces 731 8.9.1 Orientation of Mmain(M(cid:3),L(cid:3),{J } :β;top(ρ)) 731 k+1 ρ ρ 8.9.2 Orientation of Mmain({J } :β;twp(ρ);P ,...,P ) 735 k+1 ρ ρ 1 k 8.10 Homotopy units, the operators p,q, continuous families of perturbations, etc. 738 8.10.1 Homotopy unit 738 8.10.2 Operators p, q 738 8.10.3 Continuous families of perturbations 749 Appendices 753 A1 Kuranishi structures 753 A1.1 Review of the definition of the Kuranishi structure and multisections 754 A1.2 Fiber products 764 A1.3 Finite group actions and the quotient space 766 A1.4 A remark on smoothness of coordinate transforms 768 A1.5 Some counter examples 778 A1.6 Some errors in the earlier versions and corrections thereof 779 A2 Singular chains with local coefficients 780 A3 Filtered L∞ algebras and symmetrization of filtered A∞ algebras 782 A4 The differential graded Lie algebra homomorphism in Theorem 7.4.132 787 Bibliography 791 Index 801 Preface With the advent of the method of pseudo-holomorphic curves developed by Gromov in the 80’s and the subsequent Floer’s invention of elliptic Morse theory resulted in Floer cohomology, the landscape of symplectic geometry has changed drastically. Many previously intractable problems in symplectic geometry were solved by the techniques of pseudo-holomorphic curves, and the concept of symplectic topology gradually began totakeshape. This progress was accompanied by parallel developments in physics most notably in closed string theory. In1993,partiallymotivatedbyDonaldson’spantsproductconstructioninFloer cohomology, the first named author introduced the structure of an A∞-category in symplectic geometry whose objects are Lagrangian submanifolds and whose morphisms are the Floer cohomologies (or complexes). Based on this algebraic framework, Kontsevich proposed the celebrated homological mirror symmetry between the derived category of coherent sheaves and the Fukaya category of Lagrangian submanifolds in his 1994 ICM talk in Zu¨rich. Enhanced by the later development in open string theory of D-branes, this homological mirror symmetry has been a sourceofmanynewinsightsandprogressesinbothalgebraicgeometryandsymplec- ticgeometryaswellasinphysics. Howevertherigorousformulationofhomological mirrorsymmetryhasnotbeenmade,largelyduetolackofunderstandingtheFloer theory of Lagrangian submanifolds itself. In this book, we explain how the obstruction to and anomaly in the construction of Floer cohomology arise, provide a precise formulation of the obstructions and then carry out detailed algebraic and analytic study of deformation theory of Floer cohomology. It turns out that even a description of such an obstruction (in a mathematically precise way) requires new homological algebra of filtered A∞- algebras. In addition, verification of existence of such an algebraic structure in the geometriccontextofLagrangiansubmanifoldsrequiresnon-trivialanalyticstudyof the corresponding moduli space of pseudo-holomorphic discs. We also provide var- iousimmediateapplicationsofthesoconstructedFloercohomologytoproblemsin symplectic topology. Many of these improve the previously known results obtained via Floer theory and some firsthand applications to homological mirror symmetry are new. We expect more nontrivial applications of the theory will soon follow as its true potential is unveiled and then realized. While we have been preparing this book, there have been several important developments in symplectic geometry and in related fields. The relationship be- tweentopologicalstrings, D-branesandpseudo-holomorphic curvesandsymplectic Floer theory is now more clearly understood. The usage of higher algebraic structures in Floer theory, which we have been promoting while writing this book, has now become a popular and essential area of research. Furthermore advances of the techniques handling various moduli spaces of solutions to nonlinear PDE’s, xi xii PREFACE intertwined with the formalism of higher algebraic structures, has now made the geometricpicturemoretransparent. Thiswillhelpfacilitatethefurtherprogression of the geometric theory. In this book we take full advantage of these developments and provide the Floer theory of Lagrangian submanifolds in the most general form available at this time. We hope that this book will be a stepping stone for future advancements in symplectic geometry and homological mirror symmetry. Our collaboration which has culminated in completion of this book started during the 1996 (8 July–12 July) conference held in Ascona, Switzerland. We hardly imagined then that our project would continue to span more than 10 years. We have greatlyenjoyedthis collaborationandhope tocontinue it intothecoming decades. In fact our second journey into newly landscaped field of symplectic topology and mirror symmetry has already begun, and we hope to garner more fruits of collaboration: The scene in front looks very different and much more exciting than the one we left behind 13 years ago! June 9 2009. Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono. https://doi.org/10.1090/amsip/046.1/01 CHAPTER 1 Introduction 1.1. What is the Floer (co)homology Floer homology is a semi-infinite (∞/2) dimensional homology theory on an infinite (∞) dimensional space in general. It has been defined in various contexts and proven to be an extremely deep and useful tool to study many problems aris- ing in the various areas including symplectic geometry, gauge theory and the low dimensional topology. Floer’s main motivation to introduce Floer homology in symplectic geometry [Flo88IV] was to prove Arnold’s conjecture [Arn65] for the fixed points of Hamiltonian diffeomorphisms (or exact symplectic diffeomorphisms). One version ofArnold’sconjectureisthatanyHamiltoniandiffeomorphismφ:(M,ω)→(M,ω) on a compact M satisfies (cid:4) (1.1) #Fixφ≥ rankH (M) k k provided all of its fixed points are nondegenerate. A Hamiltonian diffeomorphism (or exact symplectic diffeomorphism) is a symplectic diffeomorphism obtained as the time one map of a (time-dependent) Hamiltonian flow. By considering the diagonal Δ ⊂ (M ×M,ω ⊕−ω), which is a Lagrangian submanifold in the product, one can instead ask its intersection theoretic version, which is indeed the original approach taken by Floer in [Flo88IV]. Let L ,L ⊂ (M,ω) be two compact Lagrangian submanifolds. The ideal 1 0 statement one might expect for Floer homology is as follows: (1.2.1) Wecanassigna(graded)abeliangroup(orZ vectorspace),denotedby 2 HF(L ,L ),toeachpair(L ,L )ofLagrangiansubmanifolds. WecallHF(L ,L ) 1 0 1 0 1 0 the Floer homology. (1.2.2) Floer homology is invariant under the Hamiltonian diffeomorphisms. Namely if φ , i=0,1 are two Hamiltonian diffeomorphisms, then i HF∗(L1,L0)(cid:8)HF∗(φ1L1,φ0L0). (1.2.3) If L = L = L, then the Floer homology group coincides with the 0 1 standard homology group of Lagrangian submanifold L. Namely HF∗(L,L) (cid:8) H∗(L;Z2). (1.2.4) WhenL0 istransversal toL1, Floer homology HF∗(L1,L0)isahomol- ogy group of the chain complex that is given by (cid:4) (1.3) CF∗(L1,L0)= Z2[p] p∈L1∩L0 as a group. 1 Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms