Kuranishi Structures and Virtual Fundamental Chains

Preface
Acknowledgments
Contents
1 Introduction
1.1 Background: Why Virtual Fundamental Chains?
1.2 The Story of Kuranishi Structures and Virtual Fundamental Chains
1.3 Main Results of Part I
1.3.1 Elementary Material
1.3.2 Multisections
1.3.3 CF-Perturbation and Integration
1.3.4 Stokes' Formula
1.3.5 Smooth Correspondence and Composition Formula
1.3.6 Proof of Existence Theorems
1.3.7 Virtual Fundamental Chain Over Q
1.4 Related Works
1.4.1 Papers Appearing in the Year 1996
1.4.2 Issues in Developing the Theory of Virtual Fundamental Chains and Cycles
1.4.3 The Works of the Authors of This Book
1.4.4 The Development from the Work by Li–Tian LiTi98 and Liu-Tian LiuTi98
1.4.5 The Work by Joyce
1.4.6 Polyfolds
1.4.7 The Work by Pardon
1.4.8 Other Works
2 Notations and Conventions
Conventions on the Way to Use Several Notations
Conventions on Orientation and Sign
List of Notations in Part I
List of Notations in Part II
List of Notations in Appendices
Part I Abstract Theory of Kuranishi Structures, Fiber Products and Perturbations
3 Kuranishi Structures and Good Coordinate Systems
3.1 Kuranishi Structures
3.2 Good Coordinate Systems
3.3 Embedding of Kuranishi Structures I
3.4 Notes on Various Versions of the Definitions
3.4.1 Tangent Bundle Condition
3.4.2 Global Quotient
3.4.3 Germ of Kuranishi Chart
3.4.4 Definition 3.15 (7), (8)
3.4.5 `Hausdorffness' Issue
4 Fiber Product of Kuranishi Structures
4.1 Fiber Product
4.2 Boundaries and Corners I
4.3 A Basic Property of Fiber Products
5 Thickening of a Kuranishi Structure
5.1 Background to Introducing the Notion of Thickening
5.2 Definition of Thickening
5.3 Embedding of Kuranishi Structures II
5.4 Support System and Existence of Thickening
6 Multivalued Perturbations
6.1 Multisections and Multivalued Perturbations
6.1.1 Multivalued Perturbations on an Orbifold
6.1.2 Multivalued Perturbations on a Good Coordinate System
6.2 Properties of the Zero Set of Multivalued Perturbations
6.3 Transversality of the Multisection
6.4 Embedding of Kuranishi Structures and Multivalued Perturbations
6.5 General Strategy of Construction of Virtual FundamentalChains
7 CF-Perturbations and Integration Along the Fiber (Pushout)
7.1 Introduction to Chaps.7, 8, 9, 10, and 12
7.2 CF-Perturbation on a Single Kuranishi Chart
7.2.1 CF-Perturbation on a Kuranishi Chart Restricted to One Orbifold Chart
7.2.2 CF-Perturbation on a Single Kuranishi Chart
7.3 Integration Along the Fiber (Pushout) on a Single Kuranishi Chart
7.4 CF-Perturbations of a Good Coordinate System
7.4.1 Embedding of Kuranishi Chartsand CF-Perturbations
7.4.2 CF-Perturbations on Good Coordinate Systems
7.4.3 Extension of a Good Coordinate System and Relative Version of the Existence Theorem of CF-Perturbations
7.5 Partition of Unity Associated to a Good Coordinate System
7.6 Differential Forms on a Good Coordinate System and a Kuranishi Structure
7.7 Integration Along the Fiber (pushout) on a Good Coordinate System
8 Stokes' Formula
8.1 Boundaries and Corners II
8.2 Stokes' Formula for a Good Coordinate System
8.3 Well-Definedness of Virtual Fundamental Cycle
9 From Good Coordinate Systems to Kuranishi Structures and Back with CF-Perturbations
9.1 CF-Perturbations and Embedding of Kuranishi Structures
9.2 Integration Along the Fiber (pushout) for KuranishiStructures
9.3 Composition of GK-and KG-Embeddings: Proof of Definition-Lemma 5.17
9.4 GG-Embedding and Integration: Proof of Proposition 9.16
9.5 CF-Perturbations of Correspondences
9.6 Stokes' Formula for a Kuranishi Structure
9.7 Uniformity of CF-Perturbations on a Kuranishi Structure
10 Composition Formula of Smooth Correspondences
10.1 Direct Product and CF-Perturbation
10.2 Fiber Product and CF-Perturbation
10.3 Composition of Smooth Correspondences
10.4 Composition Formula
11 Construction of Good Coordinate Systems
11.1 Construction of Good Coordinate Systems: The AbsoluteCase
11.2 Construction of Good Coordinate Systems: When Thickening Is Given
11.3 KG-Embeddings and Compatible Perturbations
11.4 Extension of Good Coordinate Systems: The Relative Case
12 Construction of CF-Perturbations
12.1 Construction of CF-Perturbations on a Single Chart
12.2 Sheaf CFK of CF-Perturbations on Hetero-Dimensional Compactum
13 Construction of Multivalued Perturbations
13.1 Sheaf MVK of Multivalued Perturbations
13.2 Construction of Multivalued Perturbations
13.3 Extending Multivalued Perturbations from One Chart to Another: Remarks
14 Zero-and One-Dimensional Cases via Multivalued Perturbation
14.1 Virtual Fundamental Chain for a Good Coordinate System with Multivalued Perturbation
14.2 Virtual Fundamental Chain of 0-Dimensional K-Space with Multivalued Perturbation
14.3 A Simple Morse Theory on a Space with a Good Coordinate System
14.4 Denseness of the Set of Morse Functions on an Orbifold
14.5 Similarity and Difference Between CF-Perturbation and Multivalued Perturbation: Remarks
Part II System of K-Spaces and Smooth Correspondences
15 Introduction to Part II
15.1 Outline of the Story of Linear K-Systems
15.1.1 Floer Cohomology of Periodic HamiltonianSystems
15.1.2 Periodic Hamiltonian Systems and Axiom of Linear K-Systems
15.1.3 Construction of Floer Cochain Complex
15.1.4 Corner Compatibility Conditions
15.1.5 Well-Definedness of Floer Cohomology and Morphism of Linear K-Systems
15.1.6 Identity Morphism
15.1.7 Homotopy Limit
15.1.8 Story in the Case with Rational Coefficients
15.2 Outline of the Story of Tree-Like K-Systems
15.2.1 Moduli Space of Pseudo-holomorphic Disks:Review
15.2.2 Axiom of Tree-Like K-System and the Construction of the Filtered A∞ Algebra
15.2.3 Bifurcation Method and Pseudo-isotopy
15.2.4 Bifurcation Method and Self-Gluing
15.3 Discussion Deferred to the Appendices
15.3.1 Orbifolds and Covering Space ofOrbifolds/K-Spaces
15.3.2 Admissibility of Orbifolds and of KuranishiStructures
15.3.3 Stratified Submersion
15.3.4 Integration Along the Fiber and Local System
16 Linear K-Systems: Floer Cohomology I – Statement
16.1 Axiom of Linear K-Systems
16.2 Floer Cohomology Associated to a Linear K-System
16.3 Morphism of Linear K-Systems
16.4 Homotopy and Higher Homotopy of Morphisms of Linear K-Systems
16.5 Composition of Morphisms of Linear K-Systems
16.6 Inductive System of Linear K-Systems
17 Extension of a Kuranishi Structure and Its Perturbation from Boundary to Its Neighborhood
17.1 Introduction to Chap.17
17.2 Outer Collaring on One Chart
17.3 Outer Collaring and Embedding
17.4 Outer Collaring of Kuranishi Structures
17.5 Collared Kuranishi Structure
17.6 Products of Collared Kuranishi Structures
17.7 Extension of Collared Kuranishi Structures
17.7.1 Statement
17.7.2 Extension Theorem for a Single Collared Kuranishi Chart
17.7.3 Construction of Kuranishi Chart U+
17.7.4 Completion of the Proof of Lemma 17.60
17.7.5 Proof of Proposition 17.58
17.8 Extension of Collared CF-Perturbations
17.9 Extension of Kuranishi Structures and CF-Perturbations from a Neighborhood of a Compact Set
17.10 Conclusion of Chap.17
18 Corner Smoothing and Composition of Morphisms
18.1 Why Corner Smoothing?
18.2 Introduction to Chap.18
18.3 Partial Outer Collaring of Cornered K-Spaces
18.4 In Which Sense Is Corner Smoothing Canonical?
18.5 Corner Smoothing of [0,∞)k
18.6 Corner Smoothing of Collared Orbifolds and of Kuranishi Structures
18.7 Composition of Morphisms of Linear K-Systems
18.8 Associativity of the Composition
18.9 Parametrized Version of Morphism: Composition and Gluing
18.9.1 Compositions of Parametrized Morphisms
18.9.2 Gluing Parametrized Morphisms
18.10 Identity Morphism
18.11 Geometric Origin of the Definition of the Identity Morphism
18.11.1 Interpolation Space of the Identity Morphism
18.11.2 Identification of the Interpolation Space of the Identity Morphism with Direct Product
18.11.3 Interpolation Space of the Homotopy
18.11.4 Identification of the Interpolation Space of the Homotopy with Direct Product
19 Linear K-Systems: Floer Cohomology II – Proof
19.1 Construction of Cochain Complexes
19.2 Construction of Cochain Maps
19.3 Proof of Theorem 16.9 (1) and Theorem 16.39 (1)
19.4 Composition of Morphisms and of Induced Cochain Maps
19.5 Construction of Homotopy
19.6 Proof of Theorem 16.9 (2)(except (f)), Theorem 16.31 (1) and Theorem 16.39 (2)(except (e)), (3)
19.7 Construction of Higher Homotopy
19.8 Proof of Theorem 16.39 (2)(e), (4)–(6)and Theorem 16.9 (2)(f)
20 Linear K-Systems: Floer Cohomology III – Morse Case by Multisection
20.1 Extension of a Multisection from Boundaryto Its Neighborhood
20.2 Completion of the Proof of Theorem 20.2
21 Tree-Like K-Systems: A∞ Structure I – Statement
21.1 Axiom of Tree-Like K-Systems: A∞ Correspondence
21.2 Filtered A∞ Algebra and its Pseudo-Isotopy
21.3 Statement of the Results
22 Tree-Like K-Systems: A∞ Structure II – Proof
22.1 Existence of CF-Perturbations
22.2 Algebraic Lemmas: Promotion Lemmas via Pseudo-Isotopy
22.3 Pointwiseness of Parametrized Family of Smooth Correspondences
22.4 Proof of Theorem 21.35
Part III Appendices
23 Orbifolds and Orbibundles by Local Coordinates
23.1 Orbifolds and Embeddings Between Them
23.2 Vector Bundles on Orbifolds
24 Covering Space of Effective Orbifolds and K-Spaces
24.1 Covering Space of an Orbifold
24.2 Covering Space of a K-Space
24.3 Covering Spaces Associated to the Corner Structure Stratification
24.4 Finite Group Action on a K-Space
25 Admissible Kuranishi Structures
25.1 Admissible Orbifolds
25.2 Admissible Vector Bundles
25.3 Admissibility of the Moduli Spaces of Pseudo-holomorphic Curves
26 Stratified Submersion to a Manifold with Corners
27 Local System and Smooth Correspondence in de Rham Theory with Twisted Coefficients
28 Composition of KG-and GG-Embeddings: Proof of Lemma 3.34
29 Global Quotients and Orbifolds
References
Index