Polyfold and Fredholm Theory

Preface
Goals
Background
Historical Context
Outlook
Foundations
Remarks About the Index
Acknowledgements
Contents
Part I Basic Theory in M-Polyfolds
Chapter 1 Sc-Calculus
1.1 Sc-Structures and Differentiability
1.2 Properties of Sc-Differentiability
1.3 The Chain Rule and Boundary Recognition
1.4 Appendix
1.4.1 Proof of the sc-Fredholm Stability Result
1.4.2 Proof of the Chain Rule
1.4.3 Proof Lemma 1.3.4
1.4.4 A Useful Example
Chapter 2 Retracts
2.1 Retractions and Retracts
2.2 Some Basic Properties of Sc-Smooth Retracts
2.3 M-Polyfolds and Sub-M-Polyfolds
2.4 The Degeneracy Index and Boundary Geometry
2.5 Tame M-polyfolds
2.6 Strong Bundles
2.7 Appendix
2.7.1 Proof of Proposition 2.1.2
2.7.2 Proof of Theorem 2.3.10
2.7.3 Proof of Proposition 2.3.15
2.7.4 Formalism Associated to a Boundary with Corners
2.7.4.1 The Boundary Structure Functor
2.7.4.2 Construction of the Tame Boundary
2.7.4.3 Proof of Theorem 2.7.13
Chapter 3 Basic Sc-Fredholm Theory
3.1 Sc-Fredholm Sections
3.2 Subsets with Tangent Structure
3.3 Contraction Germs
3.4 Stability of Basic Germs
3.5 The Geometry of Basic Germs
3.6 Implicit Function Theorems
3.7 Conjugation to a Basic Germ
3.8 Appendix
3.8.1 Proof of Proposition 3.1.25
3.8.1.1 Introduction
3.8.1.2 Cones and Quadrants in Finite Dimensions
3.8.1.3 Cones and Partial Quadrants in Infinite Dimensions
3.8.1.4 Finite-dimensional Subspaces and Partial Quadrants in Infinite Dimensions
3.8.1.5 Proof of Proposition 3.1.25
3.8.2 Proof of Theorem 3.3.3
3.8.3 Proof of Lemma 3.5.10
3.8.1-4 Proof of Lemma 3.6.9
3.8.5 Diffeomorphisms Between Partial Quadrants
3.8.6 An Implicit Function Theorem in Partial Quadrants
Chapter 4 Manifolds and Strong Retracts
4.1 Characterization
4.2 Smooth Finite-Dimensional Submanifolds
4.3 Families and an Application of Sard's Theorem
4.4 Sc-Differential Forms
4.5 Appendix
4.5.1 Definition of the Lie Bracket
4.5.2 Proof of Proposition 4.4.5
4.5.3 Proof of the Poincaré Lemma
Chapter 5 The Fredholm Package for M-Polyfolds
5.1 Auxiliary Norms
5.2 Compactness Results
5.3 Perturbation Theory and Transversality
5.4 Remark on Extensions of Sc+-Sections
5.5 Notes on Partitions of Unity and Bump Functions
Chapter 6 Orientations
6.1 An Overview
6.2 Linearizations of Sc-Fredholm Sections
6.3 Linear Algebra and Conventions
6.4 The Determinant of a Fredholm Operator
6.5 Classical Local Determinant Bundles
6.6 Local Orientation Propagation
6.7 Invariants
6.8 Appendix
6.8.1 Proof of Lemma 6.3.3
6.8.2 Proof of Proposition 6.4.11
Part II Ep-Groupoids
Chapter 7 Ep-Groupoids
7.1 Ep-Groupoids and Basic Properties
7.2 Effective and Reduced Ep-Groupoids
7.3 Topological Properties of Ep-Groupoids
7.4 Regularity Assumptions and the Zhou Condition
The Local Unique Continuation Property
The Zhou Condition
The Local Regularity Condition
7.5 Paracompact Orbit Spaces
7.6 Appendix
7.6.1 The Natural Representation
7.6.2 Sc-Smooth Partitions of Unity
7.6.3 On the metrizability of TR
7.6.4 Reduced Ep-Groupoids and Raising the Index
7.6.5 Boundary Structure of Tame Ep-Groupoids
7.6.5.1 Recollections of the Tame M-polyfold Case
7.6.5.2 The Case of Tame Ep-Groupoids
Chapter 8 Bundles and Covering Functors
8.1 The Tangent of an Ep-Groupoid
The Tangent Construction
Regularity Properties and Tangents
8.2 Sc-Differential Forms on Ep-Groupoids
8.3 Strong Bundles over Ep-Groupoids
8.4 Topological and Regularity Properties of Strong Bundles
8.5 Proper Covering Functors
8.6 Appendix
8.6.1 Local Structure of Proper Coverings
8.6.2 The Structure of Strong Bundle Coverings
Chapter 9 Branched Ep+-Subgroupoids
9.1 Basic Definitions
9.2 The Tangent and Boundary of Θ
9.3 Orientations
9.4 The Geometry of Local Branching Structures
9.5 Integration and Stokes
9.6 Appendix
9.6.1 Proof of Proposition 9.1.12
9.6.2 Questions about M+-Polyfolds
9.6.3 Questions about Branched Objects
Chapter 10 Equivalences and Localization
10.1 Equivalences
10.2 The Weak Fibered Product
10.3 Localization at the System of Equivalences
10.4 Strong Bundles and Equivalences
10.5 Localization in the Strong Bundle Case
10.6 Appendix
10.6.1 Proof of Theorem 10.3.8
10.6.2 Proof of Theorem 10.3.10
10.6.3 Another Useful Example
Chapter 11 Geometry up to Equivalences
11.1 Ep-Groupoids and Equivalences
11.2 Sc-Differential Forms and Equivalences
11.3 Branched Ep+-Subgroupoids and Equivalences
11.4 Equivalences and Integration
11.5 Strong Bundles up to Equivalence
11.6 Coverings and Equivalences
Part III Fredholm Theory in Ep-Groupoids
Chapter 12 Sc-Fredholm Sections
12.1 Introduction and Basic Definition
12.2 Auxiliary Norms
12.3 Sc+-Section Functors
12.4 Compactness Properties
12.5 Orientation Bundles
Chapter 13 Sc+-Multisections
13.1 Structure Result
13.2 General Sc+-Multisections
13.3 Structurable Sc+-Multisections
13.4 Equivalences, Coverings and Structurability
13.5 Constructions of Sc+-Multisections
Chapter 14 Extension of Sc+-Multisections
14.1 Definitions and Main Result
14.2 A Good Structured Version of Λ
14.3 Extension of Correspondences
14.4 Implicit Structures and Local Extension
14.5 Extension of the Sc+-Multisection
14.6 Remarks on Inductive Constructions
Chapter 15 Transversality and Invariants
15.1 Natural Constructions
15.2 Transversality and Local Solution Sets
15.3 Perturbations
15.4 Orientations and Invariants
Chapter 16 Polyfolds
16.1 Polyfold Structures
16.2 Tangent of a Polyfold
16.3 Strong Polyfold Bundles
16.4 Branched Finite-Dimensional Orbifolds
16.5 Sc+-Multisections
16.6 Fredholm Theory
Part IV Fredholm Theory in Groupoidal Categories
Chapter 17 Polyfold Theory for Categories
17.1 Polyfold Structures and Categories
17.2 Tangent Construction
17.3 Subpolyfolds
17.4 Boundary Formalism for Tame Polyfolds
The Structure of M(Ψθ, Ψ'θ')
Topology
17.5 Branched Ep+-Subcategories
17.6 Sc-Differential Forms and Stokes
17.7 Strong Bundle Structures
17.8 Proper Covering Constructions
Chapter 18 Fredholm Theory in Polyfolds
18.1 Basic Concepts
18.2 Compactness Properties
18.3 Sc+-Multisection Functors
18.4 Constructions and Extensions
18.5 Orientations
18.6 Perturbation Theory
Chapter 19 General Constructions
19.1 The Basic Constructions
19.2 The Natural Topology T for S
19.3 The Natural Topology for M(Ψ.Ψ')
19.4 Metrizability Criteria
19.5 The Polyfold Structure for (S,T)
19.6 A Strong Bundle Version
Natural Topology T
Natural Strong Bundle Structures for M(Ψ,Ψ')
Strong Bundle Structure for P:E→S
Alternative Approach
19.7 Covering Constructions
Topological Considerations for M(Ψ,Ψ') and M(Ψ,Ψ')
Topological Considerations for A and B
Sc-Smoothness Properties
19.8 Covering Constructions for Strong Bundles
Constructions for Pull-back diagrams
Pull-back diagrams
Uniformizers for Pull-back Diagrams
Transition Construction for Pull-Back Diagrams
The General Case
Bundle Covering Squares
Strong Ep-Bundle Covering Square
Uniformizers for E
Basic Construction
Appendix A Construction Cheatsheet
A.1 Groupoidal Categories
A.1.1 Uniformizer Construction for S
A.1.2 Basic Construction
The Properness Condition
Metrizability and Polyfold Structure
The Tame Boundary from a Tame (F,F)
A.1.3 Summary
A.2 Strong Bundle Structures
A.2.1 Uniformizer Construction for PS : E→S
Uniformizers
Coherency Condition
A.2.2 Basic Construction
Strong Polyfold Bundle Structure
A Remark on Coherency
A.2.3 Summary
A.3 Finite-to-One Covering Functors
A.3.1 Uniformizer Construction for P : A→B
A.3.2 Basic Construction
A.3.3 Summary
A.4 Coverings of Strong Bundles
A.4.1 Uniformizers for Covering Squares E
A.4.2 Basic Construction
A.4.3 Summary
References
Notation and List of Frequently Occurring Symbols
Part I
Part II
Part III
Part IV
Index