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