Polyfold and Fredholm Theory

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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