CONTENTS

Introduction 1 Chapter 1. Basic Notation and Background Definitions 8

A. Basic Notation 8 B. The Conley Index 10 C. Homology and Cohomology of Conley Indices 13 D. Sign Conventions for Products in Homology and Coho-

mology 15 Chapter 2. The Intersection Pairings L, 𝔏, and ^{#}𝔏 16

A. Pairs of Index Pairs Admissible for the Intersection Pair-

ing 16 B. The Euclidean Case: the Homology Intersection Number

Pairing L 17 C. The Manifold Case: the Intersection Class and Number

Pairings 𝔏 and ^{#}𝔏 20 Chapter 3. Statement of the Continuation Results and Examples 31

A. Invariance of Intersection Numbers under Continuation 31 B. Continuation of 𝔏 over a Path of Isolated Invariant Sets 36

Chapter 4. Construction of Bilinear Pairings on Conley Indices 42 A. The Existence of Admissible Pairs of Index Pairs 42 B. Functorially Produced Pairings on the Conley Indices 47 C. The Proofs of Theorems 2.4 and 2.11 54

Chapter 5. Proofs of the Continuation Results 57 A. Maps between Conley Indices from Paths of Invariant Sets 57 B. The Proofs of Theorems 3.1, 3.2, 3.3, and 3.7 63

Chapter 6. Some Basic Computational Tools 73 A. Conditions on Singular Cycles for Computing 𝔏 and ^{#}𝔏 73 B. The Behavior of 𝔏 under Orbit Preserving Maps 82

Chapter 7. 𝔏 for Normally Hyperbolic Invariant Submanifolds 86 A. Summary of Results 86 B. Computational Preliminaries 89 C. Results Leading to the Proof of Theorem 7.5 103 D. Results Leading to the Proof of Theorem 7.6 109

Chapter 8. Products of Intersection Pairings 131 A. Preliminary Observations and Definitions 131 B. Conley Indices of Product Invariant Sets 132 C. A Kunneth Theorem for Conley Indices 133

D. Factor and Product Intersection Pairings 137 Chapter 9. The Cap Product Representation of 𝔏 and the Non-

singularity of ^{#}𝔏 143 A. The Cap Product Representation and Corollaries 143 B. Some Technical Propositions on Poincaré Duality Isomor-

phisms and Čech Cap Products 145 C. Results Leading to the Proof of Theorem 9.4 147 D. The Case S ∩ ∂M ≠ ∅ 165

Appendix A. Intersection Numbers and Existence Results for Two-Point Boundary Value Problems of Singularly Perturbed Systems 168

Appendix B. Proofs of the Propositions in §9.B 176 References 183