# Intersection Pairings on Conley Indices

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*Henry L. Kurland*

Given an isolated invariant set of a flow on a manifold of dimension \(m\) oriented over a PID \(R\), Kurland defines an intersection class pairing of degree \(-m\) on the tensor product of the singular homology modules of the forward and reverse time Conley indices of the isolated invariant set with values in the Čech homology of the invariant set. Restricting the pairing to elements of degree \(m\) results in an intersection number pairing that is invariant under continuation along a continuous path of flows and isolated invariant sets. More generally, the unrestricted pairing defines continuous lifts to a space of Čech homology classes along such a path. Further, when the homology modules of the Conley indices are torsion free, the intersection number pairing is non-singular. Also, the pairing associated to an isolated invariant set of a product flow is, modulo torsion, the product (up to sign) of the pairings associated to the factor isolated invariant sets. Intersection classes of lowest and highest dimension are computed for any \(R\)-orientable, normally hyperbolic invariant submanifold whose expanding and contracting normal subbundles are also \(R\)-orientable. These computations yield, due to dimensionable considerations, a complete computation of the intersection class and number pairings for hyperbolic critical points and hyperbolic closed orbits. Application is made in an appendix to the existence of solution of a class of singularly perturbed two-point boundary value problems such problems having provided strong motivation for the present study.

#### Table of Contents

# Table of Contents

## Intersection Pairings on Conley Indices

- Contents v6 free
- Introduction 110 free
- Chapter 1. Basic Notation and Background Definitions 817 free
- Chapter 2. TheIntersection Pairings L, L, and [sup(#)]L 1625
- Chapter 3. Statement of the Continuation Results and Examples 3140
- Chapter 4. Construction of Bilinear Pairings on Conley Indices 4251
- Chapter 5. Proofs of the Continuation Results 5766
- Chapter 6. Some Basic Computational Tools 7382
- Chapter 7. L for Normally Hyperbolic Invariant Submanifolds 8695
- Chapter 8. Products of Intersection Pairings 131140
- Chapter 9. The Cap Product Representation of L and the Nonsingularity of [sup(#)]L 143152
- Appendix A. Intersection Numbers and Existence Results for Two-Point Boundary Value Problems of Singularly Perturbed Systems 168177
- Appendix B. Proofs of the Propositions in §9.B 176185
- References 183192