Homotopy invariance of the Conley index and local Morse homology in Hilbert spaces

TL;DR

This paper develops a new Conley index theory for flows in metric spaces satisfying a compactness condition, applying it to LS-flows and connecting it with Morse homology and E-cohomology.

Contribution

It introduces Property (C) for flows, establishes the existence of index pairs, and links the E-cohomological Conley index with Morse homology in Hilbert spaces.

Findings

Property (C) enables Conley theory in non-locally compact spaces.

E-cohomology provides a cohomological Conley index for LS-flows.

Morse homology computes the E-cohomological Conley index.

Abstract

In this paper we introduce a new compactness condition - Property (C) - for flows in (not necessary locally compact) metric spaces. For such flows a Conley type theory can be developed. For example (regular) index pairs always exist for Property-(C) flows and a Conley index can be defined. An important class of flows satisfying this compactness condition are LS-flows. We apply E-cohomology to index pairs of LS-flows and obtain the E-cohomological Conley index. We formulate a continuation principle for the E-cohomological Conley index and show that all LS-flows can be continued to LS-gradient flows. We show that the Morse homology of LS-gradient flows computes the E-cohomological Conley index. We use Lyapunov functions to define the Morse-Conley-Floer cohomology in this context, and show that it is also isomorphic to the E-cohomological Conley index.