Morse-Conley-Floer Homology

TL;DR

This paper introduces Morse-Conley-Floer homology, a new homological invariant for isolated invariant sets of flows, extending Morse homology and Conley index concepts to a broader dynamical systems context.

Contribution

It develops a homology theory combining Morse, Conley index, and Floer homologies for isolated invariant sets, generalizing classical Morse homology.

Findings

Defines Morse-Conley-Floer homology for flows

Establishes isomorphism with classical homology in Morse-Smale cases

Provides Morse-Conley relations analogous to Morse relations

Abstract

For Morse-Smale pairs on a smooth, closed manifold the Morse-Smale-Witten chain complex can be defined. The associated Morse homology is isomorphic to the singular homology of the manifold and yields the classical Morse relations for Morse functions. A similar approach can be used to define homological invariants for isolated invariant sets of flows on a smooth manifold, which gives an analogue of the Conley index and the Morse-Conley relations. The approach will be referred to as Morse-Conley-Floer homology.