Stable foliations and semi-flow Morse homology

TL;DR

This paper establishes a novel connection between Morse homology for semi-flows on loop spaces and singular homology, using stable foliations and Conley pairs, with implications for finite-dimensional cases.

Contribution

It introduces a new construction of Morse filtration via Conley pairs and stable foliations, and demonstrates non-triviality of Morse homology for heat flow on loop spaces.

Findings

Morse homology is isomorphic to singular homology of the loop space.

Constructed stable foliations for Conley pairs using backward λ-Lemma.

Extended the construction to finite-dimensional cases.

Abstract

In case of the heat flow on the free loop space of a closed Riemannian manifold non-triviality of Morse homology for semi-flows is established by constructing a natural isomorphism to singular homology of the loop space. The construction is also new in finite dimensions. The main idea is to build a Morse filtration using Conley pairs and their pre-images under the time-$T$-map of the heat flow. A crucial step is to contract each Conley pair onto its part in the unstable manifold. To achieve this we construct stable foliations for Conley pairs using the recently found backward $\lambda$-Lemma [31]. These foliations are of independent interest [23].