The backward {\lambda}-Lemma and Morse filtrations

TL;DR

This paper surveys the use of the backward { extbackslash lambda}-Lemma and Conley theory to construct Morse filtrations in the loop space of a Riemannian manifold, linking dynamical systems and Morse theory.

Contribution

It introduces a Morse filtration construction in infinite-dimensional loop spaces using the backward { extbackslash lambda}-Lemma, connecting dynamical systems with Morse homology.

Findings

Morse filtration complex matches the Morse complex of the action functional

Utilizes backward { extbackslash lambda}-Lemma in infinite-dimensional setting

Provides a framework linking heat flow dynamics with Morse theory

Abstract

Consider the infinite dimensional hyperbolic dynamical system provided by the (forward) heat semi-flow on the loop space of a closed Riemannian manifold M. We use the recently discovered backward {\lambda}-Lemma and elements of Conley theory to construct a Morse filtration of the loop space whose cellular filtration complex represents the Morse complex associated to the downward L2-gradient of the classical action functional. This paper is a survey. Details and proofs will be given in [6].