Morse homology and equivariance

TL;DR

This paper develops methods to compute equivariant homology using Morse functions on manifolds with finite group actions, introducing stable Morse functions and spectral sequences for better analysis.

Contribution

It introduces a process to modify Morse functions to be stable with well-behaved cell structures, and relates Morse theory to equivariant homology computations.

Findings

Stable Morse functions have a dense subset in the space of equivariant functions.

A Morse spectral sequence computes equivariant homology from critical point data.

Equivariant, stable Morse functions lead to a Thom-Smale-Witten complex.

Abstract

In this paper, we develop methods for calculating equivariant homology from equivariant Morse functions on a closed manifold with the action of a finite group. We show how to alter $G$-equivariant Morse functions to a stable one, where the descending manifold from a critical point $p$ has the same stabilizer group as $p$, giving a better-behaved cell structure on $M$. For an equivariant, stable Morse function, we show that a generic equivariant metric satisfies the Morse--Smale condition. In the process, we give a proof that a generic equivariant function is Morse, and that equivariant, stable Morse functions form a dense subset in the $C^0$-topology within the space of all equivariant functions. Finally, we give an expository account of equivariant homology and cohomology theories, as well as their interaction with Morse theory. We show that any equivariant Morse function gives a filtration of $M$ that induces a Morse spectral sequence, computing the equivariant homology of $M$ from information about how the stabilizer group of a critical point acts on its tangent space. In the case of a stable Morse function, we show that this can be further reduced to a Thom-Smale-Witten complex.