Cascades and perturbed Morse-Bott functions

TL;DR

This paper proves that the cascade chain complex for Morse-Bott functions on closed manifolds is isomorphic to the Morse chain complex of a perturbed Morse-Smale function, establishing a link to classical homology.

Contribution

It demonstrates the equivalence of cascade and perturbed Morse-Smale chain complexes for Morse-Bott functions, confirming their homology isomorphic to singular homology.

Findings

Cascade chain complex matches Morse-Smale chain complex for small perturbations

Homology of cascade complex is isomorphic to singular homology of the manifold

Provides a rigorous connection between Morse-Bott functions and classical homology

Abstract

Let $f:M \rightarrow \mathbb{R}$ be a Morse-Bott function on a finite dimensional closed smooth manifold $M$. Choosing an appropriate Riemannian metric on $M$ and Morse-Smale functions $f_j:C_j \rightarrow \mathbb{R}$ on the critical submanifolds $C_j$, one can construct a Morse chain complex whose boundary operator is defined by counting cascades \cite{FraTheA}. Similar data, which also includes a parameter $\epsilon > 0$ that scales the Morse-Smale functions $f_j$, can be used to define an explicit perturbation of the Morse-Bott function $f$ to a Morse-Smale function $h_\epsilon:M \rightarrow \mathbb{R}$ \cite{AusMor} \cite{BanDyn}. In this paper we show that the Morse-Smale-Witten chain complex of $h_\epsilon$ is the same as the Morse chain complex defined using cascades for any $\epsilon >0$ sufficiently small. That is, the two chain complexes have the same generators, and their boundary operators are the same (up to a choice of sign). Thus, the Morse Homology Theorem implies that the homology of the cascade chain complex of $f:M \rightarrow \mathbb{R}$ is isomorphic to the singular homology $H_\ast(M;\mathbb{Z})$.