Three approaches to Morse-Bott homology

TL;DR

This paper reviews three methods for computing Morse-Bott homology on smooth manifolds, comparing their approaches and relationships, including perturbation, cascades, and multicomplex techniques.

Contribution

It introduces and compares three distinct approaches to Morse-Bott homology, highlighting their differences and connections, especially the novel multicomplex method.

Findings

First two approaches produce the same chain complex up to sign.

The third approach uses a multicomplex combining singular and Morse chains.

The methods relate to Morse-Bott inequalities and chain complex structures.

Abstract

In this paper we survey three approaches to computing the homology of a finite dimensional compact smooth closed manifold using a Morse-Bott function and discuss relationships among the three approaches. The first approach is to perturb the function to a Morse function, the second approach is to use moduli spaces of cascades, and the third approach is to use the Morse-Bott multicomplex. With respect to an explicit perturbation (which can be used to derived the Morse-Bott inequalities), the first two approaches yield the same chain complex up to sign. The third approach is fundamentally different. It combines singular cubical chains and Morse chains in the same multicomplex, which provides a way of interpolating between the singular cubical chain complex and the Morse-Smale-Witten chain complex.