Morse-Bott cohomology from homological perturbation theory

TL;DR

This paper develops a framework for Morse-Bott cohomology using homological perturbation theory, establishing new constructions, relations, and equivariant extensions, and demonstrating their consistency with classical cohomology.

Contribution

It introduces a novel approach to Morse-Bott cohomology via homological perturbation theory, unifying different constructions and extending to equivariant settings.

Findings

Constructed cochain complexes from critical manifolds.

Proved independence of approximations and spectral sequence existence.

Recovered classical cohomology for Morse-Bott functions on closed manifolds.

Abstract

In this paper, we construct cochain complexes generated by the cohomology of critical manifolds in the abstract setup of flow categories for Morse-Bott theories under minimum transversality assumptions. We discuss the relations between different constructions of Morse-Bott theories. In particular, we explain how homological perturbation theory is used in Morse-Bott theories, and both our construction and the cascades construction can be interpreted as applications of homological perturbations. In the presence of group actions, we construct cochain complexes for the equivariant theory. Expected properties like the independence of approximations of classifying spaces and the existence of the action spectral sequence are proven. We carry out our construction for Morse-Bott functions on closed manifolds and prove it recovers the regular cohomology. We outline the project of combining our construction with the polyfold theory.