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Morse-Bott Cohomology from Homological Perturbation Theory

In this paper, we construct cochain complexes generated by cohomology of critical manifolds for Morse-Bott theory under minimum transversality assumptions. We discuss the relations between different constructions of cochain complexes for Morse-Bott theory. In particular, we explain how homological perturbation theory is used in Morse-Bott cohomology, and both our construction and the cascade construction can be interpreted in that way. In the presence of group actions, we construct cochain complexes for the equivariant theory. Expected properties like the independence of approximation of classifying spaces and existence of the action spectral sequence are proven. We carry out our construction for finite dimensional Morse-Bott cohomology using a generic metric and prove it recovers the regular cohomology. We outline the project of combining our construction with polyfold perturbation theory.