Morse-Bott cohomology from homological perturbation theory (Q6592998)

While Morse theory analyzes the topology of a manifold by studying critical points and gradient flow lines of Morse functions on the manifold, Morse-Bott theory studies for this purpose critical manifolds of Morse-Bott functions and gradient flow lines from one critical manifold to another.\N\NCritical manifolds and compactified moduli spaces of gradient flow lines determine a flow category; roughly speaking, its objects come from critical manifolds, and the morphisms are flow lines. Given a flow category, there are several methods to get a chain or cochain complex [\textit{D. M. Austin} and \textit{P. J. Braam}, Progr. Math. 133, 123--183 (1995; Zbl 0834.57017); \textit{K. Fukaya}, Topology 35, 89--136 (1996; Zbl 0848.58010); \textit{F. Bourgeois}, A Morse-Bott approach to contact homology. Stanford University (PhD thesis) (2002); \textit{U. Frauenfelder}, Int. Math. Res. Not. 2004, No. 42, 2179--2269 (2004; Zbl 1088.53058)]. The author unifies these methods using the homological perturbation lemma, and provides the minimal Morse-Bott construction.\N\NBased on this construction, the author assigns to an oriented flow category a minimal Morse-Bott cochain complex generated by the cohomology of the critical manifolds. In particular, for a closed manifold, the cohomology of such a complex is the cohomology of the manifold. The construction also provides explicit formulae for terms from moduli spaces related to the boundaries and corners of the manifold.\N\NIn addition, in the presence of group actions, the author constructs cochain complexes for the equivariant theory, and applies the minimal Morse-Bott construction to the polyfold theory. The paper contains a detailed review of previous work on the topic.