Geodesic
Morse–Bott cohomology from homological perturbation theory
Zhou, Zhengyi1
1 Morningside Center of Mathematics and Institute of Mathematics, Chinese Academy of Sciences, Beijing, China
Algebraic and Geometric Topology, Tome 24 (2024) no. 3, pp. 1321-1429

Voir la notice de l'article provenant de la source Mathematical Sciences Publishers

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 polyfold theory.

DOI : 10.2140/agt.2024.24.1321
Keywords: Morse–Bott theory, homological perturbation theory, Floer cohomology, equivariant theory

Zhou, Zhengyi 1

1 Morningside Center of Mathematics and Institute of Mathematics, Chinese Academy of Sciences, Beijing, China 
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Zhou, Zhengyi. Morse–Bott cohomology from homological perturbation theory. Algebraic and Geometric Topology, Tome 24 (2024) no. 3, pp. 1321-1429. doi: 10.2140/agt.2024.24.1321

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