Morse-Bott-Smale chain complex

Ryuma Orita, Kanon Yashiro
arXiv:2504.16962

Overview
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Morse–Bott theory studies the topology of a manifold using functions whose critical points may form submanifolds. To turn this idea into homology, the associated chain complex must be defined consistently. We revisit the construction of Banyaga and Hurtubise, which uses five degeneracy relations to form a quotient of a larger chain complex.

We replace those five relations with a single degeneracy condition and resolve the problem of well-definedness. The resulting construction gives a suitable definition of Morse–Bott homology and makes more examples accessible to computation. For a Morse–Smale function, we also prove that our complex is quasi-isomorphic to the usual Morse–Smale–Witten complex, so the two constructions give the same homology. This yields another proof of the Morse Homology Theorem.

BibTeX

@misc{orita2025morsebottsmale,
  author        = {Orita, Ryuma and Yashiro, Kanon},
  title         = {{Morse-Bott-Smale chain complex}},
  year          = {2025},
  eprint        = {2504.16962},
  archivePrefix = {arXiv},
  url           = {https://arxiv.org/abs/2504.16962}
}

Overview#

Overview
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