Morse-Bott-Smale chain complex

Ryuma Orita, Kanon Yashiro
arXiv:2504.16962

Abstract

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Banyaga and Hurtubise defined the Morse–Bott–Smale chain complex as a quotient of a large chain complex by introducing five degeneracy relations. However, their five degeneracy relations are in fact redundant. In the present paper, we unify these five conditions into a single degeneracy condition and resolve the issue of the well-definedness of the Morse–Bott–Smale chain complex. This provides an appropriate definition of the Morse–Bott homology and more computable examples. Moreover, we show that our chain complex for a Morse–Smale function is quasi-isomorphic to the usual Morse–Smale–Witten chain complex. As a consequence, we obtain an alternative proof of the Morse Homology Theorem.

@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}
}