Morse-Bott-Smale chain complex
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908557545308160 |
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| author | Orita, Ryuma Yashiro, Kanon |
| author_facet | Orita, Ryuma Yashiro, Kanon |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_16962 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Morse-Bott-Smale chain complex Orita, Ryuma Yashiro, Kanon Algebraic Topology Dynamical Systems Geometric Topology 57R70, 58K05, 55N10, 37D15 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. |
| title | Morse-Bott-Smale chain complex |
| topic | Algebraic Topology Dynamical Systems Geometric Topology 57R70, 58K05, 55N10, 37D15 |
| url | https://arxiv.org/abs/2504.16962 |