Minimal projective resolution and magnitude homology of geodetic metric spaces
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910573596246016 |
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| author | Asao, Yasuhiko Wakatsuki, Shun |
| author_facet | Asao, Yasuhiko Wakatsuki, Shun |
| contents | Asao-Ivanov showed that magnitude homology is a Tor functor, hence we can compute it by giving a projective resolution of a certain module. In this article, we compute magnitude homology by constructing a minimal projective resolution. As a consequence, we determine magnitude homology of geodetic metric spaces. We show that it is a free $\mathbb Z$-module, and give a recursive algorithm for constructing all cycles. As a corollary, we show that a finite geodetic metric space is diagonal if and only if it contains no 4-cuts. Moreover, we give explicit computations for cycle graphs, Petersen graph, Hoffman-Singleton graph, and a missing Moore graph. It includes another approach to the computation for cycle graphs, which has been studied by Hepworth--Willerton and Gu. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_12147 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minimal projective resolution and magnitude homology of geodetic metric spaces Asao, Yasuhiko Wakatsuki, Shun Algebraic Topology 55N35, 05C31, 51F99 Asao-Ivanov showed that magnitude homology is a Tor functor, hence we can compute it by giving a projective resolution of a certain module. In this article, we compute magnitude homology by constructing a minimal projective resolution. As a consequence, we determine magnitude homology of geodetic metric spaces. We show that it is a free $\mathbb Z$-module, and give a recursive algorithm for constructing all cycles. As a corollary, we show that a finite geodetic metric space is diagonal if and only if it contains no 4-cuts. Moreover, we give explicit computations for cycle graphs, Petersen graph, Hoffman-Singleton graph, and a missing Moore graph. It includes another approach to the computation for cycle graphs, which has been studied by Hepworth--Willerton and Gu. |
| title | Minimal projective resolution and magnitude homology of geodetic metric spaces |
| topic | Algebraic Topology 55N35, 05C31, 51F99 |
| url | https://arxiv.org/abs/2408.12147 |