Singular homology of roots of unity
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866915159391338496 |
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| author | Milićević, Nikola |
| author_facet | Milićević, Nikola |
| contents | We extend some basic results from the singular homology theory of topological spaces to the setting of Čech's closure spaces. We prove analogues of the excision and Mayer-Vietoris theorems and the Hurewicz theorem in dimension one. We use these results to calculate examples of singular homology groups of spaces that are not topological but are often encountered in applied topology, such as simple undirected graphs. We focus on the singular homology of roots of unity with closure structures arising from considering nearest neighbors. These examples can then serve as building blocks along with our Mayer-Vietoris and excision theorems for computing the singular homology of more complex closure spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_00660 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Singular homology of roots of unity Milićević, Nikola Algebraic Topology Category Theory General Topology 55N10, 54A05, 05C20 We extend some basic results from the singular homology theory of topological spaces to the setting of Čech's closure spaces. We prove analogues of the excision and Mayer-Vietoris theorems and the Hurewicz theorem in dimension one. We use these results to calculate examples of singular homology groups of spaces that are not topological but are often encountered in applied topology, such as simple undirected graphs. We focus on the singular homology of roots of unity with closure structures arising from considering nearest neighbors. These examples can then serve as building blocks along with our Mayer-Vietoris and excision theorems for computing the singular homology of more complex closure spaces. |
| title | Singular homology of roots of unity |
| topic | Algebraic Topology Category Theory General Topology 55N10, 54A05, 05C20 |
| url | https://arxiv.org/abs/2301.00660 |