Homotopy, homology, and persistent homology using closure spaces
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866929718313353216 |
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| author | Bubenik, Peter Milićević, Nikola |
| author_facet | Bubenik, Peter Milićević, Nikola |
| contents | We develop persistent homology in the setting of filtrations of (Cech) closure spaces. Examples of filtrations of closure spaces include metric spaces, weighted graphs, weighted directed graphs, and filtrations of topological spaces. We use various products and intervals for closure spaces to obtain six homotopy theories, six cubical singular homology theories, and three simplicial singular homology theories. Applied to filtrations of closure spaces, these homology theories produce persistence modules. We extend the definition of Gromov-Hausdorff distance from metric spaces to filtrations of closure spaces and use it to prove that any persistence module obtained from a homotopy-invariant functor on closure spaces is stable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_10206 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Homotopy, homology, and persistent homology using closure spaces Bubenik, Peter Milićević, Nikola Algebraic Topology Combinatorics General Topology Primary 55N31, Secondary 54A05, 05C20 We develop persistent homology in the setting of filtrations of (Cech) closure spaces. Examples of filtrations of closure spaces include metric spaces, weighted graphs, weighted directed graphs, and filtrations of topological spaces. We use various products and intervals for closure spaces to obtain six homotopy theories, six cubical singular homology theories, and three simplicial singular homology theories. Applied to filtrations of closure spaces, these homology theories produce persistence modules. We extend the definition of Gromov-Hausdorff distance from metric spaces to filtrations of closure spaces and use it to prove that any persistence module obtained from a homotopy-invariant functor on closure spaces is stable. |
| title | Homotopy, homology, and persistent homology using closure spaces |
| topic | Algebraic Topology Combinatorics General Topology Primary 55N31, Secondary 54A05, 05C20 |
| url | https://arxiv.org/abs/2104.10206 |