On $\ell_p$-Vietoris-Rips complexes
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910847050186752 |
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| author | Ivanov, Sergei O. Xu, Xiaomeng |
| author_facet | Ivanov, Sergei O. Xu, Xiaomeng |
| contents | We study the concepts of the $\ell_p$-Vietoris-Rips simplicial set and the $\ell_p$-Vietoris-Rips complex of a metric space, where $1\leq p \leq \infty.$ This theory unifies two established theories: for $p=\infty,$ this is the classical theory of Vietoris-Rips complexes, and for $p=1,$ this corresponds to the blurred magnitude homology theory. We prove several results that are known for the Vietoris-Rips complex in the general case: (1) we prove a stability theorem for the corresponding version of the persistent homology; (2) we show that, for a compact Riemannian manifold and a sufficiently small scale parameter, all the "$\ell_p$-Vietoris-Rips spaces" are homotopy equivalent to the manifold; (3) we demonstrate that the $\ell_p$-Vietoris-Rips spaces are invariant (up to homotopy) under taking the metric completion. Additionally, we show that the limit of the homology groups of the $\ell_p$-Vietoris-Rips spaces, as the scale parameter tends to zero, does not depend on $p$; and that the homology groups of the $\ell_p$-Vietoris-Rips spaces commute with filtered colimits of metric spaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_01857 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On $\ell_p$-Vietoris-Rips complexes Ivanov, Sergei O. Xu, Xiaomeng Algebraic Topology Computational Geometry Metric Geometry We study the concepts of the $\ell_p$-Vietoris-Rips simplicial set and the $\ell_p$-Vietoris-Rips complex of a metric space, where $1\leq p \leq \infty.$ This theory unifies two established theories: for $p=\infty,$ this is the classical theory of Vietoris-Rips complexes, and for $p=1,$ this corresponds to the blurred magnitude homology theory. We prove several results that are known for the Vietoris-Rips complex in the general case: (1) we prove a stability theorem for the corresponding version of the persistent homology; (2) we show that, for a compact Riemannian manifold and a sufficiently small scale parameter, all the "$\ell_p$-Vietoris-Rips spaces" are homotopy equivalent to the manifold; (3) we demonstrate that the $\ell_p$-Vietoris-Rips spaces are invariant (up to homotopy) under taking the metric completion. Additionally, we show that the limit of the homology groups of the $\ell_p$-Vietoris-Rips spaces, as the scale parameter tends to zero, does not depend on $p$; and that the homology groups of the $\ell_p$-Vietoris-Rips spaces commute with filtered colimits of metric spaces. |
| title | On $\ell_p$-Vietoris-Rips complexes |
| topic | Algebraic Topology Computational Geometry Metric Geometry |
| url | https://arxiv.org/abs/2411.01857 |