Contractibility of Vietoris-Rips Complexes of dense subsets in $(\mathbb{R}^n, \ell_1)$ via hyperconvex embeddings
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909223066009600 |
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| author | Wang, Qingsong |
| author_facet | Wang, Qingsong |
| contents | We consider the contractibility of Vietoris-Rips complexes of dense subsets of $(\mathbb{R}^n,\ell_1)$ with sufficiently large scales. This is motivated by a question by Matthew Zaremsky regarding whether for each $n$ natural there is a $r_n>0$ so that the Vietoris-Rips complex of $(\mathbb{Z}^n,\ell_1)$ at scale $r$ is contractible for all $r\geq r_n$. We approach this question using results that relates to the neighborhood of embeddings into hyperconvex metric space of a metric space $X$ and its connection to the Vietoris-Rips complex of $X$. In this manner, we provide positive answers to the question above for the case $n=2$ and $3$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_08664 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Contractibility of Vietoris-Rips Complexes of dense subsets in $(\mathbb{R}^n, \ell_1)$ via hyperconvex embeddings Wang, Qingsong Algebraic Topology 51F99, 55N31 We consider the contractibility of Vietoris-Rips complexes of dense subsets of $(\mathbb{R}^n,\ell_1)$ with sufficiently large scales. This is motivated by a question by Matthew Zaremsky regarding whether for each $n$ natural there is a $r_n>0$ so that the Vietoris-Rips complex of $(\mathbb{Z}^n,\ell_1)$ at scale $r$ is contractible for all $r\geq r_n$. We approach this question using results that relates to the neighborhood of embeddings into hyperconvex metric space of a metric space $X$ and its connection to the Vietoris-Rips complex of $X$. In this manner, we provide positive answers to the question above for the case $n=2$ and $3$. |
| title | Contractibility of Vietoris-Rips Complexes of dense subsets in $(\mathbb{R}^n, \ell_1)$ via hyperconvex embeddings |
| topic | Algebraic Topology 51F99, 55N31 |
| url | https://arxiv.org/abs/2406.08664 |