The connectivity of Vietoris-Rips complexes of spheres
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914925526384640 |
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| author | Adams, Henry Bush, Johnathan Virk, Žiga |
| author_facet | Adams, Henry Bush, Johnathan Virk, Žiga |
| contents | We survey what is known and unknown about Vietoris-Rips complexes and thickenings of spheres. Afterwards, we show how to control the homotopy connectivity of Vietoris-Rips complexes of spheres in terms of coverings of spheres and projective spaces. Let $S^n$ be the $n$-sphere with the geodesic metric, and of diameter $π$, and let $δ> 0$. Suppose that the first nontrivial homotopy group of the Vietoris-Rips complex $\mathrm{VR}(S^n;π-δ)$ of the $n$-sphere at scale $π-δ$ occurs in dimension $k$, i.e., suppose that the connectivity is $k-1$. Then $\mathrm{cov}_{S^n}(2k+2) \le δ< 2\cdot \mathrm{cov}_{\mathbb{R}P^n}(k)$. In other words, there exist $2k+2$ balls of radius $δ$ that cover $S^n$, and no set of $k$ balls of radius $\fracδ{2}$ cover the projective space $\mathbb{R}P^n$. As a corollary, the homotopy type of $\mathrm{VR}(S^n;r)$ changes infinitely many times as the scale $r$ increases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_15818 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The connectivity of Vietoris-Rips complexes of spheres Adams, Henry Bush, Johnathan Virk, Žiga Algebraic Topology Metric Geometry 55N31, 55Q52, 52C17 We survey what is known and unknown about Vietoris-Rips complexes and thickenings of spheres. Afterwards, we show how to control the homotopy connectivity of Vietoris-Rips complexes of spheres in terms of coverings of spheres and projective spaces. Let $S^n$ be the $n$-sphere with the geodesic metric, and of diameter $π$, and let $δ> 0$. Suppose that the first nontrivial homotopy group of the Vietoris-Rips complex $\mathrm{VR}(S^n;π-δ)$ of the $n$-sphere at scale $π-δ$ occurs in dimension $k$, i.e., suppose that the connectivity is $k-1$. Then $\mathrm{cov}_{S^n}(2k+2) \le δ< 2\cdot \mathrm{cov}_{\mathbb{R}P^n}(k)$. In other words, there exist $2k+2$ balls of radius $δ$ that cover $S^n$, and no set of $k$ balls of radius $\fracδ{2}$ cover the projective space $\mathbb{R}P^n$. As a corollary, the homotopy type of $\mathrm{VR}(S^n;r)$ changes infinitely many times as the scale $r$ increases. |
| title | The connectivity of Vietoris-Rips complexes of spheres |
| topic | Algebraic Topology Metric Geometry 55N31, 55Q52, 52C17 |
| url | https://arxiv.org/abs/2407.15818 |