Homotopy connectivity of Čech complexes of spheres
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866914465199423488 |
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| author | Adams, Henry Jauhari, Ekansh Mallick, Sucharita |
| author_facet | Adams, Henry Jauhari, Ekansh Mallick, Sucharita |
| contents | Let $S^n$ be the $n$-sphere with the geodesic metric and of diameter $π$. The intrinsic Čech complex of $S^n$ at scale $r$ is the nerve of all open balls of radius $r$ in $S^n$. In this paper, we show how to control the homotopy connectivity of Čech complexes of spheres at each scale between $0$ and $π$ in terms of coverings of spheres. Our upper bound on the connectivity, which is sharp in the case $n=1$, comes from the chromatic numbers of Borsuk graphs of spheres. Our lower bound is obtained using the conicity (in the sense of Barmak) of Čech complexes of the sufficiently dense, finite subsets of $S^n$. Our bounds imply the new result that for $n\ge 1$, the homotopy type of the Čech complex of $S^n$ at scale $r$ changes infinitely many times as $r$ varies over $(0,π)$; we conjecture only countably many times. Additionally, we lower bound the homological dimension of Čech complexes of finite subsets of $S^n$ in terms of their packings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_00122 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homotopy connectivity of Čech complexes of spheres Adams, Henry Jauhari, Ekansh Mallick, Sucharita Algebraic Topology Metric Geometry 55U10 (Primary) 55Q52, 05E45, 52C17 (Secondary) Let $S^n$ be the $n$-sphere with the geodesic metric and of diameter $π$. The intrinsic Čech complex of $S^n$ at scale $r$ is the nerve of all open balls of radius $r$ in $S^n$. In this paper, we show how to control the homotopy connectivity of Čech complexes of spheres at each scale between $0$ and $π$ in terms of coverings of spheres. Our upper bound on the connectivity, which is sharp in the case $n=1$, comes from the chromatic numbers of Borsuk graphs of spheres. Our lower bound is obtained using the conicity (in the sense of Barmak) of Čech complexes of the sufficiently dense, finite subsets of $S^n$. Our bounds imply the new result that for $n\ge 1$, the homotopy type of the Čech complex of $S^n$ at scale $r$ changes infinitely many times as $r$ varies over $(0,π)$; we conjecture only countably many times. Additionally, we lower bound the homological dimension of Čech complexes of finite subsets of $S^n$ in terms of their packings. |
| title | Homotopy connectivity of Čech complexes of spheres |
| topic | Algebraic Topology Metric Geometry 55U10 (Primary) 55Q52, 05E45, 52C17 (Secondary) |
| url | https://arxiv.org/abs/2502.00122 |