Contractible Vietoris-Rips complexes of $\mathbb{Z}^n$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913986785574912 |
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| author | Zaremsky, Matthew C. B. |
| author_facet | Zaremsky, Matthew C. B. |
| contents | We give a new, short proof of a result of Virk, that the Vietoris-Rips complex of the group $\mathbb{Z}^n$ with the standard word metric is contractible at large enough scales. This is inspired by a key observation in Virk's proof, but we use Bestvina-Brady discrete Morse theory to get a very short proof with better bounds. In the course of this, we get a new, general criterion for a metric space to have contractible Vietoris-Rips complexes at large enough scales, which could prove useful in the future. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_11993 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Contractible Vietoris-Rips complexes of $\mathbb{Z}^n$ Zaremsky, Matthew C. B. Group Theory Geometric Topology Metric Geometry We give a new, short proof of a result of Virk, that the Vietoris-Rips complex of the group $\mathbb{Z}^n$ with the standard word metric is contractible at large enough scales. This is inspired by a key observation in Virk's proof, but we use Bestvina-Brady discrete Morse theory to get a very short proof with better bounds. In the course of this, we get a new, general criterion for a metric space to have contractible Vietoris-Rips complexes at large enough scales, which could prove useful in the future. |
| title | Contractible Vietoris-Rips complexes of $\mathbb{Z}^n$ |
| topic | Group Theory Geometric Topology Metric Geometry |
| url | https://arxiv.org/abs/2410.11993 |