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arXiv · 2410.11993

Contractible Vietoris-Rips complexes of $\mathbb{Z}^n$

Matthew C. B. Zaremsky
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Abstract

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.

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BibTeXRIS

Matthew C. B. Zaremsky. 2024-10-15. Contractible Vietoris-Rips complexes of $\mathbb{Z}^n$. https://arxiv.org/abs/2410.11993

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