Contractibility of Vietoris-Rips Complexes of dense subsets in $(\mathbb{R}^n, \ell_1)$ via hyperconvex embeddings

TL;DR

This paper investigates the contractibility of Vietoris-Rips complexes of dense subsets in Euclidean space with the $\, ext{l}_1$ metric, providing positive results for dimensions 2 and 3 using hyperconvex embeddings.

Contribution

It introduces a novel approach linking hyperconvex embeddings to Vietoris-Rips complexes, solving the contractibility question for specific dimensions.

Findings

Contractibility established for $n=2$ and $n=3$ cases.

Uses hyperconvex embeddings to analyze Vietoris-Rips complexes.

Provides partial answers to a question by Matthew Zaremsky.

Abstract

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 $3$.