On the Vietoris-Rips Complexes of Integer Lattices

Abstract

For a metric space $X$ and $r\ge 0$, the Vietoris-Rips complex $\mathcal{V}\mathcal{R}(X;r)$ is a simplicial complex whose simplices are finite subsets of $X$ with diameter at most $r$. Vietoris-Rips complexes have applications in various places, including data analysis, geometric group theory, sensor networks, etc. Consider the integer lattice ${\mathbb{Z}}^n$ as a metric space equipped with the $d_1$-metric (the Manhattan metric or standard word metric in the Cayley graph). Ziga Virk proved that if either $r\ge n^2(2n-1)$, or $1\le n\le 3$ and $r\ge n$, then the complex $\mathcal{V}\mathcal{R}({\mathbb{Z}}^n;r)$ is contractible, and posed a question if $\mathcal{V}\mathcal{R}({\mathbb{Z}}^n;r)$ is contractible for all $r\ge n$. Recently, Matthew Zaremsky improved Ziga's result and proved that $\mathcal{V}\mathcal{R}({\mathbb{Z}}^n;r)$ is contractible if $r\ge n^2+n-1$. Further, he conjectured that $\mathcal{V}\mathcal{R}({\mathbb{Z}}^n;r)$ is contractible for all $r\ge n$. We prove Zaremsky's conjecture for $n\le 5$, i.e., we prove that $\mathcal{V}\mathcal{R}({\mathbb{Z}}^n;r)$ is contractible if $n\le 5$ and $r\ge n$. Further, we prove that $\mathcal{V}\mathcal{R}({\mathbb{Z}}^n;r)$ is contractible for $r\ge 10$. We determine the homotopy type of $\mathcal{V}\mathcal{R}({\mathbb{Z}}^n;2)$, and show that these complexes are homotopy equivalent to a wedge of countably infinite copies of ${\mathbb{S}}^3$. We also show that $\mathcal{V}\mathcal{R}({\mathbb{Z}}^n;r)$ is simply connected for $r\ge 2$.

32 pages