On the Vietoris-Rips Complexes of Integer Lattices

For a metric space $X$ and $r \geq 0$, the Vietoris-Rips complex $\mathcal{VR}(X;r)$ 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 $r \geq n^2(2n-1)$, or $1\leq n \leq 3$ and $r \geq n$, then the complex $\mathcal{VR}(\mathbb{Z}^n;r)$ is contractible, and posed a question if $\mathcal{VR}(\mathbb{Z}^n;r)$ is contractible for all $r \geq n$. Recently, Matthew Zaremsky improved Ziga's result and proved that $\mathcal{VR}(\mathbb{Z}^n;r)$ is contractible if $r \geq n^2+ n-1$. Further, he conjectured that $\mathcal{VR}(\mathbb{Z}^n;r)$ is contractible for all $r \geq n$. We prove Zaremsky's conjecture for $n \leq 5$, i.e., we prove that $\mathcal{VR}(\mathbb{Z}^n;r)$ is contractible if $n \leq 5$ and $r \geq n$. Further, we prove that $\mathcal{VR}(\mathbb{Z}^n;r)$ is contractible for $r \geq 10$. We determine the homotopy type of $\mathcal{VR}(\mathbb{Z}^n;2)$, and show that these complexes are homotopy equivalent to a wedge of countably infinite copies of $\mathbb{S}^3$. We also show that $\mathcal{VR}(\mathbb{Z}^n;r)$ is simply connected for $r \geq 2$.