The connectivity of Vietoris-Rips complexes of spheres

TL;DR

This paper explores the homotopy connectivity of Vietoris-Rips complexes of spheres, establishing bounds based on sphere coverings and showing that their homotopy type changes infinitely often with scale.

Contribution

It provides new bounds linking Vietoris-Rips complex connectivity of spheres to covering properties of spheres and projective spaces, and demonstrates the infinite homotopy type changes with scale.

Findings

Connectivity bounds related to sphere coverings and projective spaces.

Homotopy type of VR complexes changes infinitely many times with scale.

Establishment of inequalities involving covering numbers and homotopy groups.

Abstract

We survey what is known and unknown about Vietoris-Rips complexes and thickenings of spheres. Afterwards, we show how to control the homotopy connectivity of Vietoris-Rips complexes of spheres in terms of coverings of spheres and projective spaces. Let $S^n$ be the $n$-sphere with the geodesic metric, and of diameter $\pi$, and let $\delta > 0$. Suppose that the first nontrivial homotopy group of the Vietoris-Rips complex $\mathrm{VR}(S^n;\pi-\delta)$ of the $n$-sphere at scale $\pi-\delta$ occurs in dimension $k$, i.e., suppose that the connectivity is $k-1$. Then $\mathrm{cov}_{S^n}(2k+2) \le \delta < 2\cdot \mathrm{cov}_{\mathbb{R}P^n}(k)$. In other words, there exist $2k+2$ balls of radius $\delta$ that cover $S^n$, and no set of $k$ balls of radius $\frac{\delta}{2}$ cover the projective space $\mathbb{R}P^n$. As a corollary, the homotopy type of $\mathrm{VR}(S^n;r)$ changes infinitely many times as the scale $r$ increases.