The connectivity of Vietoris-Rips complexes of spheres

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.