Homotopy connectivity of Čech complexes of spheres

Abstract

Let $S^n$ be the $n$-sphere with the geodesic metric and of diameter $\pi$. The intrinsic Čech complex of $S^n$ at scale $r$ is the nerve of all open balls of radius $r$ in $S^n$. In this paper, we show how to control the homotopy connectivity of Čech complexes of spheres at each scale between $0$ and $\pi$ in terms of coverings of spheres. Our upper bound on the connectivity, which is sharp in the case $n=1$, comes from the chromatic numbers of Borsuk graphs of spheres. Our lower bound is obtained using the conicity (in the sense of Barmak) of Čech complexes of the sufficiently dense, finite subsets of $S^n$. Our bounds imply the new result that for $n\ge 1$, the homotopy type of the Čech complex of $S^n$ at scale $r$ changes infinitely many times as $r$ varies over $(0,\pi )$; we conjecture only countably many times. Additionally, we lower bound the homological dimension of Čech complexes of finite subsets of $S^n$ in terms of their packings.

Minor changes made based on the two referee reports. To appear in Discrete & Computational Geometry