Homotopy connectivity of \v{C}ech complexes of spheres

TL;DR

This paper investigates the homotopy connectivity of cech complexes of spheres, providing bounds based on sphere coverings and graph chromatic numbers, revealing infinitely many homotopy type changes as scale varies.

Contribution

It introduces new bounds on the homotopy connectivity of cech complexes of spheres, linking them to sphere coverings and Borsuk graph chromatic numbers, and shows the homotopy type changes infinitely often.

Findings

Upper bounds on connectivity are sharp for n=1.

Homotopy type of complexes changes infinitely many times with scale.

Lower bounds relate to packings and homological dimension.

Abstract

Let $S^n$ be the $n$-sphere with the geodesic metric and of diameter $\pi$. The intrinsic \v{C}ech 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 \v{C}ech 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 \v{C}ech 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 \v{C}ech 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 \v{C}ech complexes of finite subsets of $S^n$ in terms of their packings.