Maximum Betti numbers of Čech complexes

Abstract

The Upper Bound Theorem for convex polytopes implies that the $p$-th Betti number of the Čech complex of any set of $N$ points in ${\mathbb{R}}^d$ and any radius satisfies ${\beta }_p=O(N^m)$, with $m=\min \{p+1,\lceil d/2\rceil \}$. We construct sets in even and odd dimensions that prove this upper bound is asymptotically tight. For example, we describe a set of $N=2(n+1)$ points in ${\mathbb{R}}^3$ and two radii such that the first Betti number of the Čech complex at one radius is $(n+1{)}^2-1$, and the second Betti number of the Čech complex at the other radius is $n^2$.

22 pages, 3 figures