Maximum Betti numbers of \v{C}ech complexes

Abstract

The Upper Bound Theorem for convex polytopes implies that the $p$-th Betti number of the \v{C}ech 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 \v{C}ech complex at one radius is $(n+1)^2 - 1$, and the second Betti number of the \v{C}ech complex at the other radius is $n^2$.