Let $S$ be a set of $n$ points in $\mathbb{R}^d$, where $d \ge 2$. Then let $C=C(S)$ denote the union of closed balls of radius $1$ centered at points of $S$.
For $0 \le j \le d-1$, how large can the Betti number $b_j ( C)$ be? Let $M=M(n,j,d)$ be the maximum possible Betti number over all configurations of $n$ points. We're mostly interested in the asymptotic rate of growth of $M$ as $j$ and $d$ are fixed and $n \to \infty$. By the Nerve Theorem, the union of balls $C$ is homotopy equivalent to the Čech complex, so you can consider the Čech complex instead of the union of balls if you like.
In all the examples we know of, we have that $M(n,j,d) \le c n$. Here $c=c(j,d)$ is a constant that depends on $j$ and $d$, but not on $n$. Does this hold in general?
Here are a few things we know.
- A linear bound certainly holds for $j=0$.
- If you consider the Vietoris–Rips complex instead of the Čech complex, it holds for $j=1$ and $d\ge 2$, but fails for larger $j,d$. In particular, according to this paper of Goff, once $j=2$ and $d=5$, the Betti number can be of order at least $n^{3/2}$.
- A linear upper bound on Betti numbers holds asymptotically almost surely for random Čech and Vietoris–Rips complexes (in any regime of parameter). A proof for the Vietoris–Rips setting can be found here.
But at the moment we don't even know if a linear bound on the Betti numbers of the Čech complex holds in the special case when $j=1$ and $d=2$.
