Configuration spaces of disks in an infinite strip

Abstract

We study the topology of the configuration spaces $C(n,w)$ of $n$ hard disks of unit diameter in an infinite strip of width $w$. We describe ranges of parameter or "regimes", where homology $H_j[C(n,w)]$ behaves in qualitatively different ways. We show that if $w\ge j+2$, then the homology $H_j[C(n,w)]$ is isomorphic to the homology of the configuration space of points in the plane, $H_j[C(n,{\mathbb{R}}^2)]$. The Betti numbers of $C(n,{\mathbb{R}}^2)$ were computed by Arnold, and so as a corollary of the isomorphism, ${\beta }_j[C(n,w)]$ is a polynomial in $n$ of degree $2j$. On the other hand, we show that if $2\le w\le j+1$, then ${\beta }_j[C(n,w)]$ grows exponentially with $n$. Most of our work is in carefully estimating ${\beta }_j[C(n,w)]$ in this regime. We also illustrate, for every $n$, the homological "phase portrait" in the $(w,j)$-plane--- the parameter values where homology $H_j[C(n,w)]$ is trivial, nontrivial, and isomorphic with $H_j[C(n,{\mathbb{R}}^2)]$. Motivated by the notion of phase transitions for hard-spheres systems, we discuss these as the "homological solid, liquid, and gas" regimes.

Mostly minor revisions in v2. The biggest change is that we now have a complete proof that the isomorphism on homology in the gas regime is induced by the inclusion map