Representation Stability for Disks in a Strip

TL;DR

This paper studies the homology of configuration spaces of disks in an infinite strip, establishing finite presentations and demonstrating various levels of representation stability as the strip width varies.

Contribution

It provides a finite presentation for the rational homology of disk configuration spaces and introduces notions of first- and higher-order representation stability for these spaces.

Findings

Finite presentation for the rational homology groups.

First-order representation stability analogous to point configurations.

Higher-order stability for large strip widths.

Abstract

We consider the ordered configuration space of $n$ open unit-diameter disks in the infinite strip of width $w$. In the spirit of Arnol'd and Cohen, we provide a finite presentation for the rational homology groups of this ordered configuration space as a twisted algebra. We use this presentation to prove that the ordered configuration space of open unit-diameter disks in the infinite strip of width $w$ exhibits a notion of first-order representation stability similar to Church--Ellenberg--Farb and Miller--Wilson's first-order representation stability for the ordered configuration space of points in a manifold. In addition, we prove that for large $w$ this disk configuration space exhibits notions of second- (and higher) order representation stability.