Configuration spaces of disks in a strip, twisted algebras, persistence, and other stories

Abstract

We give $\mathbb{Z}$-bases for the homology and cohomology of the configuration space $\mathrm{config}(n,w)$ of $n$ unit disks in an infinite strip of width $w$, first studied by Alpert, Kahle and MacPherson. We also study the way these spaces evolve both as $n$ increases (using the framework of representation stability) and as $w$ increases (using the framework of persistent homology). Finally, we include some results about the cup product in the cohomology and about the configuration space of unordered disks.

49 pages, 15 figures. Builds on and supersedes arXiv:2006.01240. v2 corrects problems with Sections 2 and 9 and expands and clarifies arguments elsewhere

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