Configuration spaces of disks in an infinite strip

Abstract

We study the topology of the configuration space 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 ≥ j + 2, then the inclusion i into the configuration space of n points in the plane C(n, R 2 ) induces an isomorphism on homology i * :The Betti numbers of C(n, R 2 ) were computed by Arnold [1], and so as a corollary of the isomorphism, if w and j are fixed then β j [C(n, w)] is a polynomial of degree 2j in n.On the other hand, we show that w and j are fixed and 2 ≤ w ≤ j + 1, then β j [C(n, w)] grows exponentially fast with n. Most of our work is in carefully estimating β j [C(n, w)] in this regime.We also illustrate for every n the "phase portrait" in the (w, j)-planethe parameter values where homology H j [C(n, w)] is trivial, nontrivial, and isomorphic with H j [C(n, R 2 )]. Motivated by the notion of phase transitions for hard-spheres systems, we discuss these as the "homological solid, liquid, and gas" regimes.

Generalized representation stability for disks in a strip and no-k-equal spaces

“…This paper is based on the technique of the paper [AKM19], which is to replace the configuration space config(n, w) by a homotopy-equivalent cell complex cell(n, w) and to estimate the homology by doing combinatorics (specifically, discrete Morse theory) on the cell complex. We use the same cell complex cell(n, w) as in that paper, and we use the same method to find a cell complex desc(n, k − 1) that is homotopy equivalent to the no-k-equal space no k (n, R).…”

“…We use the same cell complex cell(n, w) as in that paper, and we use the same method to find a cell complex desc(n, k − 1) that is homotopy equivalent to the no-k-equal space no k (n, R). In the remainder of this section we define the complexes cell(n, w) and desc(n, w), and we prove that desc(n, k − 1) is homotopy equivalent to no k (n, R) by adapting the method of [AKM19].…”

“…Theorem 3.1 of [AKM19] shows that cell(n, w) is homotopy equivalent to the configuration space config(n, w) of n disks in a strip of width w. The strategy is to find an open cover of config(n, w) indexed by the symbols in cell(n, w), where the intersections between open sets correspond to incidences in cell(n, w); the nerve theorem then implies that config(n, w) is homotopy equivalent to the barycentric subdivision of cell(n, w), and thus is also homotopy equivalent to cell(n, w).…”

“…The no-k-equal space no k (n, R) consists of all the elements of R n such that no k of the coordinates are equal. In the remainder of this section, we mimic the proof of Theorem 3.1 of [AKM19], in order to verify that no k (n, R) is homotopy equivalent to desc(n, k − 1).…”

“…The paper [AKM19] introduces the spaces config(n, w) and asks, for fixed j and w, how does H j (config(n, w)) depend on n? That paper estimates the dimension of H j (config(n, w)) up to a constant factor; it turns out to be exponential in n unless the strip is wide compared to j.…”