Homology groups of the curvature sets of ${\mathbb{S}}^1$

Abstract

For $n\ge 2$, the $n$-th curvature set of a metric space $X$ is the set consisting of all $n$-by-$n$ distance matrices of $n$ points sampled from $X$. Curvature sets can be regarded as a geometric analogue of configuration spaces. In this paper we carry out a geometric and topological study of the curvature sets of the unit circle ${\mathbb{S}}^1$ equipped with the geodesic metric. Via an inductive argument we compute the homology groups of all curvature sets of ${\mathbb{S}}^1$. We also construct an abstract simplicial complex, called the $n$-th State Complex, whose geometric realization is homeomorphic to the $n$-th Curvature Set of ${\mathbb{S}}^1$.

Reworked Section 4 to work directly with cluster structures rather than using state matrices. As a bonus, we added a proof that the curvature sets of the circle are geometric simplicial complexes (Theorem 2), and we updated the appendix with the calculations of K_3 accordingly