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

TL;DR

This paper investigates the topological properties of curvature sets derived from the unit circle, computing their homology groups and establishing a homeomorphism with a constructed simplicial complex.

Contribution

It provides the first comprehensive homological analysis of curvature sets of the circle and introduces the $n$-th State Complex as a new geometric model.

Findings

Homology groups of all curvature sets of $ ext{S}^1$ are explicitly computed.

The $n$-th State Complex is homeomorphic to the $n$-th Curvature Set of $ ext{S}^1$.

The study links curvature sets with configuration space analogues.

Abstract

For $n \geq 2$, 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 $\mathbb{S}^1$.