Configuration spaces of circles in the plane

TL;DR

This paper studies the topological structure of the space of all configurations of finitely many circles in the plane, revealing its asphericity and describing the fundamental groups as iterated semidirect products related to braid groups.

Contribution

It characterizes the topology of circle configuration spaces, computes their fundamental groups, and links these groups to braid groups and tree automorphisms, providing new insights into their structure.

Findings

The configuration space is aspherical.

Fundamental groups are iterated semidirect products of braid group subgroups.

Connections to statistical mechanics, topological data analysis, and geometric group theory.

Abstract

We consider the space of all configurations of finitely many (potentially nested) circles in the plane. We prove that this space is aspherical, and compute the fundamental group of each of its connected components. It turns out these fundamental groups are obtained as iterated semidirect products of subgroups of braid groups, with the structure for each component dictated by a finite rooted tree. These groups can be viewed as "braided" versions of the automorphism groups of such trees. We also discuss connections to statistical mechanics, topological data analysis, and geometric group theory.