Avoiding collisions: an introduction to the topology of configuration spaces

Abstract

This thesis provides a comprehensive study of configuration spaces, tracing their topological properties from classical homogeneous manifolds to the discrete setting of topological graphs. Configuration spaces serve as a fundamental bridge between algebraic topology, geometric group theory, and robotics, capturing the collision-free motion of multiple particles.

We begin by establishing the foundational properties of ordered configuration spaces on connected manifolds, with a rigorous focus on the forgetting map and the Fadell–Neuwirth fibration theorem. This geometric framework transitions naturally into the study of braid groups, where we explore their algebraic presentations and characterisations. Following this, we analyze the specific topology of the planar configuration space $\mathcal{F}_n(\mathbb{R}^2)$ by examining the classical Fox-Neuwirth theorem. Additionally, we introduce the complexities that arise when extending these techniques to compact connected surfaces, where the splitting behavior of the Fadell–Neuwirth sequence is no longer guaranteed.

Finally, we shift our focus to the distinct combinatorial nature of configuration spaces on graphs. We investigate the local structure by analyzing the configuration space of the star of a vertex, and scale these techniques to analyze the global space of a whole graph. To illustrate the concrete applications of this discrete framework, we present a fully worked example modeling a segment graph with three points.