Presentations of Graph Braid Groups

TL;DR

This paper uses discrete Morse theory to compute presentations of graph braid groups, which are fundamental groups of configuration spaces of points on graphs, for all finite connected graphs and natural numbers.

Contribution

It provides a systematic method to derive presentations of all graph braid groups for any finite connected graph and number of points, expanding understanding of their algebraic structure.

Findings

Computed presentations for all graph braid groups on finite connected graphs.

Established a uniform approach using discrete Morse theory.

Enabled further algebraic and topological analysis of configuration spaces.

Abstract

Let G be a graph. The (unlabeled) configuration space of n points on G is the space of all n-element subsets of G. The fundamental group of such a configuration space is called a graph braid group. We use a version of discrete Morse theory to compute presentations of all graph braid groups, for all finite connected graphs G and all natural numbers n.