Discrete Morse theory and graph braid groups

Abstract

If $\Gamma$ is any finite graph, then the unlabelled configuration space of $n$ points on $\Gamma$, denoted $U{\mathcal{C}}^n\Gamma$, is the space of $n$–element subsets of $\Gamma$. The braid group of $\Gamma$ on $n$ strands is the fundamental group of $U{\mathcal{C}}^n\Gamma$.

We apply a discrete version of Morse theory to these $U{\mathcal{C}}^n\Gamma$, for any $n$ and any $\Gamma$, and provide a clear description of the critical cells in every case. As a result, we can calculate a presentation for the braid group of any tree, for any number of strands. We also give a simple proof of a theorem due to Ghrist: the space $U{\mathcal{C}}^n\Gamma$ strong deformation retracts onto a CW complex of dimension at most $k$, where $k$ is the number of vertices in $\Gamma$ of degree at least 3 (and $k$ is thus independent of $n$).