On the cohomology rings of tree braid groups

Abstract

Let $Γ$ be a finite connected graph. The (unlabelled) configuration space $UC^n Γ$ of $n$ points on $Γ$ is the space of $n$-element subsets of $Γ$. The $n$-strand braid group of $Γ$, denoted $B_nΓ$, is the fundamental group of $UC^n Γ$. We use the methods and results of our paper &#34;Discrete Morse theory and graph braid groups&#34; to get a partial description of the cohomology rings $H^*(B_n T)$, where $T$ is a tree. Our results are then used to prove that $B_n T$ is a right-angled Artin group if and only if $T$ is linear or $n<4$. This gives a large number of counterexamples to Ghrist&#39;s conjecture that braid groups of planar graphs are right-angled Artin groups.