On the cohomology rings of tree braid groups

Abstract

Let $\Gamma$ be a finite connected graph. The (unlabelled) configuration space $U{C}^n\Gamma$ of $n$ points on $\Gamma$ is the space of $n$-element subsets of $\Gamma$. The $n$-strand braid group of $\Gamma$, denoted $B_n\Gamma$, is the fundamental group of $U{C}^n\Gamma$. We use the methods and results of our paper "Discrete Morse theory and graph braid groups" to get a partial description of the cohomology rings $H^{\ast }(B_{n}T)$, 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's conjecture that braid groups of planar graphs are right-angled Artin groups.

25 pages, 7 figures. Revised version, accepted by the Journal of Pure and Applied Algebra

References in corpus (1)

• Braid groups and right angled Artin groups

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