Cohomology ring of tree braid groups and exterior face rings

González, Jesús; Hoekstra-Mendoza, Teresa

doi:10.1090/btran/131

Cohomology ring of tree braid groups and exterior face rings
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Trans. Amer. Math. Soc. Ser. B 9 (2022), 1065-1101

Abstract:

For a tree $T$ and a positive integer $n$, let $B_nT$ denote the $n$-strand braid group on $T$. We use discrete Morse theory techniques to show that the cohomology ring $H^*(B_nT)$ is encoded by an explicit abstract simplicial complex $K_nT$ that measures $n$-local interactions among essential vertices of $T$. We show that, in many cases (for instance when $T$ is a binary tree), $H^*(B_nT)$ is the exterior face ring determined by $K_nT$.

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