Cohomology ring of tree braid groups and exterior face rings

Abstract

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