Cohomology ring of tree braid groups and exterior face rings

TL;DR

This paper characterizes the cohomology ring of tree braid groups using discrete Morse theory, revealing that it can often be described as an exterior face ring associated with a simplicial complex derived from the tree.

Contribution

It introduces an explicit simplicial complex that encodes the cohomology ring of tree braid groups and demonstrates when this ring is an exterior face ring, especially for binary trees.

Findings

Cohomology ring $H^*(B_nT)$ is encoded by a simplicial complex $K_nT$.

In many cases, $H^*(B_nT)$ is an exterior face ring.

The approach uses discrete Morse theory to analyze the structure.

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 $K_nT$.