On the structure of braid groups on complexes

TL;DR

This paper explores the structure of braid groups on finite simplicial complexes, establishing relationships with geometric decompositions, providing algorithms for group presentations, and offering criteria for surface embeddability and planarity.

Contribution

It introduces a method to compute braid group presentations on complexes and links geometric decompositions to algebraic properties like torsion-freeness and abelianization.

Findings

Algorithm for computing braid group presentations

Criteria for surface embeddability of complexes

Criteria for planarity based on braid group properties

Abstract

We consider the braid groups $\mathbf{B}_n(X)$ on finite simplicial complexes $X$, which are generalizations of those on both manifolds and graphs that have been studied already by many authors. We figure out the relationships between geometric decompositions for $X$ and their effects on braid groups, and provide an algorithmic way to compute the group presentations for $\mathbf{B}_n(X)$ with the aid of them. As applications, we give complete criteria for both the surface embeddability and planarity for $X$, which are the torsion-freeness of the braid group $\mathbf{B}_n(X)$ and its abelianization $H_1(\mathbf{B}_n(X))$, respectively.