On the structure of braid groups on complexes

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.

40 pages, 26 figures