Braid Groups on Triangulated Surfaces and Singular Homology

Abstract

Let $\Sigma_g$ denote the closed orientable surface of genus $g$ and fix an arbitrary simplicial triangulation of $\Sigma_g$. We construct and study a natural surjective group homomorphism from the surface braid group on $n$ strands on $\Sigma_g$ to the first singular homology group of $\Sigma_g$ with integral coefficients. In particular, we show that the kernel of this homomorphism is generated by canonical braids which arise from the triangulation of $\Sigma_g$. This provides a simple description of natural subgroups of surface braid groups which are closely tied to the homology groups of the surfaces $\Sigma_g$.