  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 $\Sigma_g$.
