Braid Groups on Triangulated Surfaces and Singular Homology

TL;DR

This paper establishes a natural surjective homomorphism from surface braid groups on triangulated surfaces to their first homology groups, with kernels generated by canonical braids from the triangulation, linking braid groups to surface homology.

Contribution

It constructs and analyzes a homomorphism from surface braid groups to homology, identifying its kernel as generated by canonical braids from the triangulation.

Findings

Surjective homomorphism from braid group to homology established

Kernel generated by canonical braids from triangulation

Provides a simple description of subgroups related to surface 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 $\Sigma_g$.