A Special Subgroup of the Surface Braid Group

TL;DR

This paper investigates the algebraic structure of the fundamental group of the configuration space of points on a Riemann surface, showing that for large n, its kernel is generated by transpositions related to the surface's edges.

Contribution

It establishes that the kernel of the fundamental group map for the configuration space is generated by transpositions, extending understanding of surface braid groups.

Findings

Kernel is generated by transpositions for large n

Edge set of the surface generates the kernel

Results apply to compact oriented Riemann surfaces

Abstract

Herein we prove that if $M$ is a compact oriented Riemann surface of genus $g$, and $M^{[n]}$ is the classifying space of $n$ distinct, unordered points on $M$, then the kernel of the map $\pi_1(M^{[n]})\to H_1(M)$ is generated by transpositions for sufficiently large $n$. Specifically, we treat $M$ as a polyhedron, and the edge set of $M$ generates this group.