Configuration space, moduli space and 3-fold covering space

TL;DR

This paper explores a specific injective homomorphism from braid groups to surface mapping class groups induced by 3-fold branched coverings, revealing new algebraic structures and homological properties.

Contribution

It provides a concrete description of the homomorphism, proves its injectivity, and demonstrates compatibility with operad actions, supporting Harer’s conjecture.

Findings

The map is injective via Birman-Hilden theory.

The induced map on classifying spaces is compatible with little 2-cube operad.

Lifted braids act as products of inverse Dehn twists.

Abstract

A function from configuration space to moduli space of surface may induce a homomorphism between their fundamental groups which are braid groups and mapping class groups of surface, respectively. This map $\phi: B_k \rightarrow \Gamma_{g,b}$ is induced by 3-fold branched covering over a disk with some branch points. In this thesis we give a concrete description of this map and show that it is injective by Birman-Hilden theory. This gives us a new interesting non-geometric embedding of braid group into mapping class group. On the other hand, we show that the map on the level of classifying spaces of groups is compatible with the action of little 2-cube operad so that it induces a trivial homomorphim between stable homology group of braid groups and that of mapping class groups(Harer conjecture). We also show how the lift $\tilde{\beta_i}$ acts on the fundamental group of the surface and through this we prove that $\tilde{\beta_i}$ equals the product of two inverse Dehn twists.