Equivariant Mapping Class Group and Orbit Braid Group

TL;DR

This paper explores the relationship between orbit braid groups and equivariant mapping class groups on surfaces with free group actions, revealing significant differences in the case of the torus compared to non-group action scenarios.

Contribution

It establishes a new connection between orbit braid groups and equivariant mapping class groups using an exact sequence derived from a fibration, especially highlighting the case of the torus.

Findings

The relationship is closely linked to the braid group of the quotient space.

A significant difference is observed when the quotient space is the torus.

The construction relies on the fibration sequence involving configuration spaces.

Abstract

Motivated by the work of Birman about the relationship between mapping class groups and braid groups, we discuss the relationship between the orbit braid group and the equivariant mapping class group on the closed surface $M$ with a free and proper group action in this article. Our construction is based on the exact sequence given by the fibration $\mathcal{F}_{0}^{G}M \rightarrow F(M/G,n)$. The conclusion is closely connected with the braid group of the quotient space. Comparing with the situation without the group action, there is a big difference when the quotient space is $\mathbb{T}^{2}$.