Equivariant Mapping Class Group and Orbit Braid Group

Abstract

Motivated by the work of Birman about the relationship between mapping class groups and braid groups, the authors 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 paper. Their construction is based on the exact sequence given by the fibration ${\cal F}_0^GM \to F\left( {M/G,n} \right)$. 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 ${\mathbb{T}^2}$.

Keywords

Equivariant mapping class group / Orbit braid group / Evaluation map / Center of braid group

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