Equivariant Mapping Class Group and Orbit Braid Group

Motivated by the work of Birman about relationship between mapping class groups and braid groups, authors discuss orbit group equivariant on closed surface M with a free proper action in this paper. Their construction is based exact sequence given fibration $${\cal F}_0^GM \to F\left( {M/G,n} \right)$$ . The conclusion closely connected quotient space. Comparing situation without action, there big difference when space $${\mathbb{T}^2}$$