Embeddings and the (virtual) cohomological dimension of the braid and mapping class groups of surfaces

摘要

In this paper, we make use of the relations between the braid and mappingclass groups of a compact, connected, non-orientable surface N without boundaryand those of its orientable double covering S to study embeddings of thesegroups and their (virtual) cohomological dimensions. We first generaliseresults of Birman and Chillingworth and of Gonc{c}alves and Guaschi to showthat the mapping class group MCG(N ; k) of N relative to a k-point subsetembeds in the mapping class group MCG(S; 2k) of S relative to a 2k-pointsubset. We then compute the cohomological dimension of the braid groups of allcompact, connected aspherical surfaces without boundary. Finally, if the genusof N is greater than or equal to 2, we give upper bounds for the virtualcohomological dimension of MCG(N ; k).