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

TL;DR

This paper explores the relationships between braid and mapping class groups of non-orientable surfaces and their orientable double covers, providing new embeddings, cohomological dimension calculations, and bounds for these groups.

Contribution

It generalizes existing results on embeddings of mapping class groups and computes cohomological dimensions for braid groups on aspherical surfaces, offering bounds for non-orientable cases.

Findings

Embedding of MCG(N; k) into MCG(S; 2k) established

Cohomological dimension of braid groups on aspherical surfaces computed

Upper bounds for virtual cohomological dimension of MCG(N; k) provided

Abstract

In this paper, we make use of the relations between the braid and mapping class groups of a compact, connected, non-orientable surface N without boundary and those of its orientable double covering S to study embeddings of these groups and their (virtual) cohomological dimensions. We first generalise results of Birman and Chillingworth and of Gon\c{c}alves and Guaschi to show that the mapping class group MCG(N ; k) of N relative to a k-point subset embeds in the mapping class group MCG(S; 2k) of S relative to a 2k-point subset. We then compute the cohomological dimension of the braid groups of all compact, connected aspherical surfaces without boundary. Finally, if the genus of N is greater than or equal to 2, we give upper bounds for the virtual cohomological dimension of MCG(N ; k).