Embeddings and the (virtual) cohomological dimension of the braid and mapping class groups of surfaces
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).
Keywords
Cite
@article{arxiv.1610.03288,
title = {Embeddings and the (virtual) cohomological dimension of the braid and mapping class groups of surfaces},
author = {Daciberg Lima Gonçalves and John Guaschi and Miguel Maldonado},
journal= {arXiv preprint arXiv:1610.03288},
year = {2016}
}