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Article Dans Une Revue Confluentes Mathematici Année : 2018

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

Résumé

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 Gonçalves 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 $MCG(N ; k)$.
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Dates et versions

hal-01377681 , version 1 (11-10-2016)

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Daciberg Lima Gonçalves, John Guaschi, Miguel Maldonado. Embeddings and the (virtual) cohomological dimension of the braid and mapping class groups of surfaces. Confluentes Mathematici, 2018, 10 (1), pp.41-61. ⟨hal-01377681⟩
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