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

Lima Gonçalves, Daciberg; Guaschi, John; Maldonado, Miguel

Confluentes Mathematici, Tome 10 (2018) no. 1, pp. 41-61.

We use 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 [4, 14] 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, generalising results of [15]. Finally, if the genus of $N$ is at least $2$, we deduce upper bounds for the virtual cohomological dimension of $MCG(N;k)$ that are coherent with computations of Ivanov.

Classification : 57N05, 20F36, 55R80, 55P20, 20F38, 57M07, 20J06
Mots clés : Mapping class groups, surface braid groups, finite coverings, embeddings, (virtual) cohomological dimension

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