# CONFLUENTES MATHEMATICI

Embeddings and the (virtual) cohomological dimension of the braid and mapping class groups of surfaces
Confluentes Mathematici, Volume 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 $MCG\left(N;k\right)$ of $N$ relative to a $k$-point subset embeds in the mapping class group $MCG\left(S;2k\right)$ 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 $MCG\left(N;k\right)$ that are coherent with computations of Ivanov.

Accepted:
Published online:
DOI: 10.5802/cml.45
Classification: 57N05, 20F36, 55R80, 55P20, 20F38, 57M07, 20J06
Keywords: Mapping class groups, surface braid groups, finite coverings, embeddings, (virtual) cohomological dimension
Daciberg Lima Gonçalves 1; John Guaschi 2; Miguel Maldonado 3

1 Departamento de Matemática, IME, Universidade de São Paulo, Rua do Matão, 1010, CEP 05508-090 - São Paulo - SP, Brazil
2 Normandie Univ., UNICAEN, CNRS, Laboratoire de Mathématiques Nicolas Oresme UMR CNRS 6139, CS 14032, 14032 Caen Cedex 5, France
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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, Volume 10 (2018) no. 1, pp. 41-61. doi : 10.5802/cml.45. https://cml.centre-mersenne.org/articles/10.5802/cml.45/

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