Vol. 287, No. 1, 2017

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Inclusion of configuration spaces in Cartesian products, and the virtual cohomological dimension of the braid groups of $\mathbb{S}^2$ and $\mathbb{R}P^2$

Daciberg Lima Gonçalves and John Guaschi

Vol. 287 (2017), No. 1, 71–99

Let S be a surface, perhaps with boundary, and either compact or with a finite number of points removed from the interior of the surface. We consider the inclusion ι : Fn(S) 1nS of the n-th configuration space Fn(S) of S into the n-fold Cartesian product of S, as well as the induced homomorphism ι# : Pn(S) 1nπ1(S), where Pn(S) is the n-string pure braid group of S. Both ι and ι# were studied initially by J. Birman, who conjectured that Ker(ι#) is equal to the normal closure of the Artin pure braid group Pn in Pn(S). The conjecture was later proved by C. Goldberg for compact surfaces without boundary different from the 2-sphere S2 and the projective plane P2. In this paper, we prove the conjecture for S2 and P2. In the case of P2, we prove that Ker(ι#) is equal to the commutator subgroup of Pn(P2), we show that it may be decomposed in a manner similar to that of Pn(S2) as a direct sum of a torsion-free subgroup Ln and the finite cyclic group generated by the full twist braid, and we prove that Ln may be written as an iterated semidirect product of free groups. Finally, we show that the groups Bn(S2) and Pn(S2) (resp. Bn(P2) and Pn(P2)) have finite virtual cohomological dimension equal to n 3 (resp. n 2), where Bn(S) denotes the full n-string braid group of S. This allows us to determine the virtual cohomological dimension of the mapping class groups of S2 and P2 with marked points, which in the case of S2 reproves a result due to J. Harer.

configuration spaces, surface braid groups, group presentations, virtual cohomological dimension
Mathematical Subject Classification 2010
Primary: 20F36
Secondary: 20J06
Received: 12 November 2015
Revised: 8 July 2016
Accepted: 19 July 2016
Published: 6 February 2017
Daciberg Lima Gonçalves
Instituto de Matemática e Estatística da Universidade de São Paulo
Departamento de Matemática
Rua do Matão, 1010 CEP 05508-090
São Paulo-SP
John Guaschi
Laboratoire de Mathématiques Nicolas Oresme UMR CNRS 6139
Normandie Université
Université de Caen Normandie
14000 Caen