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

DOI: 10.2140/pjm.2017.287.71

Abstract

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 $\iota :F_n\left(S\right)\to {\prod }_1^{n}S$ of the $n$-th configuration space $F_n\left(S\right)$ of $S$ into the $n$-fold Cartesian product of $S$, as well as the induced homomorphism ${\iota }_{\#}:P_n\left(S\right)\to {\prod }_1^n{\pi }_1\left(S\right)$, where $P_n\left(S\right)$ is the $n$-string pure braid group of $S$. Both $\iota$ and ${\iota }_{\#}$ were studied initially by J. Birman, who conjectured that $Ker\left({\iota }_{\#}\right)$ is equal to the normal closure of the Artin pure braid group $P_n$ in $P_n\left(S\right)$. The conjecture was later proved by C. Goldberg for compact surfaces without boundary different from the $2$-sphere ${\mathbb{S}}^2$ and the projective plane $ℝP^2$. In this paper, we prove the conjecture for ${\mathbb{S}}^2$ and $ℝP^2$. In the case of $ℝP^2$, we prove that $Ker\left({\iota }_{\#}\right)$ is equal to the commutator subgroup of $P_n\left(ℝP^2\right)$, we show that it may be decomposed in a manner similar to that of $P_n\left({\mathbb{S}}^2\right)$ as a direct sum of a torsion-free subgroup $L_n$ and the finite cyclic group generated by the full twist braid, and we prove that $L_n$ may be written as an iterated semidirect product of free groups. Finally, we show that the groups $B_n\left({\mathbb{S}}^2\right)$ and $P_n\left({\mathbb{S}}^2\right)$ (resp. $B_n\left(ℝP^2\right)$ and $P_n\left(ℝP^2\right)$) have finite virtual cohomological dimension equal to $n-3$ (resp. $n-2$), where $B_n\left(S\right)$ denotes the full $n$-string braid group of $S$. This allows us to determine the virtual cohomological dimension of the mapping class groups of ${\mathbb{S}}^2$ and $ℝP^2$ with marked points, which in the case of ${\mathbb{S}}^2$ reproves a result due to J. Harer.

Keywords

configuration spaces, surface braid groups, group presentations, virtual cohomological dimension

Mathematical Subject Classification 2010

Primary: 20F36

Secondary: 20J06

Milestones

Received: 12 November 2015

Revised: 8 July 2016

Accepted: 19 July 2016

Published: 6 February 2017