#### 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
##### 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 $ℝ{P}^{2}$. In this paper, we prove the conjecture for ${\mathbb{S}}^{2}$ and $ℝ{P}^{2}$. In the case of $ℝ{P}^{2}$, we prove that $Ker\left({\iota }_{#}\right)$ is equal to the commutator subgroup of ${P}_{n}\left(ℝ{P}^{2}\right)$, 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. ${B}_{n}\left(ℝ{P}^{2}\right)$ and ${P}_{n}\left(ℝ{P}^{2}\right)$) 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 $ℝ{P}^{2}$ 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
Primary: 20F36
Secondary: 20J06