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
$\mathbb{R}{P}^{2}$. In this paper, we
prove the conjecture for
${\mathbb{S}}^{2}$
and
$\mathbb{R}{P}^{2}$. In the case
of
$\mathbb{R}{P}^{2}$, we prove that
$Ker\left({\iota}_{\#}\right)$ is equal to the
commutator subgroup of
${P}_{n}\left(\mathbb{R}{P}^{2}\right)$,
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
torsionfree 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. ${B}_{n}\left(\mathbb{R}{P}^{2}\right)$
and
${P}_{n}\left(\mathbb{R}{P}^{2}\right)$)
have finite virtual cohomological dimension equal to
$n3$
(resp. $n2$), 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
$\mathbb{R}{P}^{2}$ with marked points,
which in the case of
${\mathbb{S}}^{2}$
reproves a result due to J. Harer.
