  Let M 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 i: F\_n(M) --\textgreater{} M^n of the nth configuration space
F\_n(M) of M into the n-fold Cartesian product of M, as well as the induced
homomorphism i\_\#: P\_n(M) --\textgreater{} (\pi\_1(M))^n, where P\_n(M) is
the n-string pure braid group of M. Both i and i\_\# were studied initially by
J.Birman who conjectured that Ker(i\_\#) is equal to the normal closure of the
Artin pure braid group P\_n in P\_n(M). The conjecture was later proved by
C.Goldberg for compact surfaces without boundary different from the 2-sphere
S^2 and the projective plane RP^2. In this paper, we prove the conjecture for
S^2 and RP^2. In the case of RP^2, we prove that Ker(i\_\#) is equal to the
commutator subgroup of P\_n(RP^2), we show that it may be decomposed in a
manner similar to that of P\_n(S^2) 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 semi-direct product of free
groups. Finally, we show that the groups B\_n(S^2) and P\_n(S^2) (resp.
B\_n(RP^2) and P\_n(RP^2)) have finite virtual cohomological dimension equal to
n-3 (resp. n-2), where B\_n(M) denotes the full n-string braid group of M. This
allows us to determine the virtual cohomological dimension of the mapping class
groups of the mapping class groups of S^2 and RP^2 with marked points, which in
the case of S^2, reproves a result due to J.Harer.
