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Let M be a surface, perhaps with boundary, and either compact, or with afinite number of points removed from the interior of the surface. We considerthe inclusion i: F\_n(M) --\textgreater{} M^n of the nth configuration spaceF\_n(M) of M into the n-fold Cartesian product of M, as well as the inducedhomomorphism i\_\#: P\_n(M) --\textgreater{} (\pi\_1(M))^n, where P\_n(M) isthe n-string pure braid group of M. Both i and i\_\# were studied initially byJ.Birman who conjectured that Ker(i\_\#) is equal to the normal closure of theArtin pure braid group P\_n in P\_n(M). The conjecture was later proved byC.Goldberg for compact surfaces without boundary different from the 2-sphereS^2 and the projective plane RP^2. In this paper, we prove the conjecture forS^2 and RP^2. In the case of RP^2, we prove that Ker(i\_\#) is equal to thecommutator subgroup of P\_n(RP^2), we show that it may be decomposed in amanner similar to that of P\_n(S^2) as a direct sum of a torsion-free subgroupL\_n and the finite cyclic group generated by the full twist braid, and weprove that L\_n may be written as an iterated semi-direct product of freegroups. 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 ton-3 (resp. n-2), where B\_n(M) denotes the full n-string braid group of M. Thisallows us to determine the virtual cohomological dimension of the mapping classgroups of the mapping class groups of S^2 and RP^2 with marked points, which inthe case of S^2, reproves a result due to J.Harer.

pacific journal of mathematics

The inclusion of configuration spaces of surfaces in Cartesian products,  its induced homomorphism, and the virtual cohomological dimension of the  braid groups of S^2 and RP^2