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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) an

Author

Gonçalves, Daciberg Lima; Guaschi, John

Published

2015

Language

EN