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 (1511.02101v1)

Abstract: 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.