The braid group Bn,m(S2) and a generalisation of the Fadell-Neuwirth short exact sequence

Let m, n is an element of N. We define B-n,B-m (S-2) to be the set of (n + m)-braids of the sphere whose associated permutation lies in the subgroup S-n x S-m of the symmetric group Sn+m on n + m letters. In a previous paper [13], we showed that if n >= 3, then there exists the following generalisation of the Fadell-Neuwirth short exact sequence: 1 -> B-m(S-2\{x(1),...,x(n)}) -> B-n,B-m (S-2) ->(p*) B-n(S-2) -> 1, where p(*): B-n,B-m (S-2) -> B-n(S-2) is the group homomorphism (defined for all n is an element of N) given geometrically by forgetting the last m strings. In this paper we study the splitting of this short exact sequence, as well as the existence of a cross-section for the fibration p: D-n,D-m (S-2) -> D-n(S-2) of the quotients of the corresponding configuration spaces. Our main results are as follows: if n = 1 (respectively, n = 2) then the homomorphism p. and the fibration p admit (respectively, do not admit) a section. If n = 3, then p. and p admit a section if and only if m equivalent to 0, 2 (mod 3). If n >= 4, we show that if p(*) and p admit a section then m equivalent to epsilon(1)(n-1)(n-2) - epsilon(2)n(n-2) (mod n(n-1)(n-2)), where epsilon(1), epsilon(2) is an element of {0, 1}. Finally, we show that B-n(S-2) is generated by two of its torsion elements.