THE BRAID GROUP $B_{n,m}(\mathbb{S}^{2})$ AND A GENERALISATION OF THE FADELL–NEUWIRTH SHORT EXACT SEQUENCE.

Let m,n ∈ ℕ. We define $B_{n,m}(\mathbb{S}^{2})$ to be the set of (n+m)-braids of the sphere whose associated permutation lies in the subgroup Sn × Sm 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: \[ 1\to B_m(\mathbb{S}^{2}\setminus\{x_1,\ldots,x_n\})\to B_{n,m}(\mathbb{S}^{2})\stackrel{p_{\ast}}{\longrightarrow} B_n(\mathbb{S}^{2})\to 1, \] where ${p_{\ast}}{:}\, B_{n,m}(\mathbb{S}^{2}) \to B_n(\mathbb{S}^{2})$ is the group homomorphism (defined for all n ∈ 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 $p{:}\, D_{n,m}(\mathbb{S}^{2}) \to D_n(\mathbb{S}^{2})$ 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 ≡ 0,2 (mod 3). If n ≥ 4, we show that if p* and p admit a section then m ≡ ε1(n - 1)(n - 2) - ε2n(n - 2) (mod n(n - 1)(n - 2)), where ε1,ε2 ∈ {0,1}. Finally, we show that $B_n(\mathbb{S}^{2})$ is generated by two of its torsion elements. [ABSTRACT FROM AUTHOR]