Brunnian braids on surfaces.
A Brunnian braid is a braid that becomes trivial after removing any one of its strands, and let \(\text{Brun}_n(M)\) be the subgroup of the \(n\)-strand Brunnian braids on a connected 2-manifold \(M\). In the paper under review, the authors investigate the relationships between \(\text{Brun}_n(M)\) and the symmetric commutator subgroup \(R_n(M)\) generated by normal closures of standard generators \(A_{i,n}\) of pure braid group \(P_n(D^2)\). That is, \(\text{Brun}_n(M)\) is isomorphic to \(R_n(M)\) when \(M\) is different from a sphere and real projective space; and it is a group extension of \(\pi_{n-1}(S^2)\) by \(R_n(M)\) otherwise. Moreover, they prove the finite presentability for \(P_n(M)/\text{Brun}_n(M)\) and \(B_n(M)/\text{Brun}_n(M)\) for \(n\geq3\).
scientific article; zbMATH DE number 6068608
scientific article