Braid groups of non-orientable surfaces and the Fadell-Neuwirth short exact sequence

Abstract

Let M be a compact, connected non-orientable surface without boundary and of genus g greater than or equal to 3. We investigate the pure braid groups P_n(M) of M, and in particular the possible splitting of the Fadell-Neuwirth short exact sequence 1 --> P_m(M {x_1,...,x_n}) --> P_{n+m}(M) --> P_n(M) --> 1, where m,n are positive integers, and the homomorphism p*:P_{n+m}(M) --> P_n(M) corresponds geometrically to forgetting the last m strings. This problem is equivalent to that of the existence of a section for the associated fibration p:F_{n+m}(M)} --> F_n(M) of configuration spaces, defined by p((x_1,...,x_n,..., x_{n+m}))= (x_1, ..., x_n). We show that p and p* admit a section if and only if n=1. Together with previous results, this completes the resolution of the splitting problem for surfaces pure braid groups.