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

Let MM be a compact, connected non-orientable surface without boundary and of genus g⩾3g⩾3. We investigate the pure braid groups Pn(M)Pn(M) of MM, and in particular the possible splitting of the Fadell–Neuwirth short exact sequence 1⟶Pm(M∖{x1,…,xn})↪Pn+m(M)⟶p∗Pn(M)⟶1, where m,n⩾1m,n⩾1, and p∗p∗ is the homomorphism which corresponds geometrically to forgetting the last mm strings. This problem is equivalent to that of the existence of a section for the associated fibration p:Fn+m(M)⟶Fn(M)p:Fn+m(M)⟶Fn(M) of configuration spaces, defined by p((x1,…,xn,xn+1,…,xn+m))=(x1,…,xn)p((x1,…,xn,xn+1,…,xn+m))=(x1,…,xn). We show that pp and p∗p∗ admit a section if and only if n=1n=1. Together with previous results, this completes the resolution of the splitting problem for surface pure braid groups.