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

TL;DR

This paper studies the structure of pure braid groups on non-orientable surfaces, proving that certain algebraic sequences split only in trivial cases, thereby completing the understanding of their splitting properties.

Contribution

It proves that the Fadell-Neuwirth short exact sequence splits only when n=1 for non-orientable surfaces of genus ≥ 3, resolving a key question in the theory of braid groups.

Findings

The sequence splits only when n=1.

The result applies to non-orientable surfaces of genus ≥ 3.

Completes the classification of splitting cases for pure braid groups.

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.