The braid groups of the projective plane and the Fadell-Neuwirth short exact sequence

We study the pure braid groups Pn(RP) of the real projective plane RP2, and in particular the possible splitting of the Fadell-Neuwirth short exact sequence 1 −→ Pm(RP \ {x1, . . . , xn}) −֒→ Pn+m(RP 2) p∗ −→ Pn(RP) −→ 1, where n ≥ 2 and m ≥ 1, and p∗ is the homomorphismwhich 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 : Fn+m(RP) −→ Fn(RP) of configuration spaces. Van Buskirk proved in 1966 that p and p∗ admit a section if n = 2 and m = 1. Our main result in this paper is to prove that there is no section if n ≥ 3. As a corollary, it follows that n = 2 and m = 1 are the only values for which a section exists. As part of the proof, we derive a presentation of Pn(RP): this appears to be the first time that such a presentation has been given in the literature.