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

Abstract

We study the pure braid groups $P_n({\mathbb{R}}P^2)$ of the real projective plane ${\mathbb{R}}P^2$ , and in particular the possible splitting of the Fadell–Neuwirth short exact sequence $1 \to P_m({\mathbb{R}}P^2 \setminus \{x_1,\ldots, x_n\}) \hookrightarrow P_{n+m}({\mathbb{R}}P^2) \stackrel {p_{\ast}}{\to} P_n({\mathbb{R}}P^2) \to 1$ , where n  ≥  2 and m  ≥  1, and p _* is the homomorphism which corresponds geometrically to forgetting the last m strings. This problem is equivalent to that of the existence of a section for the associated fibration $p : F_{n+m}({\mathbb{R}}P^2) \to F_n({\mathbb{R}}P^2)$ of configuration spaces. Van Buskirk proved (1966, Trans. Am. Math. Soc., 122:81–97) 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 $P_n({\mathbb{R}}P^2)$ : this appears to be the first time that such a presentation has been given in the literature.