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

Abstract

We study the pure braid groups $P_n(R{P}^2)$ of the real projective plane $R{P}^2$, and in particular the possible splitting of the Fadell-Neuwirth short exact sequence $1\to P_m(R{P}^2{x}_1,...,x_n\to P_{n+m}(R{P}^2)\stackrel{p_{\ast }}{\to }{P}_n(R{P}^2)\to 1$, where $n\ge 2$ and $m\ge 1$, and $p_{\ast }$ 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 $p:F_{n+m}(R{P}^2)\to F_n(R{P}^2)$ of configuration spaces. Van Buskirk proved in 1966 that $p$ and $p_{\ast }$ admit a section if $n=2$ and $m=1$. Our main result in this paper is to prove that there is no section if $n\ge 3$. 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 $P_n(R{P}^2)$: this appears to be the first time that such a presentation has been given in the literature.