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

TL;DR

This paper investigates the splitting of the Fadell-Neuwirth short exact sequence for pure braid groups on the real projective plane, proving non-existence of sections for most cases and providing a new presentation of these groups.

Contribution

It establishes the non-existence of sections for the sequence when n ≥ 3 and offers the first known presentation of the pure braid groups of RP^2.

Findings

No section exists for n ≥ 3 in the Fadell-Neuwirth sequence.

Sections exist only when n=2 and m=1.

Provides a novel presentation of P_n(RP^2).

Abstract

We study the pure braid groups $P_n(RP^2)$ of the real projective plane $RP^2$, and in particular the possible splitting of the Fadell-Neuwirth short exact sequence $1 \to P_m(RP^2 {x_1,...,x_n} \to P_{n+m}(RP^2) \stackrel{p_{\ast}}{\to} P_n(RP^2) \to 1$, where $n\geq 2$ and $m\geq 1$, 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 $p: F_{n+m}(RP^2) \to F_n(RP^2)$ 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 $n\geq 3$. 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 $P_n(RP^2)$: this appears to be the first time that such a presentation has been given in the literature.