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.

References in corpus (2)

• The braid groups of the projective plane
• The quaternion group as a subgroup of the sphere braid groups

Cited by in corpus (7)

• Braid groups of non-orientable surfaces and the Fadell-Neuwirth short exact sequence
• Embeddings of the braid groups of covering spaces, classification of the finite subgroups of the braid groups of the real projective plane, and linearity of braid groups of low-genus surfaces
• The inclusion of configuration spaces of surfaces in Cartesian products, its induced homomorphism, and the virtual cohomological dimension of the braid groups of S^2 and RP^2
• The lower central and derived series of the braid groups of the projective plane
• On p-almost direct products and residual properties of pure braid groups of nonorientable surfaces
• Minimal generating and normally generating sets for the braid and mapping class groups of the disc, the sphere and the projective plane
• When the lower central series stops: a comprehensive study for braid groups and their relatives