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

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dc.creatorGonçalves, Daciberg Lima
dc.creatorGuaschi, John
dc.date2007-07-06
dc.date.accessioned2026-07-07T09:22:10Z
dc.date.available2026-07-07T09:22:10Z
dc.descriptionWe 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.
dc.identifierhttps://arxiv.org/abs/0707.0925
dc.identifierhttp://arxiv.org/abs/0707.0925
dc.identifierGeometriae Dedicata 130 (2007) 93-107
dc.identifierdoi:10.1007/s10711-007-9207-z
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/155283
dc.subjectGeometric Topology
dc.subjectGroup Theory
dc.subject20F36, 20F05, 55R80
dc.titleThe braid groups of the projective plane and the Fadell-Neuwirth short exact sequence
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