The braid groups of the projective plane and the Fadell-Neuwirth short exact sequence. (English) Zbl 1237.20028
Summary: We study the pure braid groups \(P_n(\mathbb RP^2)\) of the real projective plane \(\mathbb RP^2\), and in particular the possible splitting of the Fadell-Neuwirth short exact sequence \(1\to P_m(\mathbb RP^2\setminus\{x_1,\dots,x_n\})\hookrightarrow P_{n+m}(\mathbb RP^2)@>p_*>>P_n(\mathbb RP^2)\to 1\), where \(n\geq 2\) and \(m\geq 1\), 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 \(p\colon F_{n+m}(\mathbb RP^2)\to F_n(\mathbb RP^2)\) of configuration spaces.
J. van Buskirk proved [Trans. Am. Math. Soc. 122, 81-97 (1966; Zbl 0138.19103)] 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 \(n\geq 3\). 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 \(P_n(\mathbb RP^2)\): this appears to be the first time that such a presentation has been given in the literature.
J. van Buskirk proved [Trans. Am. Math. Soc. 122, 81-97 (1966; Zbl 0138.19103)] 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 \(n\geq 3\). 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 \(P_n(\mathbb RP^2)\): this appears to be the first time that such a presentation has been given in the literature.
MSC:
| 20F36 | Braid groups; Artin groups |
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
| 20F05 | Generators, relations, and presentations of groups |
| 55R80 | Discriminantal varieties and configuration spaces in algebraic topology |
Keywords:
pure braid groups; surface braid groups; projective plane braid groups; configuration spaces; presentations; Fadell-Neuwirth short exact sequencesCitations:
Zbl 0138.19103References:
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