The braid groups of the projective plane

Gonçalves, Daciberg Lima; Guaschi, John

doi:10.2140/agt.2004.4.757

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The braid groups of the projective plane

Daciberg Lima Gonçalves and John Guaschi

Algebraic & Geometric Topology 4 (2004) 757–780

DOI: 10.2140/agt.2004.4.757

arXiv: math.AT/0409350

Abstract

Let $B_{n} (ℝ P^{2})$ (respectively $P_{n} (ℝ P^{2})$ ) denote the braid group (respectively pure braid group) on $n$ strings of the real projective plane $ℝ P^{2}$ . In this paper we study these braid groups, in particular the associated pure braid group short exact sequence of Fadell and Neuwirth, their torsion elements and the roots of the ‘full twist’ braid. Our main results may be summarised as follows: first, the pure braid group short exact sequence

1 \to P_{m - n} (ℝ P^{2} ∖ {x_{1}, \dots, x_{n}}) \to P_{m} (ℝ P^{2}) \to P_{n} (ℝ P^{2}) \to 1

does not split if $m \geq 4$ and $n = 2, 3$ . Now let $n \geq 2$ . Then in $B_{n} (ℝ P^{2})$ , there is a $k$ –torsion element if and only if $k$ divides either $4 n$ or $4 (n - 1)$ . Finally, the full twist braid has a $k^{th}$ root if and only if $k$ divides either $2 n$ or $2 (n - 1)$ .

Keywords

braid group, configuration space, torsion, Fadell–Neuwirth short exact sequence

Mathematical Subject Classification 2000

Primary: 20F36, 55R80

Secondary: 55Q52, 20F05

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Publication

Received: 11 December 2003

Accepted: 23 August 2004

Published: 11 September 2004

Authors

Daciberg Lima Gonçalves
Departamento de Matemática – IME-USP Caixa Postal 66281 – Ag. Cidade de São Paulo 05311-970 São Paulo SP Brazil

John Guaschi
Laboratoire de Mathématiques Emile Picard UMR CNRS 5580 UFR-MIG Université Toulouse III 118, Route de Narbonne 31062 Toulouse Cedex 4 France

Volume 4, issue 2 (2004)