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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
The braid groups of the projective plane

Daciberg Lima Gonçalves and John Guaschi

Algebraic & Geometric Topology 4 (2004) 757–780

arXiv: math.AT/0409350

Abstract

Let Bn(P2) (respectively Pn(P2)) denote the braid group (respectively pure braid group) on n strings of the real projective plane P2. 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 Pmn(P2 {x 1,,xn}) Pm(P2) P n(P2) 1

does not split if m 4 and n = 2,3. Now let n 2. Then in Bn(P2), there is a k–torsion element if and only if k divides either 4n or 4(n 1). Finally, the full twist braid has a kth root if and only if k divides either 2n 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