Algebraic and Geometric Topology 4 (2004), paper no. 33, pages 757-780.

The braid groups of the projective plane

Daciberg Lima Goncalves and John Guaschi

Abstract. Let B_n(RP^2)$ (respectively P_n(RP^2)) denote the braid group (respectively pure braid group) on n strings of the real projective plane RP^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 --> P_{m-n}(RP^2 - (x_1,...,x_n)) --> P_m(RP^2) --> P_n(RP^2) --> 1
does not split if m > 3 and n=2,3. Now let n > 1. Then in B_n(RP^2), 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 k-th root if and only if k divides either 2n or 2(n-1).

Keywords. Braid group, configuration space, torsion

AMS subject classification. Primary: Primary: 20F36, 55R80. Secondary: Secondary: 55Q52, 20F05.

DOI: 10.2140/agt.2004.4.757

E-print: arXiv:math.AT/0409350

Submitted: 11 December 2003. Accepted: 23 August 2004. Published: 11 September 2004.

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Daciberg Lima Goncalves and John Guaschi
Departamento de Matematica - IME-USP
Caixa Postal 66281 - Ag. Cidade de Sao Paulo
CEP: 05311-970 - Sao Paulo - SP - Brasil
Laboratoire de Mathematiques Emile Picard, UMR CNRS 5580 UFR-MIG
Universite Toulouse III, 118, route de Narbonne
31062 Toulouse Cedex 4, France


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