The braid groups of the projective plane

Gonçalves, Daciberg Lima¹ ; Guaschi, John²

¹Departamento de Matemática – IME-USP, Caixa Postal 66281 – Ag. Cidade de São Paulo, 05311-970 São Paulo SP, Brazil
²Laboratoire de Mathématiques Emile Picard, UMR CNRS 5580 UFR-MIG, Université Toulouse III, 118, Route de Narbonne, 31062 Toulouse Cedex 4, France

Algebraic and Geometric Topology, Tome 4 (2004) no. 2, pp. 757-780

Résumé

Let Bn(ℝP2) (respectively Pn(ℝP2)) denote the braid group (respectively pure braid group) on n strings of the real projective plane ℝP2. 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

does not split if m ≥ 4 and n = 2,3. Now let n ≥ 2. Then in Bn(ℝP2), 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).

Détail

DOI : 10.2140/agt.2004.4.757

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

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