#### 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.

Notes on file formats
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

and

Laboratoire de Mathematiques Emile Picard, UMR CNRS
5580 UFR-MIG

Universite Toulouse III, 118, route de Narbonne

31062 Toulouse Cedex 4, France

Email: dlgoncal@ime.usp.br, guaschi@picard.ups-tlse.fr

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