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2004 The braid groups of the projective plane
Daciberg Lima Gonçalves, John Guaschi
Algebr. Geom. Topol. 4(2): 757-780 (2004). DOI: 10.2140/agt.2004.4.757


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 P m n ( P 2 { x 1 , , x n } ) P m ( P 2 ) P n ( P 2 ) 1

does not split if m4 and n=2,3. Now let n2. Then in Bn(P2), there is a k–torsion element if and only if k divides either 4n or 4(n1). Finally, the full twist braid has a kth root if and only if k divides either 2n or 2(n1).


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Daciberg Lima Gonçalves. John Guaschi. "The braid groups of the projective plane." Algebr. Geom. Topol. 4 (2) 757 - 780, 2004.


Received: 11 December 2003; Accepted: 23 August 2004; Published: 2004
First available in Project Euclid: 21 December 2017

zbMATH: 1056.20024
MathSciNet: MR2100679
Digital Object Identifier: 10.2140/agt.2004.4.757

Primary: 20F36, 55R80
Secondary: 20F05, 55Q52

Rights: Copyright © 2004 Mathematical Sciences Publishers


Vol.4 • No. 2 • 2004
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