The braid groups of the projective plane

By Daciberg Lima Gonçalves, John Guaschi. 2004

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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 > 4 and n = 2, 3. Now let n > 2. 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 t h root if and only if k divides either 2n or 2(n-1).

View Paper: https://www.semanticscholar.org/paper/ff68ef73ede627acc7c92f33c9ee9f9d38013f46

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  1. John Guaschi
  2. Daciberg Lima Gonçalves

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