The braid groups of the projective plane

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).

Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-33.abs.html

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• The lower central and derived series of the braid groups of the projective plane
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• Minimal generating and normally generating sets for the braid and mapping class groups of the disc, the sphere and the projective plane
• When the lower central series stops: a comprehensive study for braid groups and their relatives
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