The braid groups of the projective plane

Abstract

Let $B_n\left(ℝP^2\right)$ (respectively $P_n\left(ℝP^2\right)$) 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

$$1\to P_{m-n}\left(ℝP^2\setminus \left\{x_1,\dots ,x_n\right\}\right)\to P_m\left(ℝP^2\right)\to P_n\left(ℝP^2\right)\to 1$$

does not split if $m\ge 4$ and $n=2,3$. Now let $n\ge 2$. Then in $B_n\left(ℝP^2\right)$, there is a $k$–torsion element if and only if $k$ divides either $4n$ or $4\left(n-1\right)$. Finally, the full twist braid has a $k^{th}$ root if and only if $k$ divides either $2n$ or $2\left(n-1\right)$.