On Cohen braids

Abstract

For a general surface $M$ and an arbitrary braid $\alpha$ from the surface braid group $B_{n-1}(M)$ we study the system of equations $d_1\beta =\cdots =d_n\beta =\alpha ,$ where operation $d_i$ is deleting of $i$-th strand. We obtain that if $M\ne S^2$ or $\mathbb{R}{P}^2$ this system of equations has a solution $\beta \in B_n(M)$ if and only if $d_1\alpha =\dots =d_n\alpha .$ The set of braids satisfying the last system of equations we call Cohen braids. We also construct a set of generators for the groups of Cohen braids. In the cases of the sphere and the projective plane we give some examples for the small number of strands.

23 pages