On Cohen braids

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