On Cohen braids

For a general connected surface M and an arbitrary braid α from the surface braid group Bn−1(M), we study the system of equations d1β = … = dnβ = α, where the operation di is the removal of the ith strand. We prove that for M ≠ S2 and M ≠ ℝP2, this system of equations has a solution β ∈ Bn(M) if and only if d1α = … = dn−1α. 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.