On Cohen Braids

For a general connected surface M and an arbitrary braid a from the surface braid group Bn-1(M), we study the system of equations d(1)beta = ... = d(n)beta = alpha, where the operation d(i) is the removal of the ith strand. We prove that for M not equal S-2 and M not equal P-2, this system of equations has a solution beta is an element of B-n (M) if and only if d(1)alpha = ... = 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.