  We study bosons interacting with an abelian Chern-Simons field on Riemann
surfaces of genus $g>0$. It is shown that a singular gauge transformation
brings the hamiltonian to free form. The transformed wave functions furnish a
multi-component representation of the braid group studied by Imbo and
March-Russell. The construction constitutes a proof of the equivalence of
bosons coupled to a Chern-Simons field and anyons and generalizes the well
known equivalence of the two pictures on the plane.
