  We reconsider the problem of anyons on higher genus surfaces by embedding
them in three dimensional space. From a concrete realization based on three
dimensional flux tubes bound to charges moving on the surface, we explicitly
derive all the representations of the spinning braid group. The component
structure of the wave functions arises from winding the flux tubes around the
handles. We also argue that the anyons in our construction must fulfil the
generalized spin-statistics relation.
