Homological stability for spaces of embedded surfaces

We study the space of oriented genus g subsurfaces of a fixed manifold M, and in particular its homological properties. We construct a "scanning map" which compares this space to the space of sections of a certain fibre bundle over M associated to its tangent bundle, and show that this map induces an isomorphism on homology in a range of degrees. Our results are analogous to McDuff's theorem on configuration spaces, extended from 0-manifolds to 2-manifolds.