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-dimensional submanifolds to 2-dimensional submanifolds.