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.F. Cantero Moran was funded through FPI Grant BES-2008-002642 and by Michael Weiss Humboldt professor grant. He was partially supported by project MTM2013-42178-P funded by the Spanish Ministry of Economy. O. Randal-Williams was supported by ERC Advanced Grant No. 228082, the Danish National Research Foundation through the Centre for Symmetry and Deformation, and the Herchel Smith Fund
