Homological stability for spaces of embedded surfaces

Abstract

© 2017, Mathematical Sciences Publishers. All rights reserved.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.