Homological Stability for Spaces of Embedded Surfaces

Abstract

We study the space of oriented genus $\textit{g}$ subsurfaces of a fixed manifold $\textit{M}$, 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 $\textit{M}$ 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.