Homological stability for spaces of embedded surfaces

TL;DR

This paper investigates the homological properties of the space of embedded oriented genus g surfaces in a fixed manifold, introducing a scanning map that reveals isomorphisms in homology within a certain degree range.

Contribution

It extends McDuff's theorem from configuration spaces of points to spaces of embedded surfaces, providing new insights into their homological stability.

Findings

The scanning map induces homology isomorphisms in a specific degree range.

Homological stability results analogous to McDuff's theorem for 2-manifolds.

Connections established between embedded surface spaces and sections of fibre bundles.

Abstract

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.