Homological Stability for Spaces of Subsurfaces with Tangential Structure

Abstract

Given a manifold with boundary, one can consider the space of subsurfaces of this manifold meeting the boundary in a prescribed fashion. It is known that these spaces of subsurfaces satisfy homological stability if the manifold has at least dimension five and is simply-connected. We introduce a notion of tangential structure for subsurfaces and give a general criterion for when the space of subsurfaces with tangential structure satisfies homological stability provided that the manifold is simply-connected and has dimension $n\geq 5$. Examples of tangential structures such that the spaces of subsurface with that tangential structure satisfy homological stability are framings or spin structures of their tangent bundle, or $k$-frames of the normal bundle provided that $k\leq n-2$. Furthermore we introduce spaces of pointedly embedded subsurfaces and construct stabilization maps, as well as prove homological stability for these. This is used to prove homological stability for spaces of symplectic subsurfaces.