# Homological Stability for Spaces of Subsurfaces with Tangential Structure

## TL;DR

This paper establishes homological stability for spaces of subsurfaces with various tangential structures in high-dimensional simply-connected manifolds, including new stability results for symplectic subsurfaces.

## Contribution

It introduces a general criterion for homological stability of subsurface spaces with tangential structures and extends stability results to pointed embeddings and symplectic subsurfaces.

## Key findings

- Homological stability holds for subsurfaces with certain tangential structures in manifolds of dimension ≥5.
- Stability results are extended to spaces of pointedly embedded subsurfaces.
- Homological stability is proved for spaces of symplectic subsurfaces.

## 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.