Homological Stability for Spaces of Subsurfaces with Tangential Structure

Thorben Kastenholz

Published 2018-12-13Version 1

Given a closed, simply connected and at least $5$-dimensional manifold $M$ together with some $p\in M$ and a two-plane $A$ in $T_pM$, one can consider the space of submanifolds of $M$ that are diffeomorphic to a surface of genus $g$ and that meet $p$ tangential to $A$. We introduce a notion of tangential structure for these subsurfaces and construct a stabilization map for these spaces of subsurfaces which increases the genus by $1$. Then we proceed to prove homological stability for these stabilization maps. As an application of this general result we prove homological stability for spaces of symplectic subsurfaces.