Isotopy of surfaces in 4-manifolds after a single stabilization

TL;DR

This paper proves that homologous surfaces of the same genus in certain 4-manifolds become smoothly isotopic after a single stabilization, depending on their properties, expanding understanding of surface isotopy in 4-manifolds.

Contribution

It demonstrates that two homologous surfaces can be made isotopic after a single stabilization in specific 4-manifolds, distinguishing between ordinary and characteristic surfaces.

Findings

Surfaces become isotopic after stabilization in X # S^2×S^2 for ordinary surfaces.

Surfaces become isotopic after stabilization in X # non-trivial sphere bundle over S^2 for characteristic surfaces.

The result applies to surfaces with simply-connected complements in smooth 4-manifolds.

Abstract

Any two homologous surfaces of the same genus embedded in a smooth 4-manifold X with simply-connected complements are shown to be smoothly isotopic in the connected sum of X and the product of a 2-sphere with itself, if the surfaces are ordinary, and in the connected sum of X with the non-trivial sphere bundle over the sphere if they are characteristic.