Knotted surfaces in 4-manifolds and stabilizations

TL;DR

This paper demonstrates that exotically knotted surfaces in 4-manifolds become smoothly isotopic after stabilization, extending classical results and showing that many exotic examples unify under a single stabilization.

Contribution

It proves that all exotically knotted surfaces in the same homology class become smoothly isotopic after stabilization, generalizing Wall's stable equivalence to knotted surfaces.

Findings

Surfaces become smoothly isotopic after stabilization in many cases.

Exotic knottings become equivalent after a single stabilization.

Stabilization can be performed in a standard, unknotted manner.

Abstract

In this paper, we study stable equivalence of exotically knotted surfaces in 4-manifolds, surfaces that are topologically isotopic but not smoothly isotopic. We prove that any pair of embedded surfaces in the same homology class become smoothly isotopic after stabilizing them by handle additions in the ambient 4-manifold, which can moreover assumed to be attached in a standard way (locally and unknottedly) in many favorable situations. In particular, any exotically knotted pair of surfaces with cyclic fundamental group complements become smoothly isotopic after a same number of standard stabilizations - analogous to C.T.C. Wall's celebrated result on the stable equivalence of simply-connected 4-manifolds. We moreover show that all constructions of exotic knottings of surfaces we are aware of, which display a good variety of techniques and ideas, produce surfaces that become smoothly isotopic after a single stabilization.