For Exotic Surfaces with Boundary, One Stabilization is Not Enough

TL;DR

This paper demonstrates that for certain exotic surfaces in the four-ball, a single stabilization is insufficient to make them isotopic, highlighting limitations of stabilization techniques in 4-manifold topology.

Contribution

It provides the first examples of exotic disks in the four-ball with arbitrarily large stabilization distance, showing one stabilization is not always enough.

Findings

Exotic disks with arbitrarily large stabilization distance

Single stabilization does not always suffice for isotopy

Floer theoretic methods reveal new exotic behaviors

Abstract

A result of Baykur-Sunukjian states that homologous surfaces in a 4-manifold become isotopic after a finite number of internal stabilizations, i.e. attaching tubes to the surfaces. A natural question is how many stabilizations are needed before the surfaces become isotopic. In particular, given an exotic pair of surfaces, is a single stabilization always enough to make the pair smoothly isotopic? We answer this question by studying how the stabilization distance between surfaces with boundary changes with respect to satellite operations. Using a range of Floer theoretic techniques, we show that there are exotic disks in the four-ball which have arbitrarily large stabilization distance, giving the first examples of exotic behavior in the four-ball for which "one is not enough".