Internal and external stabilization of exotic surfaces in 4-manifolds

TL;DR

This paper demonstrates that many exotic surfaces in 4-manifolds become smoothly isotopic after stabilization with specific 4-manifolds, revealing a connection between internal and external stabilization methods.

Contribution

It introduces a relation between internal and external stabilization of exotic surfaces and applies it to various constructions, including rim-surgery and nullhomologous surfaces.

Findings

Exotic pairs become isotopic after stabilization with S^2×S^2 or CP^2#−CP^2.

Identification of stabilizing manifolds depends on construction choices.

Certain 2-links are shown to be Brunnian under specific conditions.

Abstract

We show that many explicit examples of exotic pairs of surfaces in a smooth 4-manifold become smoothly isotopic after one external stabilization with $S^2\times S^2$ or $CP^2\#\overline{CP^2}$. Our results cover surfaces produced by rim-surgery, twist-rim-surgery, annulus rim-surgery, as well as infinite families of nullhomologus surfaces and examples with non-cyclic fundamental group of the complement. A special attention is given to the identification of the stabilizing manifold and its dependence on the choices in the construction of the surface. The main idea of this note is given by relating internal and external stabilization, and most of the results, but not all, are proved using this relation. Moreover, we show that the 2-links in the exotic family from the recent work of Bais, Benyahia, Malech and Torres are brunnian, under some additional assumptions about the construction.