Exotic Surfaces in 4-manifolds and Surface Corks

Abstract

A fundamental result in 4-manifold topology asserts that any two exotic smooth structures on a simply-connected, closed 4-manifold differ by a cork twist: the operation of removing a compact, contractible, codimension-zero submanifold and regluing it by a diffeomorphism of its boundary. In this paper, we introduce the notion of a surface cork, an analogous object in the setting of smoothly embedded, closed surfaces $F$ in closed 4-manifolds $X$. This is a compact, contractible, codimension-zero submanifold intersecting $F$ in a controllable manner, whose removal and regluing via a diffeomorphism of its boundary changes the diffeomorphism type of $(X, F)$ as a pair while leaving its homeomorphism type unchanged. The way in which the surface $F$ interacts with the codimension-zero submanifold leads us to define three distinct notions of surface corks: enclosing surface corks, exterior surface corks, and transverse surface corks. We establish the existence of exterior surface corks for certain previously known examples of exotic pairs. Furthermore, we give the first explicit construction of a transverse surface cork for certain exotic families arising from Fintushel--Stern rim surgery. Notably, this transverse surface cork turns out to be diffeomorphic to a 4-ball.