Exotic Surfaces in 4-manifolds and Surface Corks

TL;DR

This paper introduces the concept of surface corks in 4-manifold topology, demonstrating their role in altering smooth structures of embedded surfaces without changing the underlying topological manifold.

Contribution

It defines surface corks as a new tool for understanding exotic surfaces in 4-manifolds and provides the first explicit example related to rim surgery.

Findings

Constructed the first example of a surface cork for certain exotic surfaces.

Surface corks can change the smooth structure of a surface in a 4-manifold.

The constructed surface cork is diffeomorphic to a 4-ball.

Abstract

A fundamental result in 4-manifold topology asserts that every exotic smooth structure on a simply connected closed 4-manifold is determined by a cork -- a codimension-zero compact, contractible submanifold together with a diffeomorphism on its boundary. In this paper, we introduce the notion of a surface cork, an analogous object for smoothly embedded surface $F$ in 4-manifold $X$. This is a compact, contractible codimension-zero submanifold intersecting the surface $F$ 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. We construct the first example of a surface cork for certain exotic families constructed from Fintushel and Stern's rim surgery. In particular, this surface cork turns out to be diffeomorphic to a 4-ball.