Surfaces in 4-Manifolds

TL;DR

This paper introduces rim surgery, a technique to modify smooth embeddings of surfaces in 4-manifolds without changing their topological type, advancing understanding of smooth structures in four-dimensional topology.

Contribution

The paper presents rim surgery, a novel method for altering smooth embeddings of surfaces in 4-manifolds while preserving their topological embedding.

Findings

Rim surgery can change smooth embeddings without affecting topological class.

The technique applies to surfaces of positive genus with nonnegative self-intersection.

It provides new tools for studying smooth structures in 4-manifolds.

Abstract

In this paper we introduce a technique, called rim surgery, which can change a smooth embedding of an orientable surface of positive genus and nonnegative self-intersection in a smooth 4-manifold while leaving the topological embedding unchanged.