Six Lectures on Four 4-Manifolds

TL;DR

This paper reviews the current state of understanding and classification challenges of simply connected smooth and symplectic 4-manifolds, emphasizing surgical techniques and Seiberg-Witten invariants, and proposes a potential classification approach.

Contribution

It provides a comprehensive review of known results, techniques, and open problems in classifying simply connected 4-manifolds, and suggests a new classification scheme.

Findings

Effective surgical techniques alter smooth structures.

Seiberg-Witten invariants distinguish different structures.

Proposed classification scheme tested on examples.

Abstract

Despite spectacular advances in defining invariants for simply connected smooth and symplectic 4-dimensional manifolds and the discovery of effective surgical techniques, we still have been unable to classify simply connected smooth manifolds up to diffeomorphism. In these notes, adapted from six lectures given at the 2006 Park City Mathematics Institute Graduate Summer School on Low Dimensional Topology, we will review what we do and do not know about the existence and uniqueness of smooth and symplectic structures on closed, simply connected 4-manifolds. We will focus on those surgical techniques that have been effective in altering smooth and symplectic structures and the Seiberg-Witten invariants that are used to distinguish them. In the last lecture we will then pose a possible classification scheme and test it on a few examples.