Seiberg-Witten Theory and the Topology of Elliptic surfaces

Abstract

The Seiberg-Witten invariant provides dramatic information about smooth manifolds-in particular it allows for the possibility of distinguishing different smooth structures on the same topological manifold. In this paper, we will outline the construction of infinitely many distinct smooth structures on one particularly simple 4-manifold: the K3 surface, and show how to use the Seiberg-Witten invariant to distinguish them. To do this, we will need two essentially different sets of tools: gauge theory on the one hand, and Kirby calculus and general techniques of 4-manifold topology on the other. We will develop both sets of techniques, and then show how, using the rational blowdown procedure discovered by Fintushel and Stern in [5], they may be combined to produce striking results. All the material covered is standard, and may be found in many references, such as [2], [8] or [17].