(i) Topology of embedded surfaces. Let X be a smooth, simply-connected 4-manifold, and ξ a 2-dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly

Abstract not found

Donaldson theory and more recently Seiberg-Witten theory have led to dramatic breakthroughs in the study of smooth 4-manifolds, and in particular of algebraic surfaces and their generalizations, symplectic 4-manifolds. In this paper, we shall survey some of the main results. Many expositions of Seiberg-Witten theory have

. We examine a family of finite energy SO(3) Yang-Mills connections over S 4 , indexed by two real parameters. This family includes both smooth connections (when both parameters are odd integers), and connections with a holonomy singularity around 1 or 2 copies of RP 2 . These singular YM connections interpolate between the smooth solutions. Depending on the parameters, the curvature may be self-dual, anti-self-dual, or neither. For the (anti)self-dual connections, we compute the formal dimension of the moduli space. For the non-self-dual connections we examine the second variation of the Yang-Mills functional, and count the negative and zero eigenvalues. Each component of the non-self-dual moduli space appears to consist only of conformal copies of a single solution. 1. Introduction and Statement of Results 1.1 Main Results Until recently, the phrase "Yang-Mills theory in four dimensions" essentially meant the study of smooth solutions to the (anti)self-duality equations F = \S...