Abstract

Donaldson’s Diagonalization Theorem states that if the intersection form of a closed oriented smooth 4-manifold X is definite, then it is diagonalizable over the integers. We present an outline of the original proof by Donaldson, which appeared in his celebrated 1983 paper.

The proof relies on some geometric features of the moduli space of anti-self-dual (ASD) connections on an SU(2)-principal bundle over X. More concretely, it provides an oriented cobordism between X and a disjoint union of complex projective spaces. This gives an estimate on the signature of X, which is the key step to the proof.

We will introduce the ASD moduli space and discuss some of its properties, such as dimension, its singularities and its compactification. If time permits, we will also discuss some of the consequences of the theorem in Freedman’s work on 4-dimensional topology, such as the existence of an exotic differentiable structure on the 4-dimensional Euclidean space.