Contents Preface . vi i Chapter I. Introduction 1 1. Connected surface s 1 2. Simply-connected 4-manifold s 2 3. Differentiate 4-manifold s . . . 4 4. Exotica 5 5. An introduction t o Donaldson's proof 10 Chapter II. The Geometry of Connections 17 1. Quaternion lin e bundles 17 2. Connections 19 3. Riemannian connections 2 0 4. Spj-connections 2 1 5. Change of connection s 2 2 6. Automorphisms (the gauge group) 2 3 7. Sobolev completions 2 5 8. Reductions 2 8 9. The action of #o n Q P{SE) 3 1 10. Equivalence classes of connections 3 3 Chapter III. The Self-dual Yang-Mills Equations 3 9 1. The Yang-Mills functiona l 3 9 2. Self-duality 4 0 3. The fundamental ellipti c complex 4 4 4. Solutions on S4 4 5 Chapter IV. The Moduli Space 4 7 1. The space of self-dual connections 4 7 2. Kuranishi's method 4 7 3. Reducible self-dual connection s 5 0 4. Perturbations 5 1 5. The orientability of th e moduli space 5 3 v
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