## The Theory of Gauge Fields in Four DimensionsLawson's expository lectures, presented at a CBMS Regional Conference held in Santa Barbara in August 1983, provide an in-depth examination of the recent work of Simon Donaldson, and is of special interest to both geometric topologists and differential geometers. This work has excited particular interest, in light of Mike Freedman's recent profound results: the complete classification, in the simply connected case, of compact topological 4-manifolds. Arguing from deep results in gauge field theory, Donaldson has proved the nonexistence of differentiable structures on certain compact 4-manifolds. Together with Freedman's results, Donaldson's work implies the existence of exotic differentiable structures in $\mathbb R^4$-a wonderful example of the results of one mathematical discipline yielding startling consequences in another. The lectures are aimed at mature mathematicians with some training in both geometry and topology, but they do not assume any expert knowledge. In addition to a close examination of Donaldson's arguments, Lawson also presents, as background material, the foundation work in gauge theory (Uhlenbeck, Taubes, Atiyah, Hitchin, Singer, et al.) which underlies Donaldson's work. |

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### Contents

1 | |

Simplyconnected 4manifolds | 2 |

Differentiable 4manifolds | 4 |

Exotica | 5 |

An introduction to Donaldsons proof | 10 |

The Geometry of Connections | 17 |

Connections | 19 |

Riemannian connections | 20 |

The Selfdual YangMills Equations | 39 |

Selfduality | 40 |

The fundamental elliptic complex | 43 |

Solutions on S | 45 |

The Moduli Space | 47 |

Reducible selfdual connections | 50 |

Perturbations | 51 |

The orientability of the moduli space | 53 |

Spiconnections | 21 |

Change of connections | 22 |

Automorphisms the gauge group | 23 |

Sobolev completions | 25 |

Reductions | 28 |

The action of on QPE | 31 |

Equivalence classes of connections | 33 |

Fundamental Results of K Uhlenbeck | 59 |

The Taubes Existence Theorem | 71 |

Final Arguments | 85 |

The Sobolev Embedding Theorems | 91 |

99 | |

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### Common terms and phrases

2-forms 4-dimensional 4-manifold adjoint anti-self-dual argument ball C*-topology Chapter choose compact oriented 4-manifold compact simply-connected compact subset completes the proof consider constant converges Corollary corresponding curvature decomposition defined denote diffeomorphism Donaldson Donaldson's theorem element elliptic embedded equivalence classes estimate euclidean Euler class exists Furthermore gauge equivalence gauge group gauge transformation geodesic normal coordinates given gives Hence Hölder's inequality Hom(E homeomorphism inequality inner product instanton instanton number intersection form isomorphism L*-norm LEMMA manifold metric quaternion line moduli space neighborhood Note operator orthogonal orthonormal perturbation pointwise positive definite principal bundle proved QP(E quaternion line bundle real-valued Recall reducible connections riemannian 4-manifold riemannian connection satisfies scalar multiplication self-dual connections sequence simply-connected singular points Sobolev Sobolev embedding theorems solution Spi-connection sufficiently small symmetric bilinear forms t'Hooft connection tensor topological reduction trivial Uhlenbeck uniformly unique vector bundle Yang–Mills equations Yang–Mills functional