## The Geometry of Four-manifoldsThis book provides the first lucid and accessible account to the modern study of the geometry of four-manifolds. It has become required reading for postgraduates and research workers whose research touches on this topic. Pre-requisites are a firm grounding in differential topology, andgeometry as may be gained from the first year of a graduate course. The subject matter of this book is the most significant breakthrough in mathematics of the last fifty years, and Professor Donaldson won a Fields medal for his work in the area. The authors start from the standpoint that thefundamental group and intersection form of a four-manifold provides information about its homology and characteristic classes, but little of its differential topology. It turns out that the classification up to diffeomorphism of four-manifolds is very different from the classification of unimodularforms and that the study of this question leads naturally to the new Donaldson invariants of four-manifolds. A central theme of this book is that the appropriate geometrical tools for investigating these questions come from mathematical physics: the Yang-Mills theory and anti-self dual connectionsover four-manifolds. One of the many consquences of this theory is that 'exotic' smooth manifolds exist which are homeomorphic but not diffeomorphic to (4, and that large classes of forms cannot be realized as intersection forms whereas distinct manifolds may share the same form. These result havehad far-reaching consequences in algebraic geometry, topology, and mathematical physics, and will continue to be a mainspring of mathematical research for years to come. |

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### Contents

FOURMANIFOLDS | 1 |

CONNECTIONS | 31 |

THE FOURIER TRANSFORM AND ADHM | 75 |

YANGMILLS MODULI SPACES | 126 |

TOPOLOGY AND CONNECTIONS | 172 |

STABLE HOLOMORPHIC BUNDLES OVER | 208 |

EXCISION AND GLUING | 263 |

NONEXISTENCE RESULTS | 317 |

INVARIANTS OF SMOOTH FOURMANIFOLDS | 341 |

THE DIFFERENTIAL TOPOLOGY | 375 |

Notes | 417 |

427 | |

437 | |

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### Common terms and phrases

2-forms ADHM adjoint argument ASD connections ASD equations ball Banach Chapter Chern class choose cohomology class compact compactification complex structure condition connected sum connection matrix consider const construction converge corresponding covariant derivative curvature defined definition denote determinant line bundle diffeomorphism differential dimension Dirac operator elliptic elliptic operator embedding equivalence classes example extends family of connections fibre fixed follows four-manifold Fredholm function gauge equivalent gauge transformations geometry given gives holomorphic holomorphic bundles holomorphic structure homology homotopy inequality integral intersection form invariant isomorphism K3 surface Kahler kernel L2 norm Lemma line bundle linear manifold metric moduli space neighbourhood non-trivial non-zero obtain open set orbit orientation parameter polynomial proof Proposition quotient represented restriction result Riemannian satisfies sequence simply connected smooth Sobolev solution stable submanifold subset subspace suppose surface surjective theorem theory topological transverse unitary connection vanishing vector bundle Yang-Mills zero set