The theory of gauge fields in four dimensions.

*(English)*Zbl 0597.53001
Regional Conference Series in Mathematics 58. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-0708-0). vii, 101 p. (1985).

This book contains a proof of the celebrated theorem of S. Donaldson on smooth 4-manifolds with positive definite intersection form. In the introductory first chapter there is a brief description of some striking topological consequences of this result, most notably the existence of exotic differentiable structures on \({\mathbb{R}}^ 4\) which are obtained by combining it with the deep results of M. Freedman on the homeomorphism type of 4-manifolds.

The main body of this book is devoted to the proof of the main theorem and is mostly concerned with the differential-geometrical and analytical machinery needed to study the moduli space of Yang-Mills fields on 4- manifolds. A more detailed and also improved version of the analytical aspects of the proof of this theorem can be found in the book ”Instantons and 4-manifolds” (1984; Zbl 0559.57001) by D. Freed and K. Uhlenbeck. The author’s book however follows Donaldson’s original arguments more closely and perhaps has the advantage of a more causal style.

The main body of this book is devoted to the proof of the main theorem and is mostly concerned with the differential-geometrical and analytical machinery needed to study the moduli space of Yang-Mills fields on 4- manifolds. A more detailed and also improved version of the analytical aspects of the proof of this theorem can be found in the book ”Instantons and 4-manifolds” (1984; Zbl 0559.57001) by D. Freed and K. Uhlenbeck. The author’s book however follows Donaldson’s original arguments more closely and perhaps has the advantage of a more causal style.

Reviewer: M.Min-Oo

##### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53C05 | Connections, general theory |

57N13 | Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) |

57R55 | Differentiable structures in differential topology |

53C80 | Applications of global differential geometry to the sciences |