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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.
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