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The geometry of four-manifolds. (English) Zbl 0820.57002
Oxford Mathematical Monographs. Oxford: Clarendon Press. ix, 440 p. (1990).
This book presents the applications of Yang-Mills theory to four- dimensional differential topology. It is written by the creator of this impressive edifice of mathematical ideas and by another leading expert in the field. Compared to the original papers by the first author and Uhlenbeck, many proofs have been simplified. The authors have successfully endeavored to provide maximal insight and to write a book in which many people on different levels and with different intentions can read with profit. It is certainly not a book of the easy to read genre; there are much more rewarding ways to use it than to try to plough through it from cover to cover, word by word.
After Chapter 1, which is of introductory nature, Chapter 2 is dedicated to connections on bundles, the main result being Uhlenbeck’s theorem on the existence of Coulomb gauges. In Chapter 3, there is a study of the anti-self-dual connections over the 4-dimensional torus and over \(S^ 4\); in the latter case, the Atiyah-Drinfeld-Hitchin-Manin construction provides a very explicit description of moduli spaces in terms of matrix theory. Chapter 4 is dedicated to the local structure and the compactification of the moduli spaces. This requires Uhlenbeck’s theorem on the removability of point singularities for finite-energy anti-self- dual connections, for which there is a new, simpler proof. Chapter 5 treats the moduli spaces from a more global point of view; among other things, they are shown to be orientable. In Chapter 6, the authors present the relations between anti-self-dual connections and holomorphic vector bundles which are stable in the sense of algebraic geometry. Chapter 7 studies connections on connected sums of manifolds. It starts by presenting the excision principle for the index of linear elliptic operators. This is an important part of the proof of the Atiyah-Singer index theorem. Much of the material in this chapter was originally developed by Taubes.
The last three chapters, 8 to 10, are devoted to the applications to 4- manifolds: In Chapter 8, it is shown that many quadratic forms cannot appear as intersection forms of differentiable 4-manifolds. In Chapter 9, the so-called Donaldson invariants are introduced; already the easiest of them shows that the \(h\)-cobordism theorem is wrong in dimension 4. In Chapter 10, these invariants are used to demonstrate that there are complex surfaces which are homeomorphic but not diffeomorphic.
In the years after Donaldson’s first great discovery, the pace of the development in this area was breathtaking. It had somewhat slowed down by 1990, so that, when the book appeared, the general impression was that it would remain the authoritative source for the foundations of 4- dimensional differential topology for a considerable time to come.
But sometimes, unpredictable things do happen in mathematics: As most readers certainly know, the Seiberg-Witten invariants, introduced in 1994, have led to another revolution in the field. This new technique, which is somewhat similar to Yang-Mills theory in that it uses connections on bundles, differential equations, and motivations from theoretical physics, has provided easier proofs of many of Donaldson’s theorems and made new results accessible. Hence, before long, there will be the need for a new book on the geometry of four-manifolds. But of course this situation does not lessen the eminent quality of the present book.
I cannot do better than end off with a quotation from its Preface: “Although the techniques described here have had notable successes, it is at present not at all clear what their full scope is, nor how essential they are to the structure of 4-manifolds. Looking to the future, one might hope that quite new ideas will emerge which will both shed light on these points and also go further in revealing the nature of differential topology in four dimensions. In any case, we hope that this book will help the reader to appreciate the fascination of these fundamental problems in geometry and topology”.

57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
57R57 Applications of global analysis to structures on manifolds
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
58D27 Moduli problems for differential geometric structures
57R55 Differentiable structures in differential topology
58J10 Differential complexes