The Geometry of Four-Manifolds (Oxford Mathematical Monographs)

The last ten years have seen rapid advances in the understanding of differentiable four-manifolds, not least of which has been the discovery of new 'exotic' manifolds. These results have had far-reaching consequences in geometry, topology, and mathematical physics, and have proven to be a mainspring of current mathematical research. This book provides a lucid and accessible account of the modern study of the geometry of four-manifolds. Consequently, it will be required reading for all those mathematicians and theoretical physicists whose research touches on this topic. The authors present both a thorough treatment of the main lines of these developments in four-manifold topology—notably the definition of new invariants of four-manifolds—and also a wide-ranging treatment of relevant topics from geometry and global analysis. All of the main theorems about Yang-Mills instantons on four-manifolds are proven in detail. On the geometric side, the book contains a new proof of the classification of instantons on the four-sphere, together with an extensive discussion of the differential geometry of holomorphic vector bundles. At the end of the book the different strands of the theory are brought together in the proofs of results which settle long-standing problems in four-manifolds topology and which are close to the frontiers of current research. Co-author Donaldson is the 1994 co-recipient of the prestigious Crafoord Prize. This text provides the first lucid and accessible account of 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. Prerequisites are a firm grounding in differential topology and geometry as may be gained from the first year of a graduate course. The subject matter of this book is one of the most significant breakthroughs in mathematics of the last fifty years, and Professor Donaldson won a Fields medal for his work in the area Provides an accessible account to the modern study of the geometry of four-manifolds. This title is suitable for postgraduates and research workers. The central theme is that the appropriate geometrical tools for investigating these questions come from mathematical physics: the Yang-Mills theory and anti-self dual connections over four-manifolds. This text provides an accessible account to the modern study of the geometry of four-manifolds. Prerequisites are a firm grounding in differential topology and geometry, as may be gained from the first year of a graduate course