The Geometry of Four-Manifolds

by S. K. Donaldson, P. B. Kronheimer

Overview

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.

Product Details

ISBN-13:	9780198502692
Publisher:	Oxford University Press
Publication date:	12/04/1997
Series:	Oxford Mathematical Monographs
Edition description:	Reprint
Pages:	456
Product dimensions:	9.22(w) x 6.40(h) x 0.94(d)

About the Author

Oxford University

Oxford University

Table of Contents

1. Four-Manifolds2. Connections3. The Fourier Transform and ADHM Construction4. Yang-Mills Moduli Spaces5. Topology and Connections6. Stable Holomorphic Bundles over Kähler Surfaces7. Excision and Gluing8. Non-Existence Results9. Invariants of Smooth Four-Manifolds10. The Differential Topology of Algebraic Surfaces

Editorial Reviews

"The idea that Yang-Mills theory (physics) might serve to illuminate deep problems in the geometry of four-manifolds (topology) has served since about 1984 to energize the work of mathematicians and physicists alike, and has stimulated a good deal of excited interaction between specialists who had appeared for awhile to have less and less to say to each other. The idea is due in large part to the principal author of this monograph. He writes here in fairly leisurely depth about the richly ramified mathematics of the situation, and for the benefit primarily of mathematicians. The beautifully produced book will, however, be of deep interest to sufficiently well-prepared physicists." —SciTech Book News

"Successfully gives a self-contained and comprehensive treatment. . . .A model at filling the gap between general textbooks and research papers. . . .If I were to be stranded on an island, wanted to understand the beauty and intricacies of the modern study of smooth 4-manifolds, and were allowed to take only one reference, the book under review would most certainly be my choice." —Choice

"A useful addition to physics libraries." —Physics Today

From the Publisher

The idea that Yang-Mills theory (physics) might serve to illuminate deep problems in the geometry of four-manifolds (topology) has served since about 1984 to energize the work of mathematicians and physicists alike, and has stimulated a good deal of excited interaction between specialists who had appeared for awhile to have less and less to say to each other. The idea is due in large part to the principal author of this monograph. He writes here in fairly leisurely depth about the richly ramified mathematics of the situation, and for the benefit primarily of mathematicians. The beautifully produced book will, however, be of deep interest to sufficiently well-prepared physicists. (NW) Annotation c. Book News, Inc., Portland, OR (booknews.com)

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