The Geometry of Four-Manifolds (Oxford Mathematical Monographs) [Paperback]

S. K. Donaldson (Author), P. B. Kronheimer (Author)

Book Description

Publication Date: December 4, 1997 | ISBN-10: 0198502699 | ISBN-13: 978-0198502692

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.

Editorial Reviews

Review

"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

About the Author

Professor S. K. Donaldson, Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford, OX1 3LB. Professor P. B. Kronheimer, Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA 02138-2901, USA. Professor Donaldson won the Fields Medal for his work in the area of four-manifolds.

Product Details

• Paperback: 456 pages
• Publisher: Oxford University Press, USA (December 4, 1997)
• Language: English
• ISBN-10: 0198502699
• ISBN-13: 978-0198502692
• Product Dimensions: 9.6 x 6.7 x 1 inches
• Shipping Weight: 1.3 pounds (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (1 customer review)
• Amazon Best Sellers Rank: #1,111,229 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

9 of 10 people found the following review helpful:

5.0 out of 5 stars An excellent summary of Donaldson theory, June 16, 2000

By 

A.X. Razza (London, England) - See all my reviews

This review is from: The Geometry of Four-Manifolds (Oxford Mathematical Monographs) (Paperback)

This book brings together the brilliant work Donaldson did at Oxford during the early 1980s. The unique properties of 4-manifolds are clearly and concisely written out with concentration on explaining field theories like Yang-Mills and gauge theory with a truly firm mathematical foundation, presented in a book for the first time. A great companion for any researcher in the field of geometry and topology, or even loop quantum gravity!