The Geometry of Four-Manifolds by S. K. DonaldsonThe Geometry of Four-Manifolds by S. K. Donaldson

The Geometry of Four-Manifolds

byS. K. Donaldson, P. B. Kronheimer

Paperback | July 1, 1997

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This book provides the first lucid and accessible account to 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. Pre-requisites are a firm grounding in differential topology, andgeometry as may be gained from the first year of a graduate course. The subject matter of this book is the most significant breakthrough in mathematics of the last fifty years, and Professor Donaldson won a Fields medal for his work in the area. The authors start from the standpoint that thefundamental group and intersection form of a four-manifold provides information about its homology and characteristic classes, but little of its differential topology. It turns out that the classification up to diffeomorphism of four-manifolds is very different from the classification of unimodularforms and that the study of this question leads naturally to the new Donaldson invariants of four-manifolds. A central theme of this book is that the appropriate geometrical tools for investigating these questions come from mathematical physics: the Yang-Mills theory and anti-self dual connectionsover four-manifolds. One of the many consquences of this theory is that 'exotic' smooth manifolds exist which are homeomorphic but not diffeomorphic to (4, and that large classes of forms cannot be realized as intersection forms whereas distinct manifolds may share the same form. These result havehad far-reaching consequences in algebraic geometry, topology, and mathematical physics, and will continue to be a mainspring of mathematical research for years to come.
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.
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Title:The Geometry of Four-ManifoldsFormat:PaperbackDimensions:450 pages, 9.21 × 6.14 × 0.98 inPublished:July 1, 1997Publisher:Oxford University Press

The following ISBNs are associated with this title:

ISBN - 10:0198502699

ISBN - 13:9780198502692

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Table of Contents

1. Four-manifolds2. Connections3. The Fourier transform and ADHM construction4. Yang-Mills moduli spaces5. Topology and connections6. Stable holomorphic bundles over Kahler surfaces7. Excision and glueing8. Non-existence results9. Invariants of smooth four-manifolds10. The differential topology of algebraic surfacesAppendixReferencesIndex

Editorial Reviews

`... authoritative and comprehensive... it must be regarded as compulsory reading for any young researcher approaching this difficult but fascinating area.' Bulletin of the London Mathematical Society