The geometry of four-manifolds

著者

書誌事項

The geometry of four-manifolds

S.K. Donaldson and P.B. Kronheimer

（Oxford mathematical monographs）

Clarendon Press, 1990

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注記

Includes index

Bibliography: p. [427]-436

内容説明・目次

内容説明

The last 10 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 proved to be a mainspring of current mathematical research. This book provides an accessible account of the modern study of the geometry of four-manifolds. Consequently, it will form required reading for all those mathematicians and theoretical physicists whose research touches on this topic. Pre-requisites are a firm grounding in differential geometry and topology as might be gained from the first year of a graduate course. 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 proved 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 geometery 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.

目次

  • Four-manifolds
  • Connections
  • The Fourier transform and ADHM construction
  • Yang-Mills moduli spaces
  • Topology and connections
  • Stable holomorphic bundles over Kahler surfaces
  • Excision and gluing
  • Non-existence results
  • Invariants of smooth four-manifolds
  • The differential topology of algebraic surfaces.

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