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Donaldson S. K., Kronheimer P.B. - The geometry of four-manifolds

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Название: The geometry of four-manifolds
Автор: Donaldson S. K., Kronheimer P.B.
Категория: Математика
Тип: Книга
Дата: 05.01.2009 13:14:29
Скачано: 76
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Описание: This book grew out of two lecture courses given by the first author in Oxford in 1985 and 1986. These dealt with the applications of Yang-Mills theory to 4-manifold topology, which, beginning in 1982, have grown to occupy an important place in current research. The content of the lectures was governed by two main aims, and although the treatment of the material has been expanded considerably in the intervening years, some of the resulting structure is preserved in the present work. The primary aim is to give a self-contained and comprehensive treatment of these new techniques as they have been applied to the study of 4-manifolds. The second aim is to bring together some of the developments in Yang-Mills theory itself, placed in the framework of contemporary differential and algebraic geometry. Leaving aside the topological applications, ideas from Yang-Mills theory—developed by many mathematicians since the late 1970's—have played a large part in fixing the direction of modern research in geometry. We have tried to present some of these ideas at a level which bridges the gap between general text books and research papers. These two aims are reflected in the organization of the book. The first provides the main thread of the material and begins in Chapter I with the mysteries of 4-manifold topology—problems which have been well-known in that field for a quarter of a century. It finishes in the last chapters, when some of these problems are, in part, resolved. On the way to this goal we make a number of detours, each with the purpose of expounding a particular area of interest. Some are only tangentially related, but none are irrelevant to our principal topic. It may help the reader to signpost here the main digressions. The first is in Chapter 3, which deals for the most part with the description of instanton solutions on the 4-sphere; some of the facts which emerge are an ingredient in later arguments (in Chapters 7 and 8, for example) and serve as a model for more general results, but their derivation is essentially independent from the rest of the book. Chapter 6 is concerned with the proof of a key theorem which provides a route from differential to algebraic geometry. This result underpins calculations in Chapters 9 and 10, but it could be taken on trust by some readers. In Chapter 7, only the last section is central to the subject matter of the book, and the main topological results can be obtained without the rather lengthy analysis which it contains. The reader who wants only to discover how Yang-Mills theory has been applied to 4-manifold topology might want to read only Chapter 1, the first part of Chapter 2, and Chapters 4, 5, 8, and 9. The ten chapters are each reasonably self-contained and could, to a large extent, be read as individual articles on different topics. In general we have tried to avoid duplicating material which is readily available elsewhere.
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