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Monopoles and three-manifolds. Paperback reprint of the 1st 2007 hardback edition. (English) Zbl 1247.57002

New Mathematical Monographs 10. Cambridge: Cambridge University Press (ISBN 978-0-521-18476-2/pbk). xii, 796 p. (2011).
This is a paperback reprint of [“Monopoles and three-manifolds”, New Mathematical Monographs 10. Cambridge: Cambridge University Press (2007; Zbl 1158.57002)].
This book is the definitive bible for anyone wanting to learn the full story of the various Seiberg-Witten Floer homology theories. The first eighty pages of the book provide an exceptional outline and sketch of the whole story. The details appear in the subsequent chapters. The analysis, the differential geometry, topology, group theory, algebraic geometry, \(K\)-theory, operator theory – it is all here, presented completely and in a most elegant way. Originating with Andreas Floer in the 1980s, Floer homology has proved to be an effective tool in tackling many important problems in 3- and 4-dimensional topology and geometry. This book provides a comprehensive treatment of Floer homology, based on the Seiberg-Witten monopole equations, assuming only a basic grounding in differential geometry and analysis. The Floer homology groups of a general 3-manifold are then defined, and their properties studied in detail. This is the main result of this book. Two final chapters of the book are devoted to the calculation of Floer homology groups, and to applications of the theory in topology.
Gauge theory and related areas of geometry have been an important tool for the study of 4-dimensional manifolds since the early 1980s, when Donaldson introduced ideas from Yang-Mills theory to solve long-standing problems in topology. In dimension 3, the same techniques formed the basis of Floer’s construction of his “instanton homology” groups of 3-manifolds. Today, Floer homology is an active area, and there are several varieties of Floer homology theory, all with closely related structures. While Floer’s construction used the anti-self-dual Yang-Mills (or instanton) equations, the theory presented in the present book is based instead on the Seiberg-Witten equations (or monopole equations).
The authors propose in this book a secure foundation for the study of the Seiberg-Witten equations on a general 3-manifold, and for the construction of the associated Floer homology groups. There are mathematics books that are classics. As such, they will never go out of date and never be improved. The present masterpiece is almost surely such a book.

MSC:

57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57R57 Applications of global analysis to structures on manifolds
57R58 Floer homology

Citations:

Zbl 1158.57002
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