Originating with Andreas Floer in the 1980s, Floer homology provides an invariant of three-dimensional manifolds and four-dimensional cobordisms between them. It has proved to be an effective tool in tackling many important problems in three- and four-dimensional geometry and topology. This book provides a comprehensive treatment of Floer homology, based on the Seiberg-Witten equations. After first providing an overview of the results, the authors develop the analytic properties of the Seiberg-Witten equations, assuming only a basic grounding in differential geometry and analysis. The Floer groups of a general three-manifold are then defined, and their properties studied in detail. Two final chapters are devoted to the calculation of Floer groups, and to applications of the theory in topology. Suitable for beginning graduate students and researchers in the field, this book provides the first full discussion of a central part of the study of the topology of manifolds since the mid 1990s.

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