Monopoles and Three-Manifolds

Peter Kronheimer; Tomasz Mrowka

doi:10.1017/CBO9780511543111

Originating with Andreas Floer in the 1980s, Floer homology has proved to be an effective tool in tackling many important problems in three- and four-dimensional geometry and topology. This 2007 book provides a comprehensive treatment of Floer homology, based on the Seiberg–Witten monopole 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, this book provides a full discussion of a central part of the study of the topology of manifolds.

'… there are mathematics books that are classics; these are books that tell a particular story in the right way. As such, they will never go out of date and never be bettered. Kronheimer and Mrowka's book is almost surely such a book. If you want to learn about Floer homology in the Seiberg–Witten context, you will do no better than to read Kronheimer and Mrowka's masterpiece Monopoles and Three-Manifolds.'

Clifford Henry Taubes Source: Bulletin of the American Mathematical Society

'This long-awaited book is a complete and detailed exposition of the Floer theory for Seiberg-Witten invariants. It is very nicely written and contains all proofs of results. This makes the book an essential tool for both researchers and students working in this area of mathematics.'

Source: Mathematical Reviews

‘This book is the definitive bible for anyone wanting to learn the full story of the various Seiberg-Witten Floer homology theories … 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 As such, they will never go out of date and never be improved.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.’

Alexander Felshtyn Source: Zentralblatt MATH

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Monopoles and Three-Manifolds

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Contents

Frontmatter
pp i-vi

Contents
pp vii-x

Preface
pp xi-xii

I - Outlines
pp 1-83

II - The Seiberg–Witten equations and compactness
pp 84-133

III - Hilbert manifolds and perturbations
pp 134-194

IV - Moduli spaces and transversality
pp 195-273

V - Compactness and gluing
pp 274-374

VI - Floer homology
pp 375-448

VII - Cobordisms and invariance
pp 449-589

VIII - Non-exact perturbations
pp 590-633

IX - Calculations
pp 634-720

X - Further developments
pp 721-778

References
pp 779-784

Glossary of notation
pp 785-791

Index
pp 792-796

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