Monopoles and Three-Manifolds

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Peter Kronheimer (Harvard University, Massachusetts) Tomasz Mrowka (Massachusetts Institute of Technology)

Monopoles and Three-Manifolds

Peter Kronheimer (Harvard University, Massachusetts), Tomasz Mrowka (Massachusetts Institute of Technology)

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English

Cambridge University Press
20 December 2007

Algebraic topology

Series: New Mathematical Monographs

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.

By: Peter Kronheimer (Harvard University Massachusetts), Tomasz Mrowka (Massachusetts Institute of Technology)
Imprint: Cambridge University Press
Country of Publication: United Kingdom
Volume: 10
Dimensions: Height: 229mm, Width: 152mm, Spine: 49mm
Weight: 1.350kg
ISBN: 9780521880220
ISBN 10: 052188022X
Series: New Mathematical Monographs
Pages: 810
Publication Date: 20 December 2007
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active

Preface; 1. Outlines; 2. The Seiberg–Witten equations and compactness; 3. Hilbert manifolds and perturbations; 4. Moduli spaces and transversality; 5. Compactness and gluing; 6. Floer homology; 7. Cobordisms and invariance; 8. Non-exact perturbations; 9. Calculations; 10. Further developments; References; Glossary of notation; Index.

Peter Kronheimer is William Caspar Graustein Professor in the Department of Mathematics at Harvard University. He is a fellow of the Royal Society and has won several international prizes. He is co-author, with S. K. Donaldson, of The Geometry of Four-Manifolds. His research interests are gauge theory, low-dimensional topology and geometry. Tomasz Mrowka is a Professor of Mathematics at Massachusetts Institute of Technology. He holds the James and Marilyn Simson Professorship of Mathematics and was the joint recipient (with Peter Kronheimer) of the 2007 Oswald Veblen Prize. His research interests are low dimensional topology, partial differential equations, and mathematical physics.

Reviews for Monopoles and Three-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, 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.' Mathematical Reviews