Monopoles and Three-Manifolds

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Cambridge University Press, Dec 20, 2007 - Mathematics
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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.
 

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Contents

I Outlines
1
II The SeibergWitten equations and compactness
84
III Hilbert manifolds and perturbations
134
IV Moduli spaces and transversality
195
V Compactness and gluing
274
VI Floer homology
375
VII Cobordisms and invariance
449
VIII Nonexact perturbations
590
IX Calculations
634
X Further developments
721
References
779
Glossary of notation
785
Index
792
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About the author (2007)

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 been awarded several distinguished prizes including the 2007 Oswald Veblen Prize. 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 Professor of Mathematics at Massachusetts Institute of Technology. He holds the James and Marilyn Simons Professorship of Mathematics and is a Member of the American Academy of Arts and Sciences. He was a joint recipient (with Peter Kronheimer) of the 2007 Oswald Veblen Prize. His research interests are low-dimensional topology, partial differential equations and mathematical physics.

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