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Moment Maps, Cobordisms, and Hamiltonian Group Actions
Victor Guillemin, Massachusetts Institute of Technology, Cambridge, MA, Viktor Ginzburg, University of California, Santa Cruz, CA, and Yael Karshon, The Hebrew University of Jerusalem, Israel
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Mathematical Surveys and Monographs
2002; 350 pp; hardcover
Volume: 98
ISBN-10: 0-8218-0502-9
ISBN-13: 978-0-8218-0502-2
List Price: US$96 Member Price: US$76.80
Order Code: SURV/98

This research monograph presents many new results in a rapidly developing area of great current interest. Guillemin, Ginzburg, and Karshon show that the underlying topological thread in the computation of invariants of G-manifolds is a consequence of a linearization theorem involving equivariant cobordisms. The book incorporates a novel approach and showcases exciting new research.

During the last 20 years, "localization" has been one of the dominant themes in the area of equivariant differential geometry. Typical results are the Duistermaat-Heckman theory, the Berline-Vergne-Atiyah-Bott localization theorem in equivariant de Rham theory, and the "quantization commutes with reduction" theorem and its various corollaries. To formulate the idea that these theorems are all consequences of a single result involving equivariant cobordisms, the authors have developed a cobordism theory that allows the objects to be non-compact manifolds. A key ingredient in this non-compact cobordism is an equivariant-geometrical object which they call an "abstract moment map". This is a natural and important generalization of the notion of a moment map occurring in the theory of Hamiltonian dynamics.

The book contains a number of appendices that include introductions to proper group-actions on manifolds, equivariant cohomology, Spin$${^\mathrm{c}}$$-structures, and stable complex structures. It is geared toward graduate students and research mathematicians interested in differential geometry. It is also suitable for topologists, Lie theorists, combinatorists, and theoretical physicists. Prerequisite is some expertise in calculus on manifolds and basic graduate-level differential geometry.

Graduate students and research mathematicians interested in differential geometry; topologists, Lie theorists, combinatorists, and theoretical physicists.

Reviews

"This monograph is a splendid account of Hamiltonian torus actions and their connection with equivariant topology. It is a useful reference for those in the field, as well as an excellent introduction for those who want to learn more about the field."

-- Mathematical Reviews

• Introduction
Part 1. Cobordism
• Hamiltonian cobordism
• Abstract moment maps
• The linearization theorem
• Reduction and applications
Part 2. Quantization
• Geometric quantization
• The quantum version of the linearization theorem
• Quantization commutes with reduction
Part 3. Appendices
• Signs and normalization conventions
• Proper actions of Lie groups
• Equivariant cohomology
• Stable complex and Spin$$^{\mathrm{c}}$$structures
• Assignments and abstract moment maps
• Assignment cohomology
• Non-degenerate abstract moment maps
• Characteristic numbers, non-degenerate cobordisms, and non-virtual quantization
• The Kawasaki Riemann-Roch formula
• Cobordism invariance of the index of a transversally elliptic operator
• Bibliography
• Index