Mathematical Surveys and Monographs 2002; 350 pp; hardcover Volume: 98 ISBN10: 0821805029 ISBN13: 9780821805022 List Price: US$91 Member Price: US$72.80 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 Gmanifolds 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 DuistermaatHeckman theory, the BerlineVergneAtiyahBott 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 noncompact manifolds. A key ingredient in this noncompact cobordism is an equivariantgeometrical 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 groupactions 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 graduatelevel differential geometry. Readership 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 Table of Contents 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
 Nondegenerate abstract moment maps
 Characteristic numbers, nondegenerate cobordisms, and nonvirtual quantization
 The Kawasaki RiemannRoch formula
 Cobordism invariance of the index of a transversally elliptic operator
 Bibliography
 Index
