eBook ISBN:  9781470413255 
Product Code:  SURV/98.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
eBook ISBN:  9781470413255 
Product Code:  SURV/98.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 

Book DetailsMathematical Surveys and MonographsVolume: 98; 2002; 350 ppMSC: Primary 53; 57; 55
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.
ReadershipGraduate students and research mathematicians interested in differential geometry; topologists, Lie theorists, combinatorists, and theoretical physicists.

Table of Contents

Chapters

1. Introduction

Part 1. Cobordism

2. Hamiltonian cobordism

3. Abstract moment maps

4. The linearization theorem

5. Reduction and applications

Part 2. Quantization

6. Geometric quantization

7. The quantum version of the linearization theorem

8. Quantization commutes with reduction

Part 3. Appendices

Appendix A. Signs and normalization conventions

Appendix 10. Proper actions of lie groups

Appendix 11. Equivariant cohomology

Appendix D. Stable complex and $\mathrm {Spin}^c$structures

Appendix E. Assignments and abstract moment maps

Appendix F. Assignment cohomology

Appendix G. Nondegenerate abstract moment maps

Appendix H. Characteristic numbers, nondegenerate cobordisms, and nonvirtual quantization

Appendix I. The Kawasaki Riemann–Roch formula

Appendix J. Cobordism invariance of the index of a transversally elliptic operator, by Maxim Braverman


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


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

Chapters

1. Introduction

Part 1. Cobordism

2. Hamiltonian cobordism

3. Abstract moment maps

4. The linearization theorem

5. Reduction and applications

Part 2. Quantization

6. Geometric quantization

7. The quantum version of the linearization theorem

8. Quantization commutes with reduction

Part 3. Appendices

Appendix A. Signs and normalization conventions

Appendix 10. Proper actions of lie groups

Appendix 11. Equivariant cohomology

Appendix D. Stable complex and $\mathrm {Spin}^c$structures

Appendix E. Assignments and abstract moment maps

Appendix F. Assignment cohomology

Appendix G. Nondegenerate abstract moment maps

Appendix H. Characteristic numbers, nondegenerate cobordisms, and nonvirtual quantization

Appendix I. The Kawasaki Riemann–Roch formula

Appendix J. Cobordism invariance of the index of a transversally elliptic operator, by Maxim Braverman

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