# Moment Maps, Cobordisms, and Hamiltonian Group Actions

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*Victor Guillemin; Viktor Ginzburg; Yael Karshon*

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.

#### Reviews & Endorsements

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

# Table of Contents

## Moment Maps, Cobordisms, and Hamiltonian Group Actions

- Contents v6 free
- Chapter 1. Introduction 110 free
- Part 1. Cobordism 1322
- Chapter 2. Hamiltonian cobordism 1524
- Chapter 3. Abstract moment maps 3140
- Chapter 4. The linearization theorem 4554
- 1. The simplest case of the linearization theorem 4554
- 2. The Hamiltonian linearization theorem 4756
- 3. The linearization theorem for abstract moment maps 5160
- 4. Linear torus actions 5261
- 5. The right-hand side of the linearization theorems 5665
- 6. The Duistermaat-Heckman and Guillemin-Lerman-Sternberg formulas 5867

- Chapter 5. Reduction and applications 6372

- Part 2. Quantization 8796
- Chapter 6. Geometric quantization 8998
- 1. Quantization and group actions 8998
- 2. Pre-quantization 9099
- 3. Pre-quantization of reduced spaces 96105
- 4. Kirillov-Kostant pre-quantization 99108
- 5. Polarizations, complex structures, and geometric quantization 102111
- 6. Dolbeault Quantization and the Riemann-Roch formula 110119
- 7. Stable complex quantization and Spinc quantization 113122
- 8. Geometric quantization as a push-forward 117126

- Chapter 7. The quantum version of the linearization theorem 119128
- Chapter 8. Quantization commutes with reduction 139148
- 1. Quantization and reduction commute 139148
- 2. Quantization of stable complex toric varieties 141150
- 3. Linearization of [Q,R]=0 145154
- 4. Straightening the symplectic and complex structures 149158
- 5. Passing to holomorphic sheaf cohomology 150159
- 6. Computing global sections; the lit set 152161
- 7. The Cech complex 155164
- 8. The higher cohomology 157166
- 9. Singular [Q,R]=0 for non-symplectic Hamiltonian G-manifolds 159168
- 10. Overview of the literature 162171

- Part 3. Appendices 165174
- Appendix A. Signs and normalization conventions 167176
- Appendix B. Proper actions of Lie groups 173182
- Appendix C. Equivariant cohomology 197206
- 1. The definition and basic properties of equivariant cohomology 197206
- 2. Reduction and cohomology 201210
- 3. Additivity and localization 203212
- 4. Formality 205214
- 5. The relation between H*[sub(G)] and H*[sub(T)] 208217
- 6. Equivariant vector bundles and characteristic classes 211220
- 7. The Atiyah–Bott–Berline–Vergne localization formula 217226
- 8. Applications of the Atiyah–Bott–Berline–Vergne localization formula 222231
- 9. Equivariant homology 226235

- Appendix D. Stable complex and Spin[sup(c)]-structures 229238
- Appendix E. Assignments and abstract moment maps 257266
- Appendix F. Assignment cohomology 279288
- Appendix G. Non-degenerate abstract moment maps 289298
- Appendix H. Characteristic numbers, non-degenerate cobordisms, and non-virtual quantization 301310
- Appendix I. The Kawasaki Riemann–Roch formula 315324
- Appendix J. Cobordism invariance of the index of a transversally elliptic operator by Maxim Braverman 327336

- Bibliography 339348
- Index 349358