Torus Actions on Symplectic Manifolds
How I have (re-)written this book The book the reader has in hand was supposed to be a new edition of [14]. I have hesitated quite a long time before deciding to do the re-writing work-the first edition has been sold out for a few years. There was absolutely no question of just correcting numerous m...
| Main Author: | |
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| Format: | eBook |
| Language: | English |
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Basel
Birkhäuser
2004, 2004
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| Edition: | 2nd ed. 2004 |
| Series: | Progress in Mathematics
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| Subjects: | |
| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- Introductory preface
- How I have (re-)written this book
- Acknowledgements
- What I have written in this book
- I. Smooth Lie group actions on manifolds
- I.1. Generalities
- I.2. Equivariant tubular neighborhoods and orbit types decomposition
- I.3. Examples: S 1-actions on manifolds of dimension 2 and 3
- I.4. Appendix: Lie groups, Lie algebras, homogeneous spaces
- Exercises
- II. Symplectic manifolds
- II.1What is a symplectic manifold?
- II.2. Calibrated almost complex structures
- II.3. Hamiltonian vector fields and Poisson brackets
- Exercises
- III. Symplectic and Hamiltonian group actions
- III.1. Hamiltonian group actions
- III.2. Properties of momentum mappings
- III.3. Torus actions and integrable systems
- Exercises
- IV. Morse theory for Hamiltonians
- IV.1. Critical points of almost periodic Hamiltonians
- IV.2. Morse functions (in the sense of Bott)
- IV.3. Connectedness of the fibers of the momentum mapping
- VIII. Hamiltonian circle actions on manifolds of dimension 4
- VIII.1. Symplectic S 1-actions, generalities
- VIII.2. Periodic Hamiltonians on 4-dimensional manifolds
- Exercises
- IV.4. Application to convexity theorems
- IV.5. Appendix: compact symplectic SU(2)-manifolds of dimension 4
- Exercises
- V. Moduli spaces of flat connections
- V.1. The moduli space of fiat connections
- V.2. A Poisson structure on the moduli space of flat connections
- V.3. Construction of commuting functions on M
- V.4. Appendix: connections on principal bundles
- Exercises
- VI. Equivariant cohomology and the Duistermaat-Heckman theorem
- VI.1. Milnor joins, Borel construction and equivariant cohomology
- VI.2. Hamiltonian actions and the Duistermaat-Heckman theorem
- VI.3. Localization at fixed points and the Duistermaat-Heckman formula
- VI.4. Appendix: some algebraic topology
- VI.5. Appendix: various notions of Euler classes
- Exercises
- VII. Toric manifolds
- VII.1. Fans and toric varieties
- VII.2. Symplectic reduction and convex polyhedra
- VII.3. Cohomology of X ?
- VII.4. Complex toric surfaces
- Exercises