The Topology of Torus Actions on Symplectic Manifolds

Cover......Page 1
Series......Page 2
Title page......Page 3
Copyright page......Page 4
Epigraph......Page 5
CONTENTS......Page 7
How I have (re-)written this book......Page 9
Acknowledgements......Page 10
What I have written in this book......Page 11
I.1. Generalities......Page 17
I.2. Equivariant tubular neighborhoods and orbit types decomposition......Page 21
I.3. Examples: $\\mathbf{S}^1$-actions on manifolds of dimension 2 and 3......Page 26
I.4. Appendix: Lie groups, Lie algebras, homogeneous spaces......Page 40
Exercises......Page 45
II.1. What is a symplectic manifold?......Page 51
II.2. Calibrated almost complex structures......Page 60
II.3. Hamiltonian vector fields and Poisson brackets......Page 66
Exercises......Page 70
III.1. Hamiltonian group actions......Page 79
III.2. Properties of momentum mappings......Page 85
III.3. Torus actions and integrable systems......Page 95
Exercises......Page 105
IV.1. Critical points of almost periodic Hamiltonians......Page 113
IV.2. Morse functions (in the sense of Bott)......Page 116
IV.3. Connectedness of the fibers of the momentum mapping......Page 119
IV.4. Application to convexity theorems......Page 121
IV.5. Appendix: compact symplectic SU(2)-manifolds of dimension 4......Page 139
Exercises......Page 144
V.1. The moduli space of flat connections......Page 155
V.2. A Poisson structure on the moduli space of flat connections......Page 162
V.3. Construction of commuting functions on $\\mathcal{M}$......Page 170
V.4. Appendix: connections on principal bundles......Page 178
Exercises......Page 183
VI. Equivariant cohomology and the Duistermaat-Heckman theorem......Page 185
VI.1. Milnor joins, Borel construction and equivariant cohomology......Page 186
VI.2. Hamiltonian actions and the Duistermaat-Heckman theorem......Page 197
VI.3. Localization at fixed points and the Duistermaat-Heckman formula......Page 209
VI.4. Appendix: some algebraic topology......Page 220
VI.5. Appendix: various notions of Euler classes......Page 226
Exercises......Page 228
VII. Toric manifolds......Page 233
VII.1. Fans and toric varieties......Page 234
VII.2. Symplectic reduction and convex polyhedra......Page 252
VII.3. Cohomology of $\\mathbf{X}_\\Sigma$......Page 265
VII.4. Complex toric surfaces......Page 270
Exercises......Page 274
VIII. Hamiltonian circle actions on manifolds of dimension 4......Page 279
VIII.1. Symplectic $\\mathbf{S}^1$-actions, generalities......Page 280
VIII.2. Periodic Hamiltonians on 4-dimensional manifolds......Page 287
Exercises......Page 313
Bibliography......Page 319
Index......Page 329