The material and references in this extended second edition of "The Topology of Torus Actions on Symplectic Manifolds", published as Volume 93 in this series in 1991, have been updated. Symplectic manifolds and torus actions are investigated, with numerous examples of torus actions, for instance on some moduli spaces. Although the book is still centered on convexity results, it contains much more material, in particular lots of new examples and exercises.

I Smooth Lie group actions on manifolds.- 1 Generalities.- 2 Equivariant tubular neighborhoods and orbit types decomposition.- 3 Examples: 2 and 3-dimensional S1-manifolds.- II Symplectic geometry.- 1 Symplectic manifolds.- 2 Hamiltonian vector fields and Poisson manifolds.- 3 Symplectic and hamiltonian actions.- III Morse theory for hamiltonians.- 1 Critical points of almost periodic hamiltonians.- 2 Morse functions (in the sense of Bott).- 3 Connectivity of the fibers of the moment map.- 4 Application to convexity theorems.- IV About manifolds of this dimension.- 1 Characterisation of those circle actions which are hamiltonian.- 2 Symplectic reduction of the regular levels for a periodic hamiltonian.- 3 Blowing up fixed points; creation of index 2 critical points.- 4 4-manifolds with periodic hamiltonians.- 5 Plumbing.- A Appendix: compact symplectic SU(2)-manifolds of dimension 4.- B Appendix: 4-dimensional S1-manifolds with no invariant symplectic form (examples).- V Equivariant cohomology and the Duistermaat-Heckman theorems.- 1 Principal and universal bundles.- 2 The Borel construction and equivariant cohomology.- 3 Equivariant cohomology and hamiltonian actions.- 4 Duistermmat-Heckman with singularities.- 5 Localisation at fixed points.- 6 The Duistermaat-Heckman formula.- A Appendix: some algebraic topology.- B Appendix: various notions of Euler classes.- VI Toric manifolds.- 1 The action of TNC and its subgroups on CN.- 2 Fans and toric varieties.- 3 Fans, symplectic reduction, convex polyhedra.- 4 Properties of the toric manifolds X?.- 5 Complex toric surfaces.- References.