Introduction to symplectic topology / Dusa McDuff, Dietmar Salamon

This is a very welcome second edition of a book that has become required reading for anyone entering the field of symplectic geometry, and a standard reference.
The major additions concern areas that were in rapid development at the time of publication of the first edition, a fact which had led to some rather sketchily written arguments. There are now expanded discussions of Donaldson's work on symplectic submanifolds and the work of Taubes on symplectic 4-manifolds and Seiberg-Witten theory. The second edition also includes a generating function proof of the Arnolʹd conjecture for Lagrangian intersections in cotangent bundles, and additional material on the topology of the symplectomorphism group. Two other useful additions are brief sections on pseudoholomorphic curves and Floer homology, even though the reader will have to turn to more detailed sources to obtain a working knowledge of these techniques. (MathSciNet)

Bibliogr. p. [400]-410. Index

Part I: Basic background material. Chapter 1: Hamiltonian systems in Euclidean space, development of modern symplectic topology. Chapter 2: linear symplectic geometry, existence of the first Chern class. Chapter 3: symplectic forms on arbitrary manifolds, Darboux’s and Moser’s theorems, contact geometry (the odd-dimensional analogue of symplectic geometry). Chapter 4: almost complex structures, Kähler and Donaldson’s manifolds.

Part II: Examples of symplectic manifolds. Chapter 5: symplectic reduction, Atiyah-Guillemin-Sternberg convexity theorem. Chapter 6: different ways of constructing symplectic manifolds, by fibrations, symplectic blowing up and down, by fibre connected sum, Gompf’s result about the fundamental group of a compact symplectic 4-manifold. Chapter 7: existence and uniqueness of the symplectic structure, Gromov’s proof that every open, almost complex manifold has a symplectic structure.

Part III: Symplectomorphisms. Chapter 8: Poincaré-Birkhoff theorem (an area-preserving twist map of the annulus has two distinct fixed points), special case (strongly monotone twist maps). Chapter 9: generating functions (modern and classical guise), discrete-time variational problems. Chapter 10: structure of the group of symplectomorphisms, properties of the subgroup of Hamiltonian symplectomorphisms.

Part IV (the heart of the book): Finite-dimensional variational methods, full proofs of the simplest versions of important new results in the subject. Chapter 11: Arnold’s conjectures for the torus, Lysternik-Schnirelmann theory, Conley index. Chapter 12: non-squeezing theorem in Euclidean space, energy-capacity inequality for symplectomorphisms of Euclidean space.