Introduction to symplectic topology / Dusa McDuff, and Dietmar Salamon

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.