Introduction to symplectic topology. (English) Zbl 0844.58029

Oxford Mathematical Monographs. Oxford: Clarendon Press. viii, 425 p. (1995).
The book (425 p.) studies the problems of symplectic topology in four parts.
Symplectic topology is the study of the global phenomenon of symplectic geometry. If the local structure of a symplectic manifold is equivalent to the standard structure on Euclidean space, then in symplectic geometry the absence of local invariants gives rise to an infinite dimensional group of symplectomorphisms and a discrete set of nonequivalent global symplectic structures in each cohomology class.
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.
At the end 226 references and Index.
Reviewer: M.Rahula (Tartu)


53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53D35 Global theory of symplectic and contact manifolds
53D40 Symplectic aspects of Floer homology and cohomology
57R17 Symplectic and contact topology in high or arbitrary dimension
57R57 Applications of global analysis to structures on manifolds
57R58 Floer homology
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