Introduction to Symplectic Topology

Abstract

Over the past number of years powerful new methods in analysis and topology have led to the development of the modern global theory of symplectic topology, including several striking and important results. The first edition of Introduction to Symplectic Topology was published in 1995. The book was the first comprehensive introduction to the subject and became a key text in the area. In 1998, a significantly revised second edition contained new sections and updates. This third edition includes both further updates and new material on this fast-developing area. All chapters have been revised to improve the exposition, new material has been added in many places, and various proofs have been tightened up. Copious new references to key papers have been added to the bibliography. In particular, the material on contact geometry has been significantly expanded, many more details on linear complex structures and on the symplectic blowup and blowdown have been added, the section on J-holomorphic curves in Chapter 4 has been thoroughly revised, there are new sections on GIT and on the topology of symplectomorphism groups, and the section on Floer homology has been revised and updated. Chapter 13 has been completely rewritten and has a new title (Questions of Existence and Uniqueness). It now contains an introduction to existence and uniqueness problems in symplectic topology, a section describing various examples, an overview of Taubes–Seiberg–Witten theory and its applications to symplectic topology, and a section on symplectic 4-manifolds. Chapter 14 on open problems has been added.

Lagrangian description, symplectization, and Eulerian dynamics of incompressible fluids

“…Note that Ω has ∂ t as its symplectic dilation and that the Reeb vector field on M defined by σ is a Hamiltonian vector field for the symplectization with the Hamiltonian function e t [3,11].…”

Lagrangian isotopy of tori in $${S^2\times S^2}$$ S 2 × S 2 and $${{\mathbb{C}}P^2}$$ C P 2

“…We use Γ η ∼ = S 1 to denote the family of periodic Reeb orbits projecting to the oriented geodesics in homology class η ∈ H 1 (T 2 ) \ {0}. By Weinstein's Lagrangian neighborhood theorem originally proven in [53] (also, see [37]), any Lagrangian embedding ψ : L ֒→ (X, ω) can be extended to a symplectic embedding…”

“…By the latter property, Moser's trick [37] can be applied to the above family ω t of symplectic forms. This yields a smooth isotopy φ t : S 2 × S 2 → S 2 × S 2 for which φ 0 = 1l S 2 ×S 2 , while (φ t ) * ω = ω t .…”

“…By Moser's trick (see e.g. [37]) together with the fact that the space of symplectic forms on a surface inducing a fixed orientation is a contractible space, we get a weak homotopy equivalence Symp(D 2 ) ∼ Diff + (D 2 ). Since the latter group of orientation preserving diffeomorphisms is contractible by [47], and since D 2 is simply connected, we thus conclude that Symp(D 2 ) = Ham(D 2 ).…”

“…Given the choice of a symplectic trivialization τ of ν, the isotropic neighborhood theorem proven by A. Weinstein in [54] (also, see [37]) provides a symplectic embedding…”