Introduction to symplectic topology (Q2827261)

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. The origin of the symplectic topology comes from the study of the equations of motion arising from the Euler-Lagrange equations of a one-dimensional variational problem. The Hamiltonian formalism arising from a Legendre transformation leads to the notion of a canonical transformation which preserves Hamilton's form of the equations of motion. NEWLINENEWLINEA symplectic manifold is defined as a smooth manifold equipped with an atlas whose transition maps are canonical transformations. Symplectic manifolds are then spaces with a global intrinsic structure on which Hamilton's differential equations can be defined and take the classical form in each coordinate chart. Since canonical transformations preserve a standard 2-form associated to Hamilton's equations, a symplectic manifold is equipped with a closed nondegenerate 2-form that is called a symplectic form and denoted by \(\omega\). The fundamental result is Darboux's Theorem which asserts that the existence of a symplectic form is equivalent to the existence of an atlas whose transition maps are canonical transformations, and that every symplectic manifold is locally isomorphic to the Euclidean space of the appropriate dimension with its standard symplectic structure. A symplectic manifold has a very rich infinite-dimensional group of diffeomorphisms, called symplectomorphisms, that preserve the symplectic structure. The third fundamental notion in the field of symplectic topology, aside from the concepts of a symplectic form and a symplectomorphism, is a Lagrangian submanifold. If \((M,\omega)\) is a \(2n\)-dimensional symplectic manifold, then an \(n\)-dimensional submanifold \(L\subset M\) such that the symplectic form \(\omega\) vanishes on each tangent space of \(L\) is called a Lagrangian submanifold.NEWLINENEWLINEThis book is divided into four parts consisting of fourteen chapters. Part I, comprising the first four chapters, develops symplectic geometry from its very beginnings, so the book can be read without any prior knowledge of symplectic geometry. In Chapter 1, Hamiltonian systems in Euclidean space and some problems motivating the development of modern symplectic topology are discussed. Chapter 2 develops linear symplectic geometry, the theory of symplectic vector spaces and vector bundles, and presents a proof of the existence of the first Chern class. Chapter 3 establishes Darboux's theorem and Moser's stability theorem. Also, it contains an introduction to contact geometry, the odd-dimensional analogue of symplectic geometry. Chapter 4 discusses the difference between symplectic and Kähler manifolds, and then outlines the theory of \(J\)-holomorphic curves and explains some of its applications. NEWLINENEWLINEPart II, consisting of Chapters 5, 6, and 7, is concerned with global symplectic topology on manifolds, group actions, and methods of construction. Chapter 5 begins with the important topic of symplectic reduction that is fundamental in the study of mechanical systems with symmetry, and leads to many significant modern applications of symplectic topology. This chapter includes proofs of the Atiyah-Guillemin-Sternberg convexity theorem for moment maps and of the Duistermaat-Heckman localization formula. Chapter 6 is devoted to symplectic fibrations, while Chapter 7 describes different ways of constructing symplectic manifolds, by symplectic blowing up and down, fiber connected sums, and Gromov's telescope construction of symplectic forms on open almost complex manifolds. This chapter also includes a brief introduction to Donaldson submanifolds, Weinstein domains, and symplectic Lefschetz fibrations. NEWLINENEWLINEPart III discusses properties of symplectomorphisms, with special emphasis on generating functions, and consists of Chapters 8, 9, and 10. In Chapter 8, the authors give a proof of the Poincaré-Birkhoff theorem, which states that an area-preserving twist map of the annulus has two distinct fixed points. Chapter 9 explores generating functions and show how they lead to discrete-time variational problems. Chapter 10 develops basic results on the structure of the group of symplectomorphisms, paying particular attention to properties of the subgroup of Hamiltonian symplectomorphisms. It includes an introduction to the flux homomorphism and to the Calabi homomorphism. NEWLINENEWLINEThe most important part of the book, where the authors give full proofs of the simplest versions of important theorems, is Part IV which consists of Chapter 11, 12, 13, and 14. Chapter 11 starts with the study of the Arnold conjectures on the existence of fixed points of Hamiltonian symplectomorphisms, and proves these conjectures for the standard torus. Also, it contains a generating function proof of the Arnold conjecture for Lagrangian intersections in cotangent bundles. Chapter 12 gives a variational proof of the existence of capacities, and applies it to establish the nonsqueezing theorem, symplectic rigidity, and the Weinstein conjecture for hypersurfaces of Euclidean space. The authors introduce the Hofer metric on the group of Hamiltonian symplectomorphisms and give a geometric proof of the energy-capacity inequality for symplectomorphisms in Euclidean space. Chapter 13 first discusses the fundamental existence and uniqueness problems for symplectic structures, and then surveys a variety of different examples of symplectic manifolds and symplectic invariants. An overview of Taubes-Seiberg-Witten theory and some of its consequences for symplectic four-manifolds are presented. The final Chapter 14 discusses various open problems and conjectures in symplectic topology, as well as some of the known results about these questions. NEWLINENEWLINEThis is the third edition of the book. Since the second edition was published in 1998 [Zbl 1066.53137], in this edition the authors made some changes and additions. The main changes include the following: The addition of Chapter 14. Section 7.1 contains an expanded and much more detailed discussion of the construction of symplectic forms on blow-ups, where some technical parts of the construction are relegated to Appendix A and to a new Section 3.3 on isotopy extensions. The material on Donaldson hypersurfaces is expanded, previously contained in Chapter 13, and moved to Chapter 7 as an additional Section 7.4. Chapter 13 now starts with a new section on existence and uniqueness problems in symplectic topology. It includes a much more detailed section on Taubes-Seiberg-Witten theory and its applications as well as a new section on symplectic four-manifolds. In the earlier chapters, Section 2.5 on linear complex structures and Section 3.5 on contact structures have been significantly expanded, Section 4.5 on \(J\)-holomorphic curves has been rewritten and updated, and Section 5.7, containing an overview of GIT, has been added. In the later chapters, Section 10.4 on the topology of symplectomorphism groups, Section 11.4 on Floer theory, and Section 12.3 on Hofer geometry have been updated and in parts rewritten.NEWLINENEWLINEThe list of references contains an impressive number of 693 entries.