Lectures on Groups of Symplectomorphisms

Abstract

These notes combine material from short lecture courses given in Paris, France, in July 2001 and in Srni, the Czech Republic, in January 2003. They discuss groups of symplectomorphisms of closed symplectic manifolds (M,\om) from various points of view. Lectures 1 and 2 provide an overview of our current knowledge of their algebraic, geometric and homotopy theoretic properties. Lecture 3 sketches the arguments used by Gromov, Abreu and Abreu-McDuff to figure out the rational homotopy type of these groups in the cases M= CP^2 and M=S^2\times S^2. We outline the needed J-holomorphic curve techniques. Much of the recent progress in understanding the geometry and topology of these groups has come from studying the properties of fibrations with the manifold M as fiber and structural group equal either to the symplectic group or to its Hamiltonian subgroup Ham(M). The case when the base is S^2 has proved particularly important. Lecture 4 describes the geometry of Hamiltonian fibrations over S^2, while Lecture 5 discusses their Gromov-Witten invariants via the Seidel representation. It ends by sketching Entov's explanation of the ABW inequalities for eigenvalues of products of special unitary matrices. Finally in Lecture 6 we apply the ideas developed in the previous two lectures to demonstrate the existence of (short) paths in Ham(M,\om) that minimize the Hofer norm over all paths with the given endpoints.

significantly revised, 36 pages; notes from summer school in Paris 2001 and winter school in Srni 2003

References in corpus (9)

• Higher genus symplectic invariants and sigma model coupled with gravity
• Growth of maps, distortion in groups and symplectic geometry
• The topology of the space of symplectic balls in rational 4-manifolds
• On the Flux Conjectures
• Symplectomorphism groups and almost complex structures
• Spectral invariants and length minimizing property of Hamiltonian paths
• Characteristic classes of smooth fibrations
• Propagation in Hamiltonian dynamics and relative symplectic homology
• Relative family Gromov-Witten invariants and symplectomorphisms

Cited by in corpus (10)

• Characteristic classes of smooth fibrations
• Remarks on invariants of hamiltonian loops
• Maximal compact tori in the Hamiltonian groups of 4-dimensional symplectic manifolds
• Classifying semi-free Hamiltonian $S^1$-manifolds
• K-theoretic invariants of Hamiltonian fibrations
• On boundedness of characteristic class via quasi-morphism
• On the Homotopy of Symplectomorphism Groups of Homogeneous Spaces
• Hamiltonian diffeomorphisms of toric manifolds
• The Dixmier-Douady class, the action homomorphism, and group cocycles on the symplectomorphism group
• Lifting Hamiltonian loops to isotopies in fibrations