Lectures on Groups of Symplectomorphisms

TL;DR

This paper provides an overview of the algebraic, geometric, and homotopy properties of groups of symplectomorphisms of closed symplectic manifolds, including recent advances using J-holomorphic curves and Hamiltonian fibrations.

Contribution

It synthesizes recent progress in understanding the topology and geometry of symplectomorphism groups, especially through the study of fibrations, Gromov-Witten invariants, and the Hofer norm.

Findings

Rational homotopy types of symplectomorphism groups for specific manifolds determined.

Gromov-Witten invariants and Seidel representation used to analyze symplectic fibrations.

Existence of minimal Hofer norm paths in Hamiltonian groups demonstrated.

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.