The group of symplectomorphisms

Abstract

This chapter discusses the basic properties of the group of symplectomorphisms of a compact connected symplectic manifold and its subgroup of Hamiltonian symplectomorphisms. It begins by showing that the group of symplectomorphisms is locally path-connected and then moves on to the flux homomorphism. The main result here is a theorem of Banyaga that characterizes the Hamiltonian symplectomorphisms in terms of the flux homomorphism. In the noncompact case there is another interesting homomorphism, called the Calabi homomorphism, that takes values in the reals and may be defined on the universal cover of the group of Hamiltonian symplectomorphisms. The chapter ends with a brief comparison of the topological properties of the group of symplectomorphisms with those of the group of diffeomorphisms.