A survey of the topological properties of symplectomorphism groups

The special structures that arise in symplectic topology (particularly Gromov--Witten invariants and quantum homology) place as yet rather poorly understood restrictions on the topological properties of symplectomorphism groups. This article surveys some recent work by Abreu, Lalonde, McDuff, Polterovich and Seidel, concentrating particularly on the homotopy properties of the action of the group of Hamiltonian symplectomorphisms on the underlying manifold M. It sketches the proof that the evaluation map \pi_1(Ham(M))\to \pi_1(M) given by {\phi_t}\mapsto {\phi_t(x_0)} is trivial, as well as explaining similar vanishing results for the action of the homology of Ham(M) on the homology of M. Applications to Hamiltonian stability are discussed.