Abstract

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 π1(Ham(M)) → π1(M) given by {øt} ↦ {øt (x0)} 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.

Overview

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 on this subject. Throughout (M, ω) will be a closed (ie compact and without boundary), smooth symplectic manifold of dimension 2n, unless it is explicitly mentioned otherwise. Background information and more references can be found in.

The symplectomorphism group Symp(M, ω) consists of all diffeomorphisms ø : M → M such that ø*(ω) = ω, and is equipped with the C∞-topology, the topology of uniform convergence of all derivatives. We will sometimes contrast this with the C0 (i.e. compact-open) topology.