Enlarging the Hamiltonian group

Abstract

This paper investigates ways to enlarge the Hamiltonian subgroup Ham of the symplectomorphism group Symp(M) of the symplectic manifold (M, ω) to a group that both intersects every connected component of Symp(M) and characterizes symplectic bundles with fiber M and closed connection form. As a consequence, it is shown that bundles with closed connection form are stable under appropriate small perturbations of the symplectic form. Further, the manifold (M,ω) has the property that every symplectic M-bundle has a closed connection form if and only if the flux group vanishes and the flux homomorphism extends to a crossed homomorphism defined on the whole group Symp(M). The latter condition is equivalent to saying that a connected component of the commutator subgroup [Symp,Symp] intersects the identity component of Symp only if it also intersects Ham. It is not yet clear when this condition is satisfied. We show that if the symplectic form vanishes on 2-tori the flux homomorphism extends to the subgroup of Symp acting trivially on π_1(M). We also give an explicit formula for the Kotschick--Morita extension of Flux in the monotone case. The results in this paper belong to the realm of soft symplectic topology, but raise some questions that may need hard methods to answer.