Enlarging the Hamiltonian group

Abstract

This paper investigates ways to enlarge the Hamiltonian subgroup {\rm Ham} of the symplectomorphism group {\rm Symp} of a symplectic manifold $(M, \omega)$ to a group that both intersects every connected component of {\rm Symp} 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 $(M,\omega)$ 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 {\rm Symp}. The latter condition is equivalent to saying that a connected component of the commutator subgroup [{\rm Symp}, {\rm Symp}] intersects the identity component of {\rm Symp} only if it also intersects {\rm 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 {\rm Symp} acting trivially on $\pi_1(M)$. We also give an explicit formula for the Kotschick--Morita extension of the flux homomorphism 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.