Symplectic diffeomorphisms and the flux homomorphism

Abstract

Let X be a connected smooth manifold without boundary. We will write Difff, oX for the group of all compactly supported diffeomorphisms of X which preserve the symplectic form o2, topologized as usual as the direct limit of the subgroups Diff~X of all diffeomorphisms with support in the fixed compact set K. Let Diff, ooX be its identity component. The flux is locally a homomorphism from a neighbourhood of the identity in Diffo, oX to H~(X;R). (See Calabi [6].) It gives rise to a global homomorphism s o from the universal cover of Difffo0X to HI(X; F,), and hence to a homomorphism