THE FLUX HOMOMORPHISM AND CENTRAL EXTENSIONS OF DIFFEOMORPHISM GROUPS

Let D be a closed unit disk in dimension two and G(rel) the group of symplectomorphisms on D preserving the origin and the boundary partial derivative D pointwise. We consider the flux homomorphism on G(rel) and construct a central R-extension called the flux extension. We determine the Euler class of this extension and investigate the relation among the extension, the group 2-cocycle defined by Ismagilov, Losik, and Michor, and the Calabi invariant of D.