The flux homomorphism and central extensions of diffeomorphism groups

Abstract

Let $D$ be a closed unit disk in dimension two and $G_{\rm rel}$ the group of symplectomorphisms on $D$ preserving the origin and the boundary $\partial D$ pointwise. We consider the flux homomorphism on $G_{\rm rel}$ and construct a central $\mathbb{R}$-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 $D$.