The Calabi Invariant and Central Extensions of Diffeomorphism Groups

Abstract

Let D be a closed unit disc in dimension two and G the group of symplectomorphisms on D. Denote by $G_{\partial }$ the group of diffeomorphisms on the boundary $\partial D$ and by $G_{\mathrm {rel}}$ the group of relative symplectomorphisms. There exists a short exact sequence involving with those groups, whose kernel is $G_{\mathrm {rel}}$. On such a group $G_{\mathrm {rel}}$ one has a celebrated homomorphism called the Calabi invariant. By dividing the exact sequence by the kernel of the Calabi invariant, one obtains a central $\mathbb R$-extension, called the Calabi extension. We determine the resulting class of the Calabi extension in $H^2( G_{\partial };\mathbb R)$ and exhibit a transgression formula that clarify the relation among the Euler cocycle for $G_{\partial }$, the Thom class and the Calabi invariant.