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.