The Calabi invariant and the Euler class

Abstract:

We show the following relationship between the Euler class for the group of the orientation preserving diffeomorphisms of the circle and the Calabi invariant for the group of area preserving diffeomorphisms of the disk which are the identity along the boundary. A diffeomorphism of the circle admits an extension which is an area preserving diffeomorphism of the disk. For a homomorphism $\psi$ from the fundamental group $\langle a_{1}, \cdots , a_{2g} ; [a_{1},a_{2}]\cdots [a_{2g-1},a_{2g}]\rangle$ of a closed surface to the group of the diffeomorphisms of the circle, by taking the extensions $\widetilde {\psi (a}_{i})$ for the generators $a_{i}$, one obtains the product $[\widetilde {\psi (a}_{1}),\widetilde {\psi (a}_{2})]\cdots [\widetilde {\psi (a}_{2g-1}),\widetilde {\psi (a}_{2g})]$ of their commutators, and this is an area preserving diffeomorphism of the disk which is the identity along the boundary. Then the Calabi invariant of this area preserving diffeomorphism is a non-zero multiple of the Euler class of the associated circle bundle evaluated on the fundamental cycle of the surface.