Advertisement
Published in:
2016 | OriginalPaper | Chapter
The Calabi Invariant and Central Extensions of Diffeomorphism Groups
Author : Hitoshi Moriyoshi
Published in: Geometry and Topology of Manifolds
Publisher: Springer Japan
Activate our intelligent search to find suitable subject content or patents.
Select sections of text to find matching patents with Artificial Intelligence. powered by
Select sections of text to find additional relevant content using AI-assisted search. powered by
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.