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The Calabi invariant and the Euler class

Author: Takashi Tsuboi
Journal: Trans. Amer. Math. Soc. 352 (2000), 515-524
MSC (1991): Primary 57R32, 53C15; Secondary 57R50, 58H10, 53C12
Published electronically: March 18, 1999
MathSciNet review: 1487633
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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.

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Additional Information

Takashi Tsuboi
Affiliation: Graduate School of Mathematical Sciences, University of Tokyo, Komaba Meguro, Tokyo 153, Japan

Keywords: Area preserving diffeomorphisms, classifying spaces, foliated circle bundles, foliated disk bundles, Calabi invariant, Euler class
Received by editor(s): June 6, 1997
Received by editor(s) in revised form: September 12, 1997
Published electronically: March 18, 1999
Additional Notes: The author was supported in part by Grant-in-Aid for Scientific Research 07454013 and 09440028, Ministry of Education, Science, Sports and Culture, Japan.
Article copyright: © Copyright 1999 American Mathematical Society

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