Structure of the automorphisms group of a toral action
We give an alternative proof of our theorem on the perfectness of the identity component in the automorphisms group A of a T/sup n/-principal bundle ..pi..:M ..-->.. N. Let B denote the identity component in the group of diffeomorphisms of N and consider the locally trivial bundle sigma(..pi..):A ..-->.. B. In the case N ahs a symplectic form ..cap omega.. with integral periods (then N is the base of a T/sup 1/-principal bundle), we exhibit a group G ..-->.. H/sup 1/ (N,Z) such that if G is not trivial, then sigma(..pi..) is not a trivial bundle. The group G happens to be a subgroup of GAMMA = S tilde (..pi../sub 1/G/sub ..cap omega..(N)), where S tilde is the Calabi homomorphism from the universal covering of the identity component G/sub ..cap omega../(N) in the group of symplectic diffeomorphisms of N to the first de Rham cohomology group of N. this follows from a new construction, we given here, for the Calabi invariant in the special case when the symplectic form ..cap omega.. has integral periods. Our new construction shows that the Calabi invariant can be extended to a possibly bigger group than G/sub ..cap omega../(N), if the symplectic form has integral periods and N is compact.
- Research Organization:
- Harvard Univ., Cambridge, MA
- OSTI ID:
- 6522142
- Journal Information:
- Hadronic J.; (United States), Vol. 2:2
- Country of Publication:
- United States
- Language:
- English
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