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Cohomologically symplectic spaces: Toral actions and the Gottlieb group. (English) Zbl 0836.57019
The purpose of this paper is to study toral actions on closed symplectic manifolds. The authors point out an algebraic topological foundation which underlies many of the results arising in the study of symplectic actions. This allows to generalize some of them to cohomologically- symplectic manifolds: there is a class $$\omega \in H^2 (M, \mathbb{Q})$$ such that $$\omega^n$$ is a top class for $$M^{2n}$$. Since compact Kähler manifolds are symplectic manifolds which satisfy the hard Lefschetz property, it is natural to introduce symplectic manifolds of Lefschetz type: $$\omega^{n - 1} : H^1 (M) \cong H^{2n - 1} (M)$$ $$(\dim M = 2n)$$. Such a manifold $$M$$ is diffeomorphic to a torus and a torus acts cohomologically freely on $$M$$ if and only if all isotropy groups are finite. Many other results of this type are proved along the paper.

##### MSC:
 57R19 Algebraic topology on manifolds and differential topology 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 55P62 Rational homotopy theory 57S25 Groups acting on specific manifolds
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