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.
Reviewer: J.C.Thomas (Villeneuve d’Ascq)
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 |
Keywords:
Sullivan minimal models; toral actions on closed symplectic manifolds; symplectic actions; cohomologically-symplectic manifolds; symplectic manifolds of Lefschetz typeReferences:
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