Pre-c-symplectic condition for the product of odd-spheres

We say that a simply connected space $X$ is pre-c-symplectic if it is the fibre of a rational fibration $X\to Y\to \C P^{\infty}$ where $Y$ is cohomologically symplectic in the sense that there is a degree 2 cohomology class which cups to a top class. It is a rational homotopical property but not a cohomological one. By using Sullivan's minimal models, we give the necessary and sufficient condition that the product of odd-spheres $X=S^{k_1}\times ... \times S^{k_n}$ is pre-c-symplectic and see some related topics. Also we give a charactarization of the Hasse diagram of rational toral ranks for a space $X$ as a necessary condition to be pre-c-symplectic and see some examples in the cases of finite-oddly generated rational homotopy groups.

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