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

Abstract

We say that a simply connected space X is pre-c-symplectic if it is the fibre of a rational fibration \({X \to Y \to \mathbb{C}P^{\infty}}\) 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 (Felix et al. in Rational homotopy theory. Graduate Texts in Mathematics, vol. 205. Springer, Berlin, 2001), we give the necessary and sufficient condition that the product of odd-spheres \({X=S^{k_1}\times \cdots \times S^{k_n}}\) 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 (Yamaguchi in Bull Belg Math Soc Simon Stevin 18:493–508, 2011) as a necessary condition to be pre-c-symplectic and see some examples in the cases of finite-oddly generated rational homotopy groups.