  The paper deals with relations between the Hard Lefschetz property,
(non)vanishing of Massey products and the evenness of odd-degree Betti numbers
of closed symplectic manifolds. It is known that closed symplectic manifolds
can violate all these properties (in contrast with the case of Kaehler
manifolds). However, the relations between such homotopy properties seem to be
not analyzed. This analysis may shed a new light on topology of symplectic
manifolds. In the paper, we summarize our knowledge in tables (different in the
simply-connected and in symplectically aspherical cases). Also, we discuss the
variation of symplectically harmonic Betti numbers on some 6-dimensional
manifolds.
