Hereditary properties of co-K\"ahler manifolds

We show how certain topological properties of co-K{"a}hler manifolds derive from those of the K"ahler manifolds which construct them. We go beyond Betti number results and describe the cohomology algebra structure of co-K"ahler manifolds. As a consequence, we prove that co-K"ahler manifolds satisfy the Toral Rank Conjecture: $\dim (H^{\ast }(M;\mathbb{Q}))\ge 2^r$, for any $r$-torus $T^r$ which acts almost freely on $M$.