Is there something well-known about which ring can be a cohomology ring of a manifold? More concretely, I would be interested in the following question: does there exists an $m$, s.t. for each $r,s\in\mathbb{N}$ and $\beta_{i,j}^k\in\mathbb{Z}$, there exists a connected compact $m$-manifold $M$ s.t. $H^2(M)\simeq \mathbb{Z}^r$, $H^4(M)\simeq \mathbb{Z}^{s}$ and $x_i\smile x_j=\sum \beta_{i,j}^k y_k$ for the cohomology generators? I would be grateful for some reference..