Sorry for the Clicbait title, but take $A$ a graded $\mathbb R$ commutative (in the graded sense) algebra of finite dimension.
Does there exist a smooth manifold $M$ having $A$ as a DeRham cohomology ring : $H^*(M) \simeq A$ ?
In the negative case, what would be a natural restriction that I missed on the structure of the DeRham ring ?
I was thinking of taking generators for the algebra and taking product of spheres. But i have problems when those generators verfies relations among them.