Question: Given a finite dimensional positively graded algebra $A$ over some ring $R$ that satisfies Poincaré duality in some dimension $n$, is there necessarily a topological space $X$ such that $H^*(X;R) \cong A$?
I recognise this is some sort of realization question but I don't know much algebraic topology.
The case I am most interested in is when $R$ is a field. As vague motivation, I'm interested in whether, given such an $A$ over $\mathbb{Q}$, there is an elliptic Sullivan algebra $(\Lambda V, d)$ such that $H(\Lambda V, d) \cong A$. The converse appears in the textbook Rational Homotopy Theory by Felix et. al.:
Theorem: If $(\Lambda V,d)$ is an elliptic Sullivan algebra (i.e. $V$ and $H(\Lambda V, d)$ are finite dimensional vector spaces) over a field of characteristic 0, then $H(\Lambda V, d)$ satisfies Poincaré duality.
There is at least some Sullivan algebras $(\Lambda V, d)$ quasi-isomorphic to $A$ (since $A^0 \cong R$) but whether any of them are elliptic is the question. I may make this another post later.