TITLE: Realization of graded algebras with Poincaré duality
QUESTION [5 upvotes]: 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.

REPLY [3 votes]: The answer is, in general, no.
For example, the following is corollary 4L.10 of Hatcher's Algebraic Topology book (freely available):

If $H^\ast(X;\mathbb{Z})$ is a polynomial algebra $ \mathbb{Z}[\alpha]$, possibly truncated by $\alpha^m = 0$ for $m > 3$, then $|\alpha| = 2$ or $4$.

Here, $|\alpha|$ denotes the degree of $\alpha$, meaning $\alpha \in H^{|\alpha|}(X;\mathbb{Z})$.