rational homotopy of a manifold

Question

Given a finite dim rational homotopy type satisfying Poincaré duality, what is the best reference to when it is the rational homotopy type of a fin dim manifold?

Answer 1

It sounds like you are asking about the Sullivan-Barge Theorem. The original references are:

J. Barge, Structures différentiables sur les types d'homotopie rationnelle simplement connexes, Ann. Sci. École Norm. Sup. (4) 9 (1976), no.4, 469–501.

D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. (1977), no.47, 269–331 (1978).

You'll find a clean statement in Chapter 3 of this book, but to paraphrase: Suppose you have a simply-connected Sullivan algebra whose cohomology $H^\ast$ is a Poincaré duality algebra of formal dimension $n$. Then it can be realised by a closed simply-connected manifold if, and only if, one of the following holds:

1. $n$ is not of the form $4k$;
2. $n$ is of the form $4k$, the signature is zero and the quadratic form on $H^{2k}$ is equivalent over $\mathbb{Q}$ to one of the form $\sum \pm x_i^2$;
3. $n$ is of the form $4k$, the signature is nonzero, the quadratic form on $H^{2k}$ is equivalent over $\mathbb{Q}$ to one of the form $\sum \pm x_i^2$, and one can find a sequence of classes $p_i\in H^{4i}$ (the Pontrjagin classes) such that the corresponding Pontrjagin numbers satisfy certain necessary congruences.

In other words, the conditions which are necessary for realization by a smooth manifold are also sufficient.