MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.