rational homotopy of a manifold - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T16:47:13Z http://mathoverflow.net/feeds/question/115911 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/115911/rational-homotopy-of-a-manifold rational homotopy of a manifold jim stasheff 2012-12-09T19:23:43Z 2013-02-09T12:55:29Z <p>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?</p> http://mathoverflow.net/questions/115911/rational-homotopy-of-a-manifold/115921#115921 Answer by Mark Grant for rational homotopy of a manifold Mark Grant 2012-12-09T20:55:02Z 2012-12-09T20:55:02Z <p>It sounds like you are asking about the Sullivan-Barge Theorem. The original references are:</p> <p>J. Barge, Structures différentiables sur les types d'homotopie rationnelle simplement connexes, Ann. Sci. École Norm. Sup. (4) 9 (1976), no.4, 469–501.</p> <p>D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. (1977), no.47, 269–331 (1978).</p> <p>You'll find a clean statement in Chapter 3 of <a href="http://www.amazon.com/Algebraic-Models-Geometry-Graduate-Mathematics/dp/019920652X" rel="nofollow">this book</a>, 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:</p> <ol> <li>$n$ is not of the form $4k$;</li> <li>$n$ 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 $\sum \pm x_i^2$;</li> <li>$n$ 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.</li> </ol> <p>In other words, the conditions which are necessary for realization by a smooth manifold are also sufficient.</p>