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What is the relationship between the Betti numbers $b_i(M;\mathbb{Q})=rkH_i(M;\mathbb{Q})$ of a simply connected closed Riemannian manifold $M$ and the dimension of rational homotopy $\dim_{\mathbb{Q}}\pi_i(M)\otimes\mathbb{Q}$ (if there is any)?

Cross positing on MSE

Thanks in advance!

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If $M$ is a closed, simply connected smooth manifold, then the rank of the rational homotopy groups $\pi_i(M)\otimes Q$ equals the number of degree $i$ generators introduced in the construction of the minimal model of $M$. For certain manifolds that are formal (e.g., if $M$ is Kahler), their cohomology ring is quasi-isomorphic as a DGA to their minimal model, so that one can use the cohomology ring through the minimal model to compute the ranks of the rational homotopy groups.

A nice example where one can use the cohomology to read off the rational homotopy is the case of the oriented Grassmannian $BSO(n)$. Its rational cohomology ring is precisely the polynomial algebra generated by the Pontryagin classes and the Euler class for even $n,$ and the Pontryagin classes alone for odd $n$. Thus, the rational homotopy groups have rank exactly 1 every $4k$ dimensions for $BSO(even)$, and rank exactly 1 every $4k$ dimensions in $BSO(odd)$ except for one index where one may have rank 2 (due to the presence of the Euler class and Pontryagin class). One also ignores the last index in this latter case, as the Euler class squares to the top Pontryagin class.

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  • $\begingroup$ I see; but, how does one find such a minimal model if $M$ is an arbitrary closed simply connected Riemannian manifold? @TobiasShin $\endgroup$
    – Sergio Charles
    Commented Jul 25, 2018 at 4:08
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    $\begingroup$ For an arbitrary closed simply connected manifold, it's hard to compute the minimal model without additional data. If the manifold you're considering is Kahler, however, (so its reducible holonomy lives in SU(n)), then it is formal, so its minimal model is quasi-isomorphic to its cohomology algebra with the trivial differential. I'm not sure about the formality of manifolds with holonomy higher up in Berger's list (e.g., G_2, Spin(7)...). $\endgroup$
    – Tobias Shin
    Commented Jul 25, 2018 at 4:46
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    $\begingroup$ In "Amann, M. and Kapovitch, V., 2012. On fibrations with formal elliptic fibers. Advances in Mathematics, 231(3-4), pp.2048-2068" it's proven that closed quaternion-Kahler manifolds (holonomy in Sp(n)•Sp(1)) that admit a metric of positive scalar curvature are formal. It's also conjectured within that manifolds with special holonomy are formal $\endgroup$
    – Tobias Shin
    Commented Jul 25, 2018 at 4:53
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    $\begingroup$ For many results relating the rational homotopy and Betti numbers, see the book "Rational Homotopy Theory" by Félix, Halperin, Thomas, as suggested in the MSE cross-post; in particular I recommend looking at Chapter 6, Sections 32 and 33. $\endgroup$
    – Aleksandar Milivojević
    Commented Jul 25, 2018 at 10:21
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    $\begingroup$ A nice theorem of Miller (see projecteuclid.org/euclid.ijm/1256048237) states that a closed manifold of dimension $\leq 4k+2$ with $b_1,b_2,…,b_k=0$ is formal. In particular, simply connected closed manifolds of dimension $\leq 6$ are formal. Cavalcanti showed that if additionally $b_{k+1}=1$, then the conclusion of formality holds for closed manifolds of dimension $\leq 4k+4$. $\endgroup$
    – Aleksandar Milivojević
    Commented Jul 25, 2018 at 10:48

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