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On ncatlab page on formality, it is stated that Deligne--Griffiths--Morgan--Sullivan proved that the real homotopy type of a closed Kaehler manifold is formal. Later, Sullivan "improved" this to $\mathbb{Q}$-formality.

My question is: are there some easy examples of closed topological manifolds whose $\mathbb{R}$-homotopy type is formal, but $\mathbb{Q}$-homotopy type isn't?

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What Sullivan proved is not just that the $\mathbb R$-formality from Deligne-Griffiths-Morgan-Sullivan can be improved to $\mathbb Q$-formality, but rather that formality over any field of characteristic zero for any space always implies formality over $\mathbb Q$. See Sullivan's paper.

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    $\begingroup$ Another reference is Proposition 2.101 of Felix-Oprea-Tanre's "Algebraic models in geometry." $\endgroup$
    – skupers
    Commented Nov 28, 2018 at 16:19

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