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Suppose we have a map $f : \mathcal{A} \to \mathcal{B}$ between two formal, simply connected CDGAs, with induced map on cohomology $H(f) : H(\mathcal{A}) \to H(\mathcal{B})$. Further, suppose we have minimal models $m _A : \mathcal{M}_\mathcal{A} \to \mathcal{A}$ and $H(\mathcal{A}$) and $m_\mathcal{B} : \mathcal{M}_\mathcal{B} \to \mathcal{B}$. Given that the spaces are formal, we can induce minimal models $m_\mathcal{A}' : \mathcal{M}_\mathcal{A} \to H(\mathcal{A})$ and $m_\mathcal{B}' : \mathcal{M}_\mathcal{B} \to H(\mathcal{B})$ directly on the cohomology ring using the zig-zag of weak equivalences between each CDGA and their respective cohomology rings.

Now let

  • $\tilde{f} : \mathcal{M}_\mathcal{A} \to \mathcal{M}_\mathcal{B}$ be a Sullivan representative for $f$ and
  • $\tilde{H(f)} : \mathcal{M}_\mathcal{A} \to \mathcal{M}_\mathcal{B}$ be a Sullivan representative for $H(f)$.

I'm wondering if the idea that the rational homotopy groups of formal spaces can be computed directly from their cohomology ring extends to the induced maps between the rational homotopy groups of formal spaces. Explicitly, is it true that the Sullivan representative $\tilde{f}$ will be in the same homotopy class as $\tilde{H(f)}$?

If it is not the case that this is always true, I imagine that there should probably exist some references that study the criteria for this kind of 'formality' of maps. If anyone could point me to such references that would be much appreciated!

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    $\begingroup$ Suciu recently wrote a very nice survey article arxiv.org/abs/2210.08310 which discusses results on formality of maps as well, see Section 10.4. An example of a non-formal map is given by the Hopf bundle $S^3 \to S^2$. The effect of the map on rational homotopy is non-trivial (iso on $\pi_3$), while on cohomology it is trivial, so one cannot model the map using only the cohomology. $\endgroup$ Commented Feb 23, 2023 at 16:20
  • $\begingroup$ Thanks - that's exactly what I was looking for! $\endgroup$ Commented Feb 24, 2023 at 4:42

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