$\require{AMScd}$ For rational fibrations $F \rightarrow E \rightarrow B$, we have a nice description in terms of sullivan minimal models, namely a commutative diagram of cdga's $$ \begin{CD} A_{PL}(B) @>>> A_{PL}(E) @>>> A_{PL}(F) \\ @AAA @AAA @AAA \\ (\wedge V ,d_{B}) @>>> (\wedge V \otimes \wedge Z,d) @>>> (\wedge Z,d_{F}) \end{CD} $$ where all vertical arrows are minimal models. This is sometimes called a $\wedge$-minimal $\wedge$-model. See for instance https://www.jstor.org/stable/pdf/1997895.pdf?refreqid=excelsior%3Aacbce1fbb0022d42e2fe7d1265efcb8c

Is this construction natural in some sense? Say for 'morphisms of fibre bundles' or 'morphisms of fibre bundles preserving a common $G$-structure'.


1 Answer 1


I don't know what you are looking for exactly. There's Proposition 15.8 in the book Rational Homotopy Theory by Félix, Halperin, and Thomas (and I don't doubt that there is an analogous proposition for non simply connected spaces in the sequel but I haven't checked). There is a little bit of setup to do so please bear with me.

Suppose that you have two fibrations and a morphism of fiber sequences: $$\require{AMScd} \begin{CD} F @>>> E @>p>> B \\ @VhVV @VgVV @VfVV \\ F' @>>> E' @>p'>> B' \end{CD}$$ where for example you have base points $x_0 \in B$, $x_0' \in B'$ such that $f(x_0) = x_0'$, $F = p^{-1}(x_0)$ and $F' = (p')^{-1}(x_0')$.

Suppose that you are given a model for the fibration $$\begin{CD} (\Lambda W, \bar{d}) @<<< (\Lambda V \otimes \Lambda W, D) @<<< (\Lambda V, d) \\ @V{\bar{m}}VV @VMVV @VmVV \\ A_{PL}(F) @<<< A_{PL}(E) @<{p^*}<< A_{PL}(B) \end{CD}$$ and the same thing with primes everywhere. Suppose also that you have a model $\psi : (\Lambda V', d') \to (\Lambda V, d)$ for $f$, i.e. $m\psi = f^* m'$.

Then you get a map from the pushout: $$\xi : (\Lambda V, d) \otimes_{(\Lambda V', d')} (\Lambda V' \otimes \Lambda W', D) \to A_{PL}(E).$$ You can rewrite the domain of this pushout as $(\Lambda V \otimes \Lambda W', \delta)$ for a differential $\delta$. You moreover have a commutative diagram: $$\begin{CD} (\Lambda V, d) @>>> (\Lambda V \otimes \Lambda W', \delta) @>>> (\Lambda W', \bar{d'} \\ @VVV @VVV @VVV \\ A_{PL}(B') @>>> A_{PL}(E) @>>> A_{PL}(F) \end{CD}$$ The proposition is that if $h^* : H^*(F') \to H^*(F)$ is an isomorphism (with rational coefficients), then this last diagram is a model for the fibration $F \to E \to B'$.

Anyway the upshot is that if you have a model for a fibration and you pull back the fibration along some map of which you also have a model, then you get a model for the pullback. (And I certainly hope that I put all the primes where they belong.)

  • $\begingroup$ Thank you! That is the kind of statement I was looking for. It covers the case where we change the base but keep the fibre (mostly) fixed. Do you know of a similar result for the case where we change the fibre but keep the base fixed? In other words the case where f is the identity but h is more complicated? $\endgroup$
    – ort96
    Commented Sep 29, 2018 at 7:28
  • $\begingroup$ I don't know, unfortunately. $\endgroup$ Commented Oct 1, 2018 at 22:11

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