Consider the pullback of a (Hurewicz) fibration $p\colon E \longrightarrow B$ along any map $f$ and let $p'$ denote the base change of $p$ in the pullback. Suppose that $H_*(p)$, the induced homomorphism in singular homology (with coefficients in any field) of $p$, is surjective . Is it true that $H_*(p')$ is also surjective? If not, when is this possible?
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5$\begingroup$ Take $S^2\times [0,1]$ and glue the boundaries by an orientation reversing isomorphism of $S^2$. This gives a nonorientable 3-manifold. A map from $S^1$ induces an isomorphism in homology with all coefficients except characteristic 2. But when you pull back to the universal cover, it is $\mathbb R\to S^2\times \mathbb R$, which is not surjective. You can modify this to work all characteristics simultaneously. $\endgroup$– Ben WielandCommented Dec 19, 2022 at 17:10
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$\begingroup$ Basically the pull-back is not the right notion to work with, you would want to work with homotopy pull-back, and if you replace pull-back with homotopy pull-back, then your question more or less comes down to asking when the Eilenberg-Moore spectral sequence collapses at $E^2$. $\endgroup$– user43326Commented Dec 21, 2022 at 15:53
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$\begingroup$ Thanks for your comment! I agree with you, the right notion is the one of homotopy pullback. If p is a fibration we have a homotopy pullback though. I've just came across this link in wikipedia which says that the Eilenberg-Moore spectral sequence collapses when the base space is simply connected. In this case I guess that everything works well en.wikipedia.org/wiki/Eilenberg%E2%80%93Moore_spectral_sequence $\endgroup$– Jose CalcinesCommented Dec 23, 2022 at 12:44
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