Timeline for Is surjectivity in singular homology stable under pullbacks?
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| Dec 23, 2022 at 12:44 | comment | added | Jose Calcines | 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 | |
| Dec 21, 2022 at 15:53 | comment | added | user43326 | 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 $E^2$. | |
| Dec 19, 2022 at 17:10 | comment | added | Ben Wieland | 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. | |
| Dec 19, 2022 at 12:10 | history | edited | Stefan Waldmann | CC BY-SA 4.0 |
LaTeX for math...
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| S Dec 19, 2022 at 11:16 | review | First questions | |||
| Dec 19, 2022 at 12:10 | |||||
| S Dec 19, 2022 at 11:16 | history | asked | Jose Calcines | CC BY-SA 4.0 |