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Via Hurewicz it is easy to see that the homotopy fiber of a weak equivalence has trivial homology. Via the Serre spectral sequence we can then see that the domain and codomain of our weak equivalence have the same homology. Can we deduce from the Serre spectral sequence that this isomorphism is actually coming from the weak equivalence?

Connor Malin
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    Yes, but for that you have to use the naturality of the Serre spectral sequence. This is often the case when you want to identify specific morphisms in a Serre spectral sequence that "should" be the right ones; you use naturality with "trivial fiber sequences". I don't have the time to write the details right now, but you should be able to get them. If you don't, I'll write something more detailed later. – Maxime Ramzi Oct 10 '19 at 08:43
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    Check that the local coefficients are trvial and that the Serre spectral sequence collapses at $E_2$. Then just use the edge homomorphisms to see that the weak equivalence induces isomorphisms on homology with any coefficients. Since the edge homorphisms can be defined using the naturality of the Sss, this is more or less the same method as Max suggests. – Tyrone Oct 10 '19 at 10:12

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