For context I am trying to reprove lemma 1.6 in Weibel's K-book chapter IV. The point about which I have question is the following. Suppose we have a fibration $F\to X\to Y$, with $F$ acyclic, then using the Serre spectral sequence we can see that the homology (with coefficient in a $\pi_1(Y)$ module M) of $X$ and of $Y$ are abstractly isomorphic.
My question is, is there any way to show that the isomorphism is given by the map $X\to Y$ of the fibration? I feel like this should be the case, but am not comfortable enough to prove it myself.
Thank you very much
Best Answer
In the Serre spectral sequence the edge homomorphism $$H_p(X;M)\to E^\infty_{p,0}\subset E^2_{p,0}=H_p(Y;M)$$ is the map induced by $X\to Y$.
If $F$ is acyclic then the Serre spectral sequence collapses at the second page, which has only a single column, so the edge homomorphism is an isomorphism.