Question about isomorphism given by the Serre Spectral sequence

Question

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

Answer 1

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.

Answer 2

$\require{AMScd}$ I think you can use the fact that the Serre spectral sequence converges naturally; here is a rough argument: Consider the following commutative diagram of fibrations

\begin{CD} F @>>> X @>f>> Y\\ @VcVV @VfVV @VV\text{id}V\\ \text{pt} @>>> Y @>>\text{id}> Y \end{CD}

This tells us that we have a "diagram" of converging spectral sequences as follows:

\begin{CD} H_p(Y; H_q(F;M)) @. \Longrightarrow @. H_{p+q}(X;M)\\ @VH_p(\text{id};H_q(c;M))VV @. @VVH_{p+q}(f;M)V\\ H_p(Y; H_q(\text{pt};M)) @. \Longrightarrow @. H_{p+q}(Y;M) \end{CD}

In particular, the left map is an isomorphism by acyclicity of $F$ and it induces isomorphisms on all further pages of the spectral sequence. Thus, the map $f_*$ on the right is an isomorphism as well. As a reference, you can use Hatcher's chapter on the Serre spectral sequence, pages 18ff.