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I have a question, which might be very basic, but I don't know enough topology to answer.

Suppose you have a map of topological spaces (or homotopy types) $f : X \to Y$, with homotopy fibre given by $F$. We get an induced morphism on cohomology $f^{*} : H^{*}(Y) \to H^{*}(X)$. If $F$ is $n$-connected (for all homotopy fibres), can we thereby conclude that $f^{*}$ is an isomorphism up to degree $n$?

I know that this holds in the case that $f$ is locally a fibration between manifolds (for example a submersion).

Another case I have in mind is the truncation map $p : X \to K(\pi_{1}(X),1)$, which has homotopy fibre given by the universal cover $\tilde{X}$ of $X$. In this case we see that if $\tilde{X}$ is $n$-connected, then $H^{k}(\pi_{1}(X)) \cong H^{k}(X)$ for $k \leq n$, where the left cohomology group refers to group cohomology.

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    $\begingroup$ You may as well assume $Y$ is connected. Because the homotopy fiber of $f$ is $n$-connected, we see that $f$ is an isomorphism on $\pi_i$ for $i \leq n$ and a surjection for $i = n+1$. Thus we may inductively attach cells of dimension at least $n+2$ to $X$ to create a space $X'$ and an extension $f': X' \to Y$ which is an equivalence, so that we may replace $f$ by the inclusion $i: X \to X'$. In particular, $f$ is an isomorphism on $H^i$ for $i \leq n$ and an injection on $H^{n+1}$. $\endgroup$
    – mme
    Jun 8, 2019 at 3:00
  • $\begingroup$ Thanks! Would you happen to have any references for this? $\endgroup$ Jun 8, 2019 at 4:50
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    $\begingroup$ The classical argument for this goes as follows. We may assume $f$ is an inclusion by taking its mapping cylinder. The homotopy groups of the homotopy fiber are then the relative groups $\pi_i(Y,X)$. If these are $0$ for $i\leq n$ then $H_i(Y,X)=0$ for $i\leq n$ by the relative Hurewicz theorem. The relative universal coefficient theorem then gives $H^i(Y,X)=0$ for $i\leq n$, so from the long exact sequence we see that $f^*$ is an isomorphism on $H^i$ for $i<n$ and a surjection on $H^n$. $\endgroup$ Jun 9, 2019 at 11:33

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