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.