Comparing cohomology using homotopy fibre

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.

The post Comparing cohomology using homotopy fibre appeared first on 100% Private Proxies - Fast, Anonymous, Quality, Unlimited USA Private Proxy!.