It is well known that for the Postnikov tower for $S^2$, $\mathbb{C}P^{\infty}$ and $P_2S^2$ are weakly homotopy equivalent. Since they are CW complexes, they are actually homotopy equivalent. Now I want to show that the homotopy fibre of this homotopy equivalence composed with the inclusion of $S^2$ into $P_2S^2$ a map $f :S^2 \to \mathbb{C}P^{\infty}$ is weakly homotopy equivalent to $S^3$.
My idea was to use the Hopf fibration and the five lemma. Namely, since the fibration $P(\mathbb{C}P^{\infty})\to\mathbb{C}P^{\infty} $ from the path space(paths starting at the base point) has fiber $\Omega(\mathbb{C}P^{\infty} )$ we also has a fibration $\Omega(\mathbb{C}P^{\infty} ) \to \text{hofib}(f) \to S^2$ where $\text{hofib}(f)$ is the pullback of the diagram $S^2\to \mathbb{C}P^{\infty}\leftarrow P(\mathbb{C}P^{\infty})$. But $ \Omega(\mathbb{C}P^{\infty})$ is weakly homotopy equivalent to $S^1$ so if I can construct a map from $S^3$ to $\text{hofib}(f)$ that induces homotopy group isomorphism on its fibers and the base space, we can conclude from the five lemma that this induces an isomorphism between the homotopy groups of $S^3$ and $\text{hofib}(f)$. I have thought long about it and I think a possible candidate is the map induced(since the homotopy fibre is a pullback) by the Hopf map $\eta$ and the homotopy form $f \circ \eta$ to the constant map. However, I have no idea how this restrict to a weak homotopy equivalence on $S^1$.
