I'm stuck with the following question:
Consider the Postnikov tower $S^2 \to \ldots \to P_3S^2 \to P_2S^2 \to P_1S^2$ of the $2$-sphere.
Show that $P_2S^2$ and $\mathbb{C} P^\infty$ are weakly homotopy equivalent.
Show that the homotopy fiber $F$ of the resulting map $p_2 : S^2 \to \mathbb{C} P^\infty$ is weakly homotopy equivalent to $S^3$.
I have solved the first part, but not the second yet. This is what I have so far.
By the properties of the Postnikov tower, we have $\pi_k(P_2S^2) \cong 0$ for $k > 2$, and for $k \leq 2$, $$ \pi_k P_2S^2 \cong \pi_k S^2 \cong \begin{cases} \mathbb{Z}, & k = 2, \\ 0, & \text{else}. \end{cases} $$ Since we know that $\pi_k(\mathbb{C} P^\infty) \cong \mathbb{Z}$ for $k = 2$ and $0$ else, it follows that both spaces are $K(\mathbb{Z},2)$. Hence, the two spaces are weakly homotopy equivalent.
Recall that the homotopy fiber $F$ of the map $p_2 : S^2 \to \mathbb{C} P^\infty$ is given by the pullback
The hopf fibration $S^3 \to S^2$ induces a fiber bundle $S^1 \to S^3 \to S^2$. Now the homotopy fiber of the map $F \to S^2$ is homotopy equivalent to $\Omega (\mathbb{C} P^\infty)$. Hence, we get a fibration $\Omega(\mathbb{C} P^\infty) \to F \to S^2$. Since $\Omega(\mathbb{C} P^\infty)$ is a $K(\mathbb{Z}, 1)$, it is weakly homotopy to $S^1$. Now we compare the long exact sequences of these two fibrations, which gives
Then I want to use the five lemma to conclude that the middle map is an isomorphism as well. However, I don't know how to obtain such a map, i.e. I want a map $F \to S^3$ or vice versa making this diagram commute. Does anyone know how to proceed from here or has an alternative approach?