It is well known that $\mathbb{C}P(n+1)$ is diffeomorphic to $U(n+1)/U(n) \times U(1)$.
Since $U(n) \times U(1)$ is a close Lie-subgroup of $U(n+1)$, the canonical projection $p:U(n+1) \to U(n+1)/U(n) \times U(1)$ is a fiber bundle with fiber $U(n) \times U(1)$.
So, for $n=1$, $\mathbb{C}P(2)$ is diffeomorphic to $U(2)/U(1) \times U(1)$.
From the long exact sequence for the fiber bundle $U(1) \times U(1) \to U(2) \to U(n+1)/U(n) \times U(1)$, we have that $\pi_n (\mathbb{C}P(2))$ is isomorphic to $\pi_n(U(2)) \cong \pi_n(S^3)$ for $n \geq 3$.
But it is well known that there is a fiber bundle $S^1 \to S^5 \to \mathbb{C}P(2)$ is a fiber bundle.
From the long exact sequence for the fiber bundle $S^1 \to S^5 \to \mathbb{C}P(2)$ we have that $\pi_n (\mathbb{C}P(2))$ is isomorphic to $\pi_n(S^5)$ for $n \geq 3$.
Now we have a contracdition.
What is false?
