Reference request: homotopy groups of $\mathbb{C}P^n$ in terms of homotopy groups of spheres?

Question

Could anyone work out/supply a reference to the computation of homotopy groups of complex projective space $\mathbb{C}P^n$ in terms of the homotopy groups of spheres?

Answer 1

There is the so-called Hopf fibration: $$U(1) = S^1 \hookrightarrow S^{2n+1} \twoheadrightarrow \mathbb{CP}^n.$$ To see it, view $S^{2n+1}$ as embedded as the unit sphere of $\mathbb{R}^{2n+2} \cong \mathbb{C}^{n+1}$, and project $z \in S^{2n+1}$ to the $\mathbb{C}$-linear subspace $[z] \in \mathbb{CP}^n$ it generates in $\mathbb{C}^{n+1}$ (recall that $\mathbb{CP}^n = (\mathbb{C}^{n+1} \setminus 0) /(z \sim \lambda z')_{\lambda \neq 0}$).

The fiber over $[z] \in \mathbb{CP}^n$ is $\{ z' \in S^{2n+1} \mid \exists \lambda \in \mathbb{C}^* \text{ s.t. } z' = \lambda z \}$; but since $|z| = |z'| = 1$, then $|\lambda| = 1$ so $\lambda$ is in the unit circle of $\mathbb{C}$. It's then a classical exercise to see that this defines a fiber bundle over $\mathbb{CP}^n$.

The long exact sequence of a fibration in homotopy is then: $$ \ldots \to \pi_k(S^1) \to \pi_k(S^{2n+1}) \to \pi_k(\mathbb{CP}^n) \to \pi_{k-1}(S^1) \to \ldots$$ Since $\pi_k(S^1) = 0$ for $k > 1$, it follows that $$\color{red}{\pi_k(\mathbb{CP}^n) = \pi_k(S^{2n+1}) \quad \text{for } k > 2.}$$ In low degrees, the long exact sequence becomes: $$0 \to \pi_2(S^{2n+1}) \to \pi_2(\mathbb{CP}^n) \to \underbrace{\pi_1(S^1)}_{\mathbb{Z}} \to \pi_1(S^{2n+1}) \to \pi_1(\mathbb{CP}^n) \to 0.$$ For $n \ge 1$, $\pi_1(S^{2n+1}) = \pi_2(S^{2n+1}) = 0$, and so finally: $$\color{red}{\pi_1(\mathbb{CP}^n) = 0, \quad \pi_2(\mathbb{CP}^n) = \mathbb{Z}.}$$