Hopf bundle

Consider $S^3\subset \R^4=\C^2$ . The structure of $\C^2$ gives a map $\C^2-\{0\}\to\C P^1$ , the complex projective line by the natural projection. Since $\C P^1$ is homeomorphic to $S^2$ , by restriction to $S^3$ , we get a map $\pi:S^3\to S^2$ . We call this the Hopf bundle.

This is a principal $S^1$ -bundle, and a generator of $\pi_3(S^2)$ . From the long exact sequence of the bundle: $$\cdots\pi_n(S^1)\to \pi_n(S^3)\to\pi_n(S^2)\to\cdots$$ we get that $\pi_n(S^3)\cong \pi_n(S^2)$ for all $n\geq 3$ . In particular, $\pi_3(S^2)\cong\pi_3(S^3)\cong\Z$ .