Hopf bundle

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

This is a principal $S^1$-bundle (http://planetmath.org/PrincipalBundle), and a generator of ${\pi }_3(S^2)$. From the long exact sequence of the bundle (http://planetmath.org/LongExactSequenceLocallyTrivialBundle):

$$\mathrm{\cdots }{\pi }_n(S^1)\to {\pi }_n(S^3)\to {\pi }_n(S^2)\to \mathrm{\cdots }$$

we get that ${\pi }_n(S^3)\cong {\pi }_n(S^2)$ for all $n\ge 3$. In particular,${\pi }_3(S^2)\cong {\pi }_3(S^3)\cong \mathbb{Z}$.