# Hopf bundle

Consider $S^{3}\subset\mathbb{R}^{4}=\mathbb{C}^{2}$. The structure of $\mathbb{C}^{2}$ gives a map $\mathbb{C}^{2}-\{0\}\to\mathbb{C}P^{1}$, the complex projective line by the natural projection. Since $\mathbb{C}P^{1}$ 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 (http://planetmath.org/PrincipalBundle), and a generator of $\pi_{3}(S^{2})$. From the long exact sequence of the bundle (http://planetmath.org/LongExactSequenceLocallyTrivialBundle):

 $\cdots\pi_{n}(S^{1})\to\pi_{n}(S^{3})\to\pi_{n}(S^{2})\to\cdots$

we get that $\pi_{n}(S^{3})\cong\pi_{n}(S^{2})$ for all $n\geq 3$. In particular, $\pi_{3}(S^{2})\cong\pi_{3}(S^{3})\cong\mathbb{Z}$.

Title Hopf bundle HopfBundle 2013-03-22 13:20:04 2013-03-22 13:20:04 bwebste (988) bwebste (988) 5 bwebste (988) Definition msc 55R25 Hopf fibration