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
Canonical name	HopfBundle
Date of creation	2013-03-22 13:20:04
Last modified on	2013-03-22 13:20:04
Owner	bwebste (988)
Last modified by	bwebste (988)
Numerical id	5
Author	bwebste (988)
Entry type	Definition
Classification	msc 55R25
Synonym	Hopf fibration