Hopf bundle

Consider S34=2. The structureMathworldPlanetmath of 2 gives a map 2-{0}P1, the complex projective line by the natural projectionMathworldPlanetmath. Since P1 is homeomorphic to S2, by restrictionPlanetmathPlanetmathPlanetmath to S3, we get a map π:S3S2. We call this the Hopf bundle.

This is a principal S1-bundle (http://planetmath.org/PrincipalBundle), and a generatorPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of π3(S2). From the long exact sequence of the bundle (http://planetmath.org/LongExactSequenceLocallyTrivialBundle):


we get that πn(S3)πn(S2) for all n3. In particular, π3(S2)π3(S3).

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