principal bundle

Let E be a topological spaceMathworldPlanetmath on which a topological groupMathworldPlanetmath G acts continuously and freely. The map π:EE/G=B is called a principal bundleMathworldPlanetmath (or principal G-bundle) if the projection map π:EB is a locally trivial bundle.

Any principal bundle with a sectionPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath σ:BE is trivial, since the map ϕ:B×GE given by ϕ(b,g)=gσ(b) is an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. In particular, any G-bundle which is topologically trivial is also isomorphic to B×G as a G-space. Thus any local trivialization of π:EB as a topological bundle is an equivariant trivialization.

Title principal bundle
Canonical name PrincipalBundle
Date of creation 2013-03-22 13:07:18
Last modified on 2013-03-22 13:07:18
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 8
Author rmilson (146)
Entry type Definition
Classification msc 55R10
Defines principal G-bundle