locally trivial bundle

A locally trivial bundle is a continuous map π:EB of topological spacesMathworldPlanetmath such that the following conditions hold. First, each point xB must have a neighborhoodMathworldPlanetmathPlanetmath U such that the inverse image U~=π-1(U) is homeomorphicMathworldPlanetmath to U×π-1(x). Second, for some homeomorphism g:U~U×π-1(x), the diagram


must be commutative (http://planetmath.org/CommutativeDiagram).

Locally trivial bundles are useful because of their covering homotopy property and because each locally trivial bundle has an associated long exact sequence (http://planetmath.org/LongExactSequenceLocallyTrivialBundle) and Serre spectral sequence. Every fibre bundle is an example of a locally trivial bundle.

Title locally trivial bundle
Canonical name LocallyTrivialBundle
Date of creation 2013-03-22 13:15:01
Last modified on 2013-03-22 13:15:01
Owner mps (409)
Last modified by mps (409)
Numerical id 10
Author mps (409)
Entry type Definition
Classification msc 55R10
Related topic FibrationMathworldPlanetmath
Related topic Fibration2
Related topic HomotopyLiftingProperty