fiber bundle

Let F be a topological spaceMathworldPlanetmath and G be a topological groupMathworldPlanetmath which acts on F on the left. A fiber bundleMathworldPlanetmath with fiber F and structure group G consists of the following data:

  • a topological space B called the base space, a space E called the total space and a continuousPlanetmathPlanetmath surjective map π:EB called the projection of the bundle,

  • an open cover {Ui} of B along with a collectionMathworldPlanetmath of continuous maps {ϕi:π-1UiF} called local trivializations and

  • a collection of continuous maps {gij:UiUjG} called transition functionsMathworldPlanetmath

which satisfy the following properties

  1. 1.

    the map π-1UiUi×F given by e(π(e),ϕi(e)) is a homeomorphism for each i,

  2. 2.

    for all indices i,j and eπ-1(UiUj), gji(π(e))ϕi(e)=ϕj(e) and

  3. 3.

    for all indices i,j,k and bUiUjUk, gij(b)gjk(b)=gik(b).

Readers familiar with Čech cohomology may recognize condition 3), it is often called the cocycle condition. Note, this imples that gii(b) is the identity in G for each b, and gij(b)=gji(b)-1.

If the total space E is homeomorphic to the productPlanetmathPlanetmathPlanetmath B×F so that the bundle projection is essentially projection onto the first factor, then π:EB is called a trivial bundle. Some examples of fiber bundles are vector bundlesMathworldPlanetmath and covering spaces.

There is a notion of morphism of fiber bundles E,E over the same base B with the same structure group G. Such a morphism is a G-equivariant map ξ:EE, making the following diagram commute


Thus we have a category of fiber bundles over a fixed base with fixed structure group.

Title fiber bundle
Canonical name FiberBundle
Date of creation 2013-03-22 13:07:06
Last modified on 2013-03-22 13:07:06
Owner bwebste (988)
Last modified by bwebste (988)
Numerical id 10
Author bwebste (988)
Entry type Definition
Classification msc 55R10
Synonym fibre bundle
Related topic ReductionOfStructureGroup
Related topic SectionOfAFiberBundle
Related topic FibrationMathworldPlanetmath
Related topic Fibration2
Related topic HomotopyLiftingProperty
Related topic SurfaceBundleOverTheCircle
Defines trivial bundle
Defines local trivializations
Defines structure group
Defines cocycle condition
Defines local trivialization