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单词 FiberBundle
释义

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

\\xymatrixE\\ar[rr]ξ\\ar[dr]π&&E\\ar[dl]π&B&.

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

Titlefiber bundle
Canonical nameFiberBundle
Date of creation2013-03-22 13:07:06
Last modified on2013-03-22 13:07:06
Ownerbwebste (988)
Last modified bybwebste (988)
Numerical id10
Authorbwebste (988)
Entry typeDefinition
Classificationmsc 55R10
Synonymfibre bundle
Related topicReductionOfStructureGroup
Related topicSectionOfAFiberBundle
Related topicFibrationMathworldPlanetmath
Related topicFibration2
Related topicHomotopyLiftingProperty
Related topicSurfaceBundleOverTheCircle
Definestrivial bundle
Defineslocal trivializations
Definesstructure group
Definescocycle condition
Defineslocal trivialization
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更新时间:2025/5/4 12:09:31