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 fiber bundle (Definition)
 Let be a topological space and be a topological group which acts on on the left. A fiber bundle with fiber and structure group consists of the following data: a topological space called the base space, a space called the total space and a continuous surjective map called the projection of the bundle, an open cover of along with a collection of continuous maps called local trivializations and a collection of continuous maps called transition functions which satisfy the following properties the map given by is a homeomorphism for each , for all indices and , and for all indices and , . Readers familiar with Čech cohomology may recognize condition 3), it is often called the cocycle condition. Note, this imples that is the identity in for each , and . If the total space is homeomorphic to the product so that the bundle projection is essentially projection onto the first factor, then is called a trivial bundle. Some examples of fiber bundles are vector bundles and covering spaces. There is a notion of morphism of fiber bundles over the same base with the same structure group . Such a morphism is a -equivariant map , making the following diagram commute Thus we have a category of fiber bundles over a fixed base with fixed structure group.

"fiber bundle" is owned by bwebste. [ full author list (2) | owner history (1) ]
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See Also: reduction of structure group, section of a fiber bundle, fibration

 Other names: fibre bundle
 Also defines: trivial bundle, local trivializations, structure group, cocycle condition, local trivialization

 Attachments: reduction of structure group (Definition) by antonio

Cross-references: fixed, category, morphism, covering spaces, vector bundles, factor, homeomorphic, identity, cohomology, homeomorphism, properties, satisfy, transition functions, collection, open cover, projection, map, surjective, continuous, base, fiber, topological group, topological space
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This is version 7 of fiber bundle, born on 2002-10-31, modified 2003-06-24.
Object id is 3551, canonical name is FiberBundle.
Accessed 7802 times total.

Classification:
 AMS MSC: 55R10 (Algebraic topology :: Fiber spaces and bundles :: Fiber bundles)

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