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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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 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, continuous maps, collection, open cover, projection, map, surjective, continuous, base, fiber, topological group, topological space
There are 15 references to this object.

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 4087 times total.

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