Total space | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-29

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Fiber_bundle


00:01:55 1 History
00:03:18 2 Formal definition
00:05:17 3 Examples
00:05:26 3.1 Trivial bundle
00:05:35 3.2 Nontrivial bundles
00:05:56 3.2.1 Möbius strip
00:06:05 3.2.2 Klein bottle
00:07:38 3.3 Covering map
00:08:01 3.4 Vector and principal bundles
00:08:19 3.5 Sphere bundles
00:09:26 3.6 Mapping tori
00:10:29 3.7 Quotient spaces
00:10:54 4 Sections
00:13:03 5 Structure groups and transition functions
00:14:13 6 Bundle maps
00:18:31 7 Differentiable fiber bundles
00:23:13 8 Generalizations
00:24:08 9 See also
00:24:49 10 Notes
00:25:50 11 References



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SUMMARY
=======
In mathematics, and particularly topology, a fiber bundle (or, in British English, fibre bundle) is a space that is locally a product space, but globally may have a different topological structure. Specifically, the similarity between a space E and a product space 
  
    
      
        B
        ×
        F
      
    
    {\displaystyle B\times F}
   is defined using a continuous surjective map

  
    
      
        π
        :
        E
        →
        B
      
    
    {\displaystyle \pi \colon E\to B}
  that in small regions of E behaves just like a projection from corresponding regions of B × F to B.  The map π, called the projection or submersion of the bundle, is regarded as part of the structure of the bundle.  The space E is known as the total space of the fiber bundle, B as the base space, and F the fiber.
In the trivial case, E is just B × F, and the map π is just the projection from the product space to the first factor. This is called a trivial bundle. Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, as well as nontrivial covering spaces.  Fiber bundles such as the tangent bundle of a manifold and more general vector bundles play an important role in differential geometry and differential topology, as do principal bundles.
Mappings between total spaces of fiber bundles that "commute" with the projection maps are known as bundle maps, and the class of fiber bundles forms a category with respect to such mappings.  A bundle map from the base space itself (with the identity mapping as projection) to E is called a section of E.  Fiber bundles can be specialized in a number of ways, the most common of which is requiring that the transitions between the local trivial patches lie in a certain topological group, known as the structure group, acting on the fiber F.