Topology MOC

Fibre bundle

Let be spaces in Category of topological spaces or an appropriate subcategory ,¹ which we call the base space and fibre space respectively. A fibre bundle²

is a “total space” equipped with a surjective morphism such that is locally the product space . topology This is formalized as follows: For every , there is an open neighbourhood of with an isomorphism

called a local trivialization at such that

commutes. An open cover of with local trivializations is called a local trivialization of .

Further terminology

• We denote thr fibre above a base point as .
• A bundle for which and is called trivial.
• A Bundle map over a fixed based space is a morphism of bundles, and we can thus form the category Category of fibre bundles.
• A Bundle section is a section (right-inverse) to , or equivalently a bundle map from to .

Further structure

• Vector bundle

Examples

• The Möbius strip is a bundle .
• The Klein bottle is a bundle .

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Footnotes

1. Often we take Category of manifolds or Holomorphic category. ↩

2. Despite the notation, which is chosen to resemble a short exact sequence, the morphism should not be taken too literally, since there exists an isomorphism for every fibre. ↩