Editing Fiber bundle (section)

{{short description|Continuous surjection satisfying a local triviality condition}}
{{distinguish|text=an [[optical fiber bundle]]}}
{{Use American English|date=March 2021}}
{{Use mdy dates|date=March 2021}}

[[File:Roundhairbrush.JPG|thumb|A cylindrical [[hairbrush]] showing the intuition behind the term ''fiber bundle''. This hairbrush is like a fiber bundle in which the base space is a cylinder and the fibers ([[bristle]]s) are line segments. The mapping <math>\pi : E \to B</math> would take a point on any bristle and map it to its root on the cylinder.]]

In [[mathematics]], and particularly [[topology]], a '''fiber bundle''' ([[English in the Commonwealth of Nations|''Commonwealth English'']]: '''fibre bundle''') is a [[Space (mathematics)|space]] that is {{em|locally}} a [[product space]], but {{em|globally}} may have a different [[topological structure]]. Specifically, the similarity between a space <math>E</math> and a product space <math>B \times F</math> is defined using a [[Continuous function (topology)|continuous]] [[Surjective function|surjective]] [[Map (mathematics)|map]], <math>\pi : E \to B,</math> that in small regions of <math>E</math> behaves just like a projection from corresponding regions of <math>B \times F</math> to <math>B.</math>  The map <math>\pi,</math> called the '''[[Projection (mathematics)|projection]]''' or [[Submersion (mathematics)|'''submersion''']] of the bundle, is regarded as part of the structure of the bundle.  The space <math>E</math> is known as the '''total space''' of the fiber bundle, <math>B</math> as the '''base space''', and <math>F</math> the '''fiber'''.

In the ''[[Triviality (mathematics)|trivial]]'' case, <math>E</math> is just <math>B \times F,</math> and the map <math>\pi</math> 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 space]]s.  Fiber bundles, such as the [[tangent bundle]] of a [[manifold]] and other more general [[vector bundle]]s, play an important role in [[differential geometry]] and [[differential topology]], as do [[principal bundle]]s.

Mappings between total spaces of fiber bundles that "commute" with the projection maps are known as [[bundle map]]s, and the [[Class (set theory)|class]] of fiber bundles forms a [[Category theory|category]] with respect to such mappings.  A bundle map from the base space itself (with the [[identity mapping]] as projection) to <math>E</math> is called a [[Section (fiber bundle)|section]] of <math>E.</math>  Fiber bundles can be specialized in a number of ways, the most common of which is requiring that the [[Transition map|transition maps]] between the local trivial patches lie in a certain [[topological group]], known as the '''structure group''', acting on the fiber <math>F</math>.