Editing Fiber bundle (section)

== Examples ==

=== Trivial bundle ===

Let <math>E = B \times F</math> and let <math>\pi : E \to B</math> be the projection onto the first factor. Then <math>\pi</math> is a fiber bundle (of <math>F</math>) over <math>B.</math> Here <math>E</math> is not just locally a product but ''globally'' one. Any such fiber bundle is called a '''{{visible anchor|trivial bundle}}'''. Any fiber bundle over a [[Contractible space|contractible]] [[CW-complex]] is trivial.

=== Nontrivial bundles ===

==== Möbius strip ====

[[File:Moebius Surface 1 Display Small.png|thumb|right|The Möbius strip is a nontrivial bundle over the circle.]]
Perhaps the simplest example of a nontrivial bundle <math>E</math> is the [[Möbius strip]]. It has the [[circle]] that runs lengthwise along the center of the strip as a base <math>B</math> and a [[line segment]] for the fiber <math>F</math>, so the Möbius strip is a bundle of the line segment over the circle. A [[neighbourhood (mathematics)|neighborhood]] <math>U</math> of <math>\pi(x) \in B</math> (where <math>x \in E</math>) is an [[circular arc|arc]]; in the picture, this is the [[length]] of one of the squares. The [[image (mathematics)|preimage]] <math>\pi^{-1}(U)</math> in the picture is a (somewhat twisted) slice of the strip four squares wide and one long (i.e. all the points that project to <math>U</math>).

A homeomorphism (<math>\varphi</math> in {{sectionlink||Formal definition}}) exists that maps the preimage of <math>U</math> (the trivializing neighborhood) to a slice of a cylinder: curved, but not twisted. This pair locally trivializes the strip. The corresponding trivial bundle <math>B\times F</math> would be a [[cylinder (geometry)|cylinder]], but the Möbius strip has an overall "twist". This twist is visible only globally; locally the Möbius strip and the cylinder are identical (making a single vertical cut in either gives the same space).

==== Klein bottle ====

A similar nontrivial bundle is the [[Klein bottle]], which can be viewed as a "twisted" circle bundle over another circle. The corresponding non-twisted (trivial) bundle is the 2-[[torus]], <math>S^1 \times S^1</math>.

{|
|[[File:KleinBottle-01.png|thumb|120px|The Klein bottle [[Immersion (mathematics)|immersed]] in three-dimensional space.]]
|[[File:Torus.png|thumb|160px|A torus.]]
|}

=== Covering map ===

A '''[[covering map|covering space]]''' is a fiber bundle such that the bundle projection is a [[local homeomorphism]]. It follows that the fiber is a [[discrete space]].

=== Vector and principal bundles ===

A special class of fiber bundles, called '''[[vector bundle]]s''', are those whose fibers are [[vector space]]s (to qualify as a vector bundle the structure group of the bundle — see below — must be a [[general linear group|linear group]]). Important examples of vector bundles include the [[tangent bundle]] and [[cotangent bundle]] of a smooth manifold. From any vector bundle, one can construct the [[frame bundle]] of [[basis (mathematics)|bases]], which is a principal bundle (see below).

Another special class of fiber bundles, called '''[[principal bundle]]s''', are bundles on whose fibers a [[free action|free]] and [[transitive action|transitive]] [[Group action (mathematics)|action]] by a group <math>G</math> is given, so that each fiber is a [[principal homogeneous space]]. The bundle is often specified along with the group by referring to it as a principal <math>G</math>-bundle. The group <math>G</math> is also the structure group of the bundle. Given a [[group representation|representation]] <math>\rho</math> of <math>G</math> on a vector space <math>V</math>, a vector bundle with <math>\rho(G) \subseteq \text{Aut}(V)</math> as a structure group may be constructed, known as the [[associated bundle]].

=== Sphere bundles ===
{{main|Sphere bundle}}
A '''sphere bundle''' is a fiber bundle whose fiber is an [[hypersphere|''n''-sphere]]. Given a vector bundle <math>E</math> with a [[metric tensor|metric]] (such as the tangent bundle to a [[Riemannian manifold]]) one can construct the associated '''unit sphere bundle''', for which the fiber over a point <math>x</math> is the set of all [[Unit vector|unit vectors]] in <math>E_x</math>. When the vector bundle in question is the tangent bundle <math>TM</math>, the unit sphere bundle is known as the '''[[unit tangent bundle]]'''. {{clear}}

A sphere bundle is partially characterized by its [[Euler class]], which is a degree <math>n + 1</math> [[cohomology]] class in the total space of the bundle. In the case <math>n = 1</math> the sphere bundle is called a [[circle bundle]] and the Euler class is equal to the first [[Chern class]], which characterizes the topology of the bundle completely. For any <math>n</math>, given the Euler class of a bundle, one can calculate its cohomology using a [[long exact sequence]] called the [[Gysin sequence]].

{{See also|Wang sequence}}

=== Mapping tori ===
If <math>X</math> is a [[topological space]] and <math>f : X \to X</math> is a [[homeomorphism]] then the [[mapping torus]] <math>M_f</math> has a natural structure of a fiber bundle over the [[circle]] with fiber <math>X.</math> Mapping tori of homeomorphisms of [[Surface (topology)|surfaces]] are of particular importance in [[3-manifold|3-manifold topology]].

=== Quotient spaces ===
If <math>G</math> is a [[topological group]] and <math>H</math> is a [[closed subgroup]], then under some circumstances, the [[Quotient space (topology)|quotient space]] <math>G/H</math> together with the quotient map <math>\pi : G \to G/H</math> is a fiber bundle, whose fiber is the topological space <math>H</math>. A [[necessary and sufficient condition]] for (<math>G,\, G/H,\, \pi,\, H</math>) to form a fiber bundle is that the mapping <math>\pi</math> admits [[#Sections|local cross-sections]] {{harv|Steenrod|1951|loc=§7}}.

The most general conditions under which the [[quotient map]] will admit local cross-sections are not known, although if <math>G</math> is a [[Lie group]] and <math>H</math> a closed subgroup (and thus a [[Lie subgroup]] by [[Closed subgroup theorem|Cartan's theorem]]), then the quotient map is a fiber bundle. One example of this is the [[Hopf fibration]], <math>S^3 \to S^2</math>, which is a fiber bundle over the sphere <math>S^2</math> whose total space is <math>S^3</math>. From the perspective of Lie groups, <math>S^3</math> can be identified with the [[special unitary group]] <math>SU(2)</math>. The [[Abelian group|abelian]] subgroup of [[diagonal matrices]] is [[Isomorphic group|isomorphic]] to the [[circle group]] <math>U(1)</math>, and the quotient <math>SU(2)/U(1)</math> is [[Diffeomorphism|diffeomorphic]] to the sphere.

More generally, if <math>G</math> is any topological group and <math>H</math> a closed subgroup that also happens to be a Lie group, then <math>G \to G/H</math> is a fiber bundle.