Editing Fiber bundle

{{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>.

== History ==
In [[topology]], the terms '''''fiber''''' (German: ''Faser'') and '''''fiber space''''' (''gefaserter Raum'') appeared for the first time in a paper by [[Herbert Seifert]] in 1933,<ref>{{cite journal|title=Topologie dreidimensionaler gefaserter Räume|first=Herbert|last= Seifert|author-link=Herbert Seifert| journal=[[Acta Mathematica]]| volume=60|year=1933|pages=147–238|doi=10.1007/bf02398271|doi-access=free}}</ref><ref>[https://projecteuclid.org/euclid.acta/1485887992 "Topologie Dreidimensionaler Gefaserter Räume"] on [[Project Euclid]].</ref><ref>{{Cite book |last=Seifert |first=H. |url=https://www.worldcat.org/oclc/5831391 |title=Seifert and Threlfall, A textbook of topology |date=1980 |publisher=Academic Press |others=W. Threlfall, Joan S. Birman, Julian Eisner |isbn=0-12-634850-2 |location=New York |oclc=5831391}}</ref> but his definitions are limited to a very special case. The main difference from the present day conception of a fiber space, however, was that for Seifert what is now called the '''base space''' (topological space) of a fiber (topological) space ''E'' was not part of the structure, but derived from it as a quotient space of ''E''. The first definition of '''fiber space''' was given by [[Hassler Whitney]] in 1935<ref>{{cite journal|title=Sphere spaces|first=Hassler|last= Whitney|author-link=Hassler Whitney| journal=[[Proceedings of the National Academy of Sciences of the United States of America]]|volume=21|issue=7|year=1935|pages=464–468|doi=10.1073/pnas.21.7.464|doi-access=free|pmid=16588001|pmc=1076627|bibcode=1935PNAS...21..464W}}</ref> under the name '''sphere space''', but in 1940 Whitney changed the name to '''sphere bundle'''.<ref>{{cite journal|title=On the theory of sphere bundles|first=Hassler|last= Whitney|author-link=Hassler Whitney|  journal= [[Proceedings of the National Academy of Sciences of the United States of America]] |volume=26|issue=2|year=1940|pages=148–153|doi=10.1073/pnas.26.2.148|pmid=16588328|pmc=1078023|bibcode=1940PNAS...26..148W|doi-access=free}}</ref>

The theory of fibered spaces, of which [[vector bundle]]s, [[principal bundle]]s, topological [[fibration]]s and [[fibered manifold]]s are a special case, is attributed to [[Herbert Seifert]], [[Heinz Hopf]], [[Jacques Feldbau]],<ref>{{cite journal|title=Sur la classification des espaces fibrés|first=Jacques| last=Feldbau|author-link=Jacques Feldbau|journal=[[Comptes rendus de l'Académie des Sciences]]|volume=208|year=1939|pages=1621–1623}}</ref> Whitney, [[Norman Steenrod]], [[Charles Ehresmann]],<ref>{{cite journal|title=Sur la théorie des espaces fibrés|first=Charles|last= Ehresmann|author-link=Charles Ehresmann |journal=Coll. Top. Alg. Paris|volume=C.N.R.S.|year=1947|pages=3–15}}</ref><ref>{{cite journal|title=Sur les espaces fibrés différentiables|first=Charles|last= Ehresmann|author-link=Charles Ehresmann | journal=[[Comptes rendus de l'Académie des Sciences]] |volume=224|year=1947|pages=1611–1612}}</ref><ref>{{cite journal|title=Les prolongements d'un espace fibré différentiable|first=Charles|last= Ehresmann|author-link=Charles Ehresmann |journal=[[Comptes rendus de l'Académie des Sciences]]|volume=240|year=1955|pages=1755–1757}}</ref> [[Jean-Pierre Serre]],<ref>{{cite journal|title=Homologie singulière des espaces fibrés. Applications|first=Jean-Pierre|last= Serre|author-link=Jean-Pierre Serre|journal=[[Annals of Mathematics]] |volume=54|issue=3|year=1951|pages=425–505|doi=10.2307/1969485|jstor=1969485}}</ref> and others.

Fiber bundles became their own object of study in the period 1935–1940. The first general definition appeared in the works of Whitney.<ref>See {{harvtxt|Steenrod|1951|loc=Preface}}</ref>

Whitney came to the general definition of a fiber bundle from his study of a more particular notion of a [[sphere bundle]],<ref>In his early works, Whitney referred to the sphere bundles as the "sphere-spaces". See, for example:
* {{cite journal
 | last = Whitney
 | first = Hassler
 | author-link = Hassler Whitney
 | title = Sphere spaces
 | journal = Proc. Natl. Acad. Sci.
 | volume = 21
 | issue = 7
 | pages = 462–468
 | date = 1935 
 | doi = 10.1073/pnas.21.7.464
 | pmid = 16588001
 | pmc = 1076627
 | bibcode = 1935PNAS...21..464W
 | doi-access = free
 }}
* {{cite journal
 | last = Whitney
 | first = Hassler
 | author-link = Hassler Whitney
 | title = Topological properties of differentiable manifolds
 | journal = Bull. Amer. Math. Soc.
 | volume = 43
 | issue = 12
 | pages = 785–805
 | date = 1937
 | doi=10.1090/s0002-9904-1937-06642-0
 | doi-access = free
 | url = https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-43/issue-12/Topological-properties-of-differentiable-manifolds/bams/1183500133.pdf
 }}
</ref> that is a fiber bundle whose fiber is a sphere of arbitrary [[dimension]].<ref>
{{cite journal
 | last = Whitney
 | first = Hassler
 | author-link = Hassler Whitney
 | title = On the theory of sphere bundles
 | journal = Proc. Natl. Acad. Sci.
 | volume = 26
 | issue = 2
 | pages = 148–153
 | date = 1940 
 | doi = 10.1073/pnas.26.2.148
 | pmid = 16588328
 | pmc = 1078023
 | bibcode = 1940PNAS...26..148W
 | url = http://www.pnas.org/content/pnas/26/2/148.full.pdf
 | doi-access = free
 }}</ref>

== Formal definition ==

A fiber bundle is a structure <math>(E,\, B,\, \pi,\, F),</math> where <math>E, B,</math> and <math>F</math> are [[topological spaces]] and <math>\pi : E \to B</math> is a [[Continuous (topology)|continuous]] [[surjection]] satisfying a ''local triviality'' condition outlined below. The space <math>B</math> is called the '''{{visible anchor|base space}}''' of the bundle, <math>E</math> the '''{{visible anchor|total space}}''', and <math>F</math> the '''{{visible anchor|fiber}}'''. The map <math>\pi</math> is called the '''{{visible anchor|projection map}}''' (or '''{{visible anchor|bundle projection}}'''). We shall assume in what follows that the base space <math>B</math> is [[Connected space|connected]].

We require that for every <math>x \in B</math>, there is an open [[Neighborhood (topology)|neighborhood]] <math>U \subseteq B</math> of <math>x</math> (which will be called a trivializing neighborhood) such that there is a [[homeomorphism]] <math>\varphi : \pi^{-1}(U) \to U \times F</math> (where <math>\pi^{-1}(U)</math> is given the [[subspace topology]], and <math>U \times F</math> is the product space) in such a way that <math>\pi</math> agrees with the projection onto the first factor. That is, the following diagram should [[Commutative diagram|commute]]:

[[image:Fibre bundle local trivial.svg|Local triviality condition|230px|center]]

where <math>\operatorname{proj}_1 : U \times F \to U</math> is the natural projection and <math>\varphi : \pi^{-1}(U) \to U \times F</math> is a homeomorphism. The [[Set (mathematics)|set]] of all <math>\left\{\left(U_i,\, \varphi_i\right)\right\}</math> is called a '''{{visible anchor|local trivialization}}''' of the bundle.

Thus for any <math>p \in B</math>, the [[preimage]] <math>\pi^{-1}(\{p\})</math> is homeomorphic to <math>F</math> (since this is true of <math>\operatorname{proj}_1^{-1}(\{p\})</math>) and is called the '''fiber over <math>p.</math>''' Every fiber bundle <math>\pi : E \to B</math> is an [[open map]], since projections of products are open maps. Therefore <math>B</math> carries the [[quotient topology]] determined by the map <math>\pi.</math>

A fiber bundle <math>(E,\, B,\, \pi,\, F)</math> is often denoted
{{NumBlk
|:| <math>\begin{matrix}
  {} \\
  F \longrightarrow E\ \xrightarrow{\,\ \pi\ }\ B \\
  {}
\end{matrix}</math>
 | {{EquationRef|1}}
}} that, in analogy with a [[short exact sequence]], indicates which space is the fiber, total space and base space, as well as the map from total to base space.

A '''{{visible anchor|smooth fiber bundle}}''' is a fiber bundle in the [[Category (mathematics)|category]] of [[smooth manifold]]s. That is, <math>E, B,</math> and <math>F</math> are required to be smooth manifolds and all the [[Function (mathematics)|functions]] above are required to be [[smooth map]]s.

== 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.

== Sections ==
{{main article|Section (fiber bundle)}}

A '''{{visible anchor|section}}''' (or '''cross section''') of a fiber bundle <math>\pi</math> is a continuous map <math>f : B \to E</math> such that <math>\pi(f(x)) = x</math> [[for all]] ''x'' in ''B''. Since bundles do not in general have globally defined sections, one of the purposes of the theory is to account for their existence. The [[obstruction theory|obstruction]] to the existence of a section can often be measured by a cohomology class, which leads to the theory of [[characteristic class]]es in [[algebraic topology]].

The most well-known example is the [[hairy ball theorem]], where the [[Euler class]] is the obstruction to the [[tangent bundle]] of the [[2-sphere]] having a nowhere vanishing section.

Often one would like to define sections only locally (especially when global sections do not exist). A '''local section''' of a fiber bundle is a continuous map <math>f : U \to E</math> where ''U'' is an [[open set]] in ''B'' and <math>\pi(f(x)) = x</math> for all ''x'' in ''U''. If <math>(U,\, \varphi)</math> is a local trivialization [[chart (mathematics)|chart]] then local sections always exist over ''U''. Such sections are in [[1-1 Correspondence|1-1 correspondence]] with continuous maps <math>U \to F</math>. Sections form a [[sheaf (mathematics)|sheaf]].

== Structure groups and transition functions ==
Fiber bundles often come with a [[Group (mathematics)|group]] of symmetries that describe the matching conditions between overlapping local trivialization charts. Specifically, let ''G'' be a [[topological group]] that [[Group action (mathematics)|acts]] continuously on the fiber space ''F'' on the left. We lose nothing if we require ''G'' to act [[Faithful action|faithfully]] on ''F'' so that it may be thought of as a group of [[homeomorphism]]s of ''F''. A '''''G''-[[Atlas (topology)|atlas]]''' for the bundle <math>(E, B, \pi, F)</math> is a set of local trivialization charts <math>\{(U_k,\, \varphi_k)\}</math> such that for any <math>\varphi_i,\varphi_j</math> for the overlapping charts <math>(U_i,\, \varphi_i)</math> and <math>(U_j,\, \varphi_j)</math> the function
<math display=block>\varphi_i\varphi_j^{-1} : \left(U_i \cap U_j\right) \times F \to \left(U_i \cap U_j\right) \times F</math>
is given by
<math display=block>\varphi_i\varphi_j^{-1}(x,\, \xi) = \left(x,\, t_{ij}(x)\xi\right)</math>
where <math>t_{ij} : U_i \cap U_j \to G</math> is a continuous map called a '''{{visible anchor|transition function}}'''. Two ''G''-atlases are equivalent if their union is also a ''G''-atlas. A '''''G''-bundle''' is a fiber bundle with an equivalence class of ''G''-atlases. The group ''G'' is called the '''{{visible anchor|structure group}}''' of the bundle; the analogous term in [[physics]] is [[gauge group]].

In the smooth category, a ''G''-bundle is a smooth fiber bundle where ''G'' is a [[Lie group]] and the corresponding action on ''F'' is smooth and the transition functions are all smooth maps.

The transition functions <math>t_{ij}</math> satisfy the following conditions
# <math>t_{ii}(x) = 1\,</math>
# <math>t_{ij}(x) = t_{ji}(x)^{-1}\,</math>
# <math>t_{ik}(x) = t_{ij}(x)t_{jk}(x).\,</math>

The third condition applies on triple overlaps ''U<sub>i</sub>'' ∩ ''U<sub>j</sub>'' ∩ ''U<sub>k</sub>'' and is called the '''[[Čech cohomology#Cocycle|cocycle]] condition''' (see [[Čech cohomology]]). The importance of this is that the transition functions determine the fiber bundle (if one assumes the Čech cocycle condition).

A [[principal bundle|principal ''G''-bundle]] is a ''G''-bundle where the fiber ''F'' is a [[principal homogeneous space]] for the left action of ''G'' itself (equivalently, one can specify that the action of ''G'' on the fiber ''F'' is free and transitive, i.e. [[Group action (mathematics)#Regular|regular]]). In this case, it is often a matter of convenience to identify ''F'' with ''G'' and so obtain a (right) action of ''G'' on the principal bundle.

== Bundle maps ==
{{Main article|Bundle map}}
It is useful to have notions of a mapping between two fiber bundles.  Suppose that ''M'' and ''N'' are base spaces, and <math>\pi_E : E \to M</math> and <math>\pi_F : F \to N</math> are fiber bundles over ''M'' and ''N'', respectively.  A '''{{visible anchor|bundle map}}''' or '''{{visible anchor|bundle morphism}}''' consists of a pair of continuous<ref>Depending on the category of spaces involved, the functions may be assumed to have properties other than continuity.  For instance, in the category of differentiable manifolds, the functions are assumed to be smooth.  In the category of algebraic varieties, they are regular morphisms.</ref> functions
<math display=block>\varphi : E \to F,\quad f : M \to N</math>
such that <math>\pi_F\circ \varphi = f \circ \pi_E.</math>  That is, the following diagram is [[Commutative diagram|commutative]]:
[[File:BundleMorphism-04.svg|150px|center]]

For fiber bundles with structure group ''G'' and whose total spaces are (right) ''G''-spaces (such as a principal bundle), bundle [[Morphism|morphisms]] are also required to be ''G''-[[equivariant]] on the fibers. This means that <math>\varphi : E \to F</math> is also ''G''-morphism from one ''G''-space to another, that is, <math>\varphi(xs) = \varphi(x)s</math> for all <math>x \in E</math> and <math>s \in G.</math>

In case the base spaces ''M'' and ''N'' coincide, then a bundle morphism over ''M'' from the fiber bundle <math>\pi_E : E \to M</math> to <math>\pi_F : F \to M</math> is a map <math>\varphi : E \to F</math> such that <math>\pi_E = \pi_F \circ \varphi.</math> This means that the bundle map <math>\varphi : E \to F</math> [[Cover (topology)|covers]] the identity of ''M''. That is, <math>f \equiv \mathrm{id}_{M}</math> and the following diagram commutes:
[[File:BundleMorphism-03.svg|120px|center]]

Assume that both <math>\pi_E : E \to M</math> and <math>\pi_F : F \to M</math> are defined over the same base space ''M''. A bundle [[isomorphism]] is a bundle map <math>(\varphi,\, f)</math> between <math>\pi_E : E \to M</math> and <math>\pi_F : F \to M</math> such that <math>f \equiv \mathrm{id}_M</math> and such that <math>\varphi</math> is also a homeomorphism.<ref> Or is, at least, invertible in the appropriate category; e.g., a diffeomorphism.</ref>

== Differentiable fiber bundles ==
In the category of [[differentiable manifold]]s, fiber bundles arise naturally as [[submersion (mathematics)|submersions]] of one manifold to another. Not every (differentiable) submersion <math>f : M \to N</math> from a differentiable manifold ''M'' to another differentiable manifold ''N'' gives rise to a differentiable fiber bundle. For one thing, the map must be surjective, and <math>(M, N, f)</math> is called a [[fibered manifold]]. However, this necessary condition is not quite sufficient, and there are a variety of sufficient conditions in common use.

If ''M'' and ''N'' are [[Compact space|compact]] and [[Connection (mathematics)|connected]], then any submersion <math>f : M \to N</math> gives rise to a fiber bundle in the sense that there is a fiber space ''F'' diffeomorphic to each of the fibers such that <math>(E, B, \pi, F) = (M, N, f, F)</math> is a fiber bundle.  (Surjectivity of <math>f</math> follows by the assumptions already given in this case.) More generally, the assumption of compactness can be relaxed if the submersion <math>f : M \to N</math> is assumed to be a surjective [[proper map]], meaning that <math>f^{-1}(K)</math> is compact for every compact [[subset]] ''K'' of ''N''. Another sufficient condition, due to {{harvtxt|Ehresmann|1951}}, is that if <math>f : M \to N</math> is a surjective [[Submersion (mathematics)|submersion]] with ''M'' and ''N'' [[differentiable manifold]]s such that the preimage <math>f^{-1}\{x\}</math> is compact and connected for all <math>x \in N,</math> then <math>f</math> admits a [[Compatible (algebra)|compatible]] fiber bundle structure {{harv|Michor|2008|loc=§17}}.

== Generalizations ==
* The notion of a [[bundle (mathematics)|bundle]] applies to many more categories in mathematics, at the expense of appropriately modifying the local triviality condition; cf. [[principal homogeneous space]] and [[torsor (algebraic geometry)]].
* In topology, a [[fibration]] is a mapping <math>\pi : E \to B</math> that has certain [[homotopy theory|homotopy-theoretic]] properties in common with fiber bundles.  Specifically, under mild technical assumptions a fiber bundle always has the [[homotopy lifting property]] or homotopy covering property (see {{harvtxt|Steenrod|1951|loc=11.7}} for details).  This is the defining property of a fibration.
* A section of a fiber bundle is a "function whose output [[Range of a function|range]] is continuously dependent on the input." This property is formally captured in the notion of [[dependent type]].

== See also ==
{{div col|colwidth=22em}}
* [[Affine bundle]]
* [[Algebra bundle]]
* [[Characteristic class]]
* [[Covering map]]
* [[Equivariant bundle]]
* [[Fibered manifold]]
* [[Fibration]]
* [[Gauge theory]]
* [[Hopf bundle]]
* [[I-bundle]]
* [[Natural bundle]]
* [[Principal bundle]]
* [[Projective bundle]]
* [[Pullback bundle]]
* [[Quasifibration]]
* [[Universal bundle]]
* [[Vector bundle]]
* [[Wu–Yang dictionary]]
{{div col end}}

== Notes ==
<references/>

== References ==
* {{citation|first=Norman|last=Steenrod|author-link=Norman Steenrod|title=The Topology of Fibre Bundles|publisher=Princeton University Press|year=1951|isbn=978-0-691-08055-0}}
* {{Steenrod The Topology of Fibre Bundles 1999}} <!-- {{sfn|Steenrod|1999|p=}} -->
* {{citation|first=David|last=Bleecker|title=Gauge Theory and Variational Principles|publisher=Addison-Wesley publishing|location=Reading, Mass|year=1981|isbn=978-0-201-10096-9|url-access=registration|url=https://archive.org/details/gaugetheoryvaria00blee_0}}
* {{cite conference | first = Charles | last = Ehresmann | author-link=Charles Ehresmann|title = Les connexions infinitésimales dans un espace fibré différentiable | book-title = Colloque de Topologie (Espaces fibrés), Bruxelles, 1950 |publisher = Georges Thone, Liège; Masson et Cie. |location=Paris |date=1951 | pages = 29–55}}
* {{citation|first=Dale|last=Husemoller|author-link=Dale Husemoller| title=Fibre Bundles|publisher=Springer Verlag|year=1994|isbn=978-0-387-94087-8}}
* {{citation|first=Peter W.|last=Michor|title=Topics in Differential Geometry|series=[[Graduate Studies in Mathematics]]|volume=93|publisher=American Mathematical Society|location=Providence|year=2008|isbn=978-0-8218-2003-2}}
* {{springer|first=M.I.|last=Voitsekhovskii|id=F/f040060|title=Fibre space|year=2001}}

== External links ==
* [https://web.archive.org/web/20040808115056/http://planetmath.org/encyclopedia/FiberBundle.html Fiber Bundle], PlanetMath
* {{MathWorld|urlname=FiberBundle|title=Fiber Bundle|author=Rowland, Todd}}
* [http://www.popmath.org.uk/sculpmath/pagesm/fibundle.html Making John Robinson's Symbolic Sculpture `Eternity']
* [[Gennadi Sardanashvily|Sardanashvily, Gennadi]], Fibre bundles, jet manifolds and Lagrangian theory. Lectures for theoreticians, {{arXiv|0908.1886}}

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[[Category:Fiber bundles| ]]