Inmathematics, and particularlytopology, afiber bundle (Commonwealth English:fibre bundle) is aspace that islocally aproduct space, butglobally may have a differenttopological structure. Specifically, the similarity between a space${\displaystyle E}$ and a product space${\displaystyle B\times F}$ is defined using acontinuoussurjectivemap,${\displaystyle \pi :E\to B,}$ that in small regions of${\displaystyle E}$ behaves just like a projection from corresponding regions of${\displaystyle B\times F}$ to${\displaystyle B.}$ The map${\displaystyle \pi ,}$ called theprojection orsubmersion of the bundle, is regarded as part of the structure of the bundle. The space${\displaystyle E}$ is known as thetotal space of the fiber bundle,${\displaystyle B}$ as thebase space, and${\displaystyle F}$ thefiber.

In thetrivial case,${\displaystyle E}$ is just${\displaystyle B\times F,}$ and the map${\displaystyle \pi }$ is just the projection from the product space to the first factor. This is called atrivial bundle. Examples of non-trivial fiber bundles include theMöbius strip andKlein bottle, as well as nontrivialcovering spaces. Fiber bundles, such as thetangent bundle of amanifold and other more generalvector bundles, play an important role indifferential geometry anddifferential topology, as doprincipal bundles.

Mappings between total spaces of fiber bundles that "commute" with the projection maps are known asbundle maps, and theclass of fiber bundles forms acategory with respect to such mappings. A bundle map from the base space itself (with theidentity mapping as projection) to${\displaystyle E}$ is called asection of${\displaystyle E.}$ Fiber bundles can be specialized in a number of ways, the most common of which is requiring that thetransition maps between the local trivial patches lie in a certaintopological group, known as thestructure group, acting on the fiber${\displaystyle F}$.

Intopology, the termsfiber (German:Faser) andfiber space (gefaserter Raum) appeared for the first time in a paper byHerbert Seifert in 1933,^[1][2][3] 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 thebase space (topological space) of a fiber (topological) spaceE was not part of the structure, but derived from it as a quotient space ofE. The first definition offiber space was given byHassler Whitney in 1935^[4] under the namesphere space, but in 1940 Whitney changed the name tosphere bundle.^[5]

The theory of fibered spaces, of whichvector bundles,principal bundles, topologicalfibrations andfibered manifolds are a special case, is attributed toHerbert Seifert,Heinz Hopf,Jacques Feldbau,^[6] Whitney,Norman Steenrod,Charles Ehresmann,^[7][8][9]Jean-Pierre Serre,^[10] 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.^[11]

Whitney came to the general definition of a fiber bundle from his study of a more particular notion of asphere bundle,^[12] that is a fiber bundle whose fiber is a sphere of arbitrarydimension.^[13]

A fiber bundle is a 4-tuple${\displaystyle (E,B,\pi ,F),}$ where${\displaystyle E,B,}$ and${\displaystyle F}$ aretopological spaces and${\displaystyle \pi :E\to B}$ is acontinuoussurjection satisfying alocal triviality condition outlined below. The space${\displaystyle B}$ is called thebase space of the bundle,${\displaystyle E}$ thetotal space, and${\displaystyle F}$ thefiber. The map${\displaystyle \pi }$ is called theprojection map (orbundle projection). We shall assume in what follows that the base space${\displaystyle B}$ isconnected.

We require that for every${\displaystyle x\in B}$, there is an openneighborhood${\displaystyle U\subseteq B}$ of${\displaystyle x}$ (which will be called a trivializing neighborhood) such that there is ahomeomorphism${\displaystyle \phi :{\pi }^{-1}(U)\to U\times F}$ (where${\displaystyle {\pi }^{-1}(U)}$ is given thesubspace topology, and${\displaystyle U\times F}$ is the product space) in such a way that${\displaystyle \pi }$ agrees with the projection onto the first factor. That is, the following diagram shouldcommute:

where${\displaystyle {\mathrm{proj}}_1:U\times F\to U}$ is the natural projection and${\displaystyle \phi :{\pi }^{-1}(U)\to U\times F}$ is a homeomorphism. Theset of all${\displaystyle \left\{\left(U_i,{\phi }_i\right)\right\}}$ is called alocal trivialization of the bundle.

Thus for any${\displaystyle p\in B}$, thepreimage${\displaystyle {\pi }^{-1}(\{p\})}$ is homeomorphic to${\displaystyle F}$ (since this is true of${\displaystyle {\mathrm{proj}}_1^{-1}(\{p\})}$) and is called thefiber over${\displaystyle p}$. Every fiber bundle${\displaystyle \pi :E\to B}$ is anopen map, since projections of products are open maps. Therefore${\displaystyle B}$ carries thequotient topology determined by the map${\displaystyle \pi .}$

A fiber bundle${\displaystyle (E,B,\pi ,F)}$ is often denoted$${\displaystyle \begin{array}{c}\\ F⟶E\stackrel{\pi }{\to }B\\ \end{array}}$$that, in analogy with ashort exact sequence, indicates which space is the fiber, total space and base space, as well as the map from total to base space.

Asmooth fiber bundle is a fiber bundle in thecategory ofsmooth manifolds. That is,${\displaystyle E}$,${\displaystyle B}$, and${\displaystyle F}$ are required to be smooth manifolds and all thefunctions above are required to besmooth maps.

Let${\displaystyle E=B\times F}$ and let${\displaystyle \pi :E\to B}$ be the projection onto the first factor. Then${\displaystyle \pi }$ is a fiber bundle (of${\displaystyle F}$) over${\displaystyle B.}$ Here${\displaystyle E}$ is not just locally a product butglobally one. Any such fiber bundle is called atrivial bundle. Any fiber bundle over acontractibleCW-complex is trivial.

Perhaps the simplest example of a nontrivial bundle${\displaystyle E}$ is theMöbius strip. It has thecircle that runs lengthwise along the center of the strip as a base${\displaystyle B}$ and aline segment for the fiber${\displaystyle F}$, so the Möbius strip is a bundle of the line segment over the circle. Aneighborhood${\displaystyle U}$ of${\displaystyle \pi (x)\in B}$ (where${\displaystyle x\in E}$) is anarc; in the picture, this is thelength of one of the squares. Thepreimage${\displaystyle {\pi }^{-1}(U)}$ 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${\displaystyle U}$).

A homeomorphism (${\displaystyle \phi }$ in§ Formal definition) exists that maps the preimage of${\displaystyle U}$ (the trivializing neighborhood) to a slice of a cylinder: curved, but not twisted. This pair locally trivializes the strip. The corresponding trivial bundle${\displaystyle B\times F}$ would be acylinder, 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).

A similar nontrivial bundle is theKlein bottle, which can be viewed as a "twisted" circle bundle over another circle. The corresponding non-twisted (trivial) bundle is the 2-torus,${\displaystyle S^1\times S^1}$.

Acovering space is a fiber bundle such that the bundle projection is alocal homeomorphism. It follows that the fiber is adiscrete space.

A special class of fiber bundles, calledvector bundles, are those whose fibers arevector spaces (to qualify as a vector bundle the structure group of the bundle—see below—must be alinear group). Important examples of vector bundles include thetangent bundle andcotangent bundle of a smooth manifold. From any vector bundle, one can construct theframe bundle ofbases, which is a principal bundle (see below).

Another special class of fiber bundles, calledprincipal bundles, are bundles on whose fibers afree andtransitiveaction by a group${\displaystyle G}$ is given, so that each fiber is aprincipal homogeneous space. The bundle is often specified along with the group by referring to it as a principal${\displaystyle G}$-bundle. The group${\displaystyle G}$ is also the structure group of the bundle. Given arepresentation${\displaystyle \rho }$ of${\displaystyle G}$ on a vector space${\displaystyle V}$, a vector bundle with${\displaystyle \rho (G)\subseteq \text{Aut}(V)}$ as a structure group may be constructed, known as theassociated bundle.

Asphere bundle is a fiber bundle whose fiber is ann-sphere. Given a vector bundle${\displaystyle E}$ with ametric (such as the tangent bundle to aRiemannian manifold) one can construct the associatedunit sphere bundle, for which the fiber over a point${\displaystyle x}$ is the set of allunit vectors in${\displaystyle E_x}$. When the vector bundle in question is the tangent bundle${\displaystyle TM}$, the unit sphere bundle is known as theunit tangent bundle.

A sphere bundle is partially characterized by itsEuler class, which is a degree${\displaystyle n+1}$cohomology class in the total space of the bundle. In the case${\displaystyle n=1}$ the sphere bundle is called acircle bundle and the Euler class is equal to the firstChern class, which characterizes the topology of the bundle completely. For any${\displaystyle n}$, given the Euler class of a bundle, one can calculate its cohomology using along exact sequence called theGysin sequence.

If${\displaystyle X}$ is atopological space and${\displaystyle f:X\to X}$ is ahomeomorphism then themapping torus${\displaystyle M_f}$ has a natural structure of a fiber bundle over thecircle with fiber${\displaystyle X.}$ Mapping tori of homeomorphisms ofsurfaces are of particular importance in3-manifold topology.

If${\displaystyle G}$ is atopological group and${\displaystyle H}$ is aclosed subgroup, then under some circumstances, thequotient space${\displaystyle G/H}$ together with the quotient map${\displaystyle \pi :G\to G/H}$ is a fiber bundle, whose fiber is the topological space${\displaystyle H}$. Anecessary and sufficient condition for (${\displaystyle G,G/H,\pi ,H}$) to form a fiber bundle is that the mapping${\displaystyle \pi }$ admitslocal cross-sections (Steenrod 1951, §7).

The most general conditions under which thequotient map will admit local cross-sections are not known, although if${\displaystyle G}$ is aLie group and${\displaystyle H}$ a closed subgroup (and thus aLie subgroup byCartan's theorem), then the quotient map is a fiber bundle. One example of this is theHopf fibration,${\displaystyle S^3\to S^2}$, which is a fiber bundle over the sphere${\displaystyle S^2}$ whose total space is${\displaystyle S^3}$. From the perspective of Lie groups,${\displaystyle S^3}$ can be identified with thespecial unitary group${\displaystyle SU(2)}$. Theabelian subgroup ofdiagonal matrices isisomorphic to thecircle group${\displaystyle U(1)}$, and the quotient${\displaystyle SU(2)/U(1)}$ isdiffeomorphic to the sphere.

More generally, if${\displaystyle G}$ is any topological group and${\displaystyle H}$ a closed subgroup that also happens to be a Lie group, then${\displaystyle G\to G/H}$ is a fiber bundle.

Asection (orcross section) of a fiber bundle${\displaystyle \pi }$ is a continuous map${\displaystyle f:B\to E}$ such that${\displaystyle \pi (f(x))=x}$for allx inB. Since bundles do not in general have globally defined sections, one of the purposes of the theory is to account for their existence. Theobstruction to the existence of a section can often be measured by a cohomology class, which leads to the theory ofcharacteristic classes inalgebraic topology.

The most well-known example is thehairy ball theorem, where theEuler class is the obstruction to thetangent bundle of the2-sphere having a nowhere vanishing section.

Often one would like to define sections only locally (especially when global sections do not exist). Alocal section of a fiber bundle is a continuous map${\displaystyle f:U\to E}$ whereU is anopen set inB and${\displaystyle \pi (f(x))=x}$ for allx inU. If${\displaystyle (U,\phi )}$ is a local trivializationchart then local sections always exist overU. Such sections are in1-1 correspondence with continuous maps${\displaystyle U\to F}$. Sections form asheaf.

Fiber bundles often come with agroup of symmetries that describe the matching conditions between overlapping local trivialization charts. Specifically, letG be atopological group thatacts continuously on the fiber spaceF on the left. We lose nothing if we requireG to actfaithfully onF so that it may be thought of as a group ofhomeomorphisms ofF. AG-atlas for the bundle${\displaystyle (E,B,\pi ,F)}$ is a set of local trivialization charts${\displaystyle \{(U_k,{\phi }_k)\}}$ such that for any${\displaystyle {\phi }_i,{\phi }_j}$ for the overlapping charts${\displaystyle (U_i,{\phi }_i)}$ and${\displaystyle (U_j,{\phi }_j)}$ the function$${\displaystyle {\phi }_i{\phi }_j^{-1}:\left(U_i\cap U_j\right)\times F\to \left(U_i\cap U_j\right)\times F}$$is given by$${\displaystyle {\phi }_i{\phi }_j^{-1}(x,\xi )=\left(x,t_{ij}(x)\xi \right)}$$where${\displaystyle t_{ij}:U_i\cap U_j\to G}$ is a continuous map called atransition function. TwoG-atlases are equivalent if their union is also aG-atlas. AG-bundle is a fiber bundle with an equivalence class ofG-atlases. The groupG is called thestructure group of the bundle. This is related to thegauge group inphysics, which is the group of automorphisms of the principalG-bundle that leave the base space unchanged.

In the smooth category, aG-bundle is a smooth fiber bundle whereG is aLie group and the corresponding action onF is smooth and the transition functions are all smooth maps.

The transition functions${\displaystyle t_{ij}}$ satisfy the following conditions

1. ${\displaystyle t_{ii}(x)=1}$
2. ${\displaystyle t_{ij}(x)=t_{ji}(x{)}^{-1}}$
3. ${\displaystyle t_{ik}(x)=t_{ij}(x)t_{jk}(x).}$

The third condition applies on triple overlapsU_i ∩U_j ∩U_k and is called the (Čech)cocycle condition (see alsoČech cohomology). The importance of this is that the transition functions determine the fiber bundle (if one assumes the cocycle condition).

AprincipalG-bundle is aG-bundle where the fiberF is aprincipal homogeneous space for the left action ofG itself (equivalently, one can specify that the action ofG on the fiberF is free and transitive, i.e.regular). In this case, it is often a matter of convenience to identifyF withG and so obtain a (right) action ofG on the principal bundle.

It is useful to have notions of a mapping between two fiber bundles. Suppose that${\displaystyle M}$ and${\displaystyle N}$ are base spaces, and${\displaystyle {\pi }_E:E\to M}$ and${\displaystyle {\pi }_F:F\to N}$ are fiber bundles over${\displaystyle M}$ and${\displaystyle N}$, respectively. Abundle map orbundle morphism consists of a pair of continuous^[14] functions$${\displaystyle \phi :E\to F,f:M\to N}$$such that${\displaystyle {\pi }_F\circ \phi =f\circ {\pi }_E}$. That is, the following diagram iscommutative:

For fiber bundles with structure group${\displaystyle G}$ and whose total spaces are (right)${\displaystyle G}$-spaces (such as a principal bundle), bundlemorphisms are also required to be${\displaystyle G}$-equivariant on the fibers. This means that${\displaystyle \phi :E\to F}$ is also${\displaystyle G}$-morphism from one${\displaystyle G}$-space to another, that is,${\displaystyle \phi (xs)=\phi (x)s}$ for all${\displaystyle x\in E}$ and${\displaystyle s\in G}$.

In case the base spaces${\displaystyle M}$ and${\displaystyle N}$ coincide, then a bundle morphism over${\displaystyle M}$ from the fiber bundle${\displaystyle {\pi }_E:E\to M}$ to${\displaystyle {\pi }_F:F\to M}$ is a map${\displaystyle \phi :E\to F}$ such that${\displaystyle {\pi }_E={\pi }_F\circ \phi }$. This means that the bundle map${\displaystyle \phi :E\to F}$covers the identity of${\displaystyle M}$. That is,${\displaystyle f\equiv {\mathrm{i}\mathrm{d}}_M}$ and the following diagram commutes:

Assume that both${\displaystyle {\pi }_E:E\to M}$ and${\displaystyle {\pi }_F:F\to M}$ are defined over the same base space${\displaystyle M}$. A bundleisomorphism is a bundle map${\displaystyle (\phi ,f)}$ between${\displaystyle {\pi }_E:E\to M}$ and${\displaystyle {\pi }_F:F\to M}$ such that${\displaystyle f\equiv {\mathrm{i}\mathrm{d}}_M}$ and such that${\displaystyle \phi }$ is also a homeomorphism.^[15]

In the category ofdifferentiable manifolds, fiber bundles arise naturally assubmersions of one manifold to another. Not every (differentiable) submersion${\displaystyle f:M\to N}$ from a differentiable manifoldM to another differentiable manifoldN gives rise to a differentiable fiber bundle. For one thing, the map must be surjective, and${\displaystyle (M,N,f)}$ is called afibered manifold. However, this necessary condition is not quite sufficient, and there are a variety of sufficient conditions in common use.

IfM andN arecompact andconnected, then any submersion${\displaystyle f:M\to N}$ gives rise to a fiber bundle in the sense that there is a fiber spaceF diffeomorphic to each of the fibers such that${\displaystyle (E,B,\pi ,F)=(M,N,f,F)}$ is a fiber bundle. (Surjectivity of${\displaystyle f}$ follows by the assumptions already given in this case.) More generally, the assumption of compactness can be relaxed if the submersion${\displaystyle f:M\to N}$ is assumed to be a surjectiveproper map, meaning that${\displaystyle f^{-1}(K)}$ is compact for every compactsubsetK ofN. Another sufficient condition, due toEhresmann (1951), is that if${\displaystyle f:M\to N}$ is a surjectivesubmersion withM andNdifferentiable manifolds such that the preimage${\displaystyle f^{-1}\{x\}}$ is compact and connected for all${\displaystyle x\in N,}$ then${\displaystyle f}$ admits acompatible fiber bundle structure (Michor 2008, §17).

• The notion of abundle applies to many more categories in mathematics, at the expense of appropriately modifying the local triviality condition; cf.principal homogeneous space andtorsor (algebraic geometry).
• In topology, afibration is a mapping${\displaystyle \pi :E\to B}$ that has certainhomotopy-theoretic properties in common with fiber bundles. Specifically, under mild technical assumptions a fiber bundle always has thehomotopy lifting property or homotopy covering property (seeSteenrod (1951, 11.7) for details). This is the defining property of a fibration.
• A section of a fiber bundle is a "function whose outputrange is continuously dependent on the input." This property is formally captured in the notion ofdependent type.

• Steenrod, Norman (1951),The Topology of Fibre Bundles, Princeton University Press,ISBN 978-0-691-08055-0{{citation}}:ISBN / Date incompatibility (help)
• Steenrod, Norman (April 5, 1999).The Topology of Fibre Bundles. Princeton Mathematical Series. Vol. 14. Princeton, N.J.: Princeton University Press.ISBN 978-0-691-00548-5.OCLC 40734875.
• Bleecker, David (1981),Gauge Theory and Variational Principles, Reading, Mass: Addison-Wesley publishing,ISBN 978-0-201-10096-9
• Ehresmann, Charles (1951). "Les connexions infinitésimales dans un espace fibré différentiable".Colloque de Topologie (Espaces fibrés), Bruxelles, 1950. Paris: Georges Thone, Liège; Masson et Cie. pp. 29–55.
• Husemoller, Dale (1994),Fibre Bundles, Springer Verlag,ISBN 978-0-387-94087-8
• Michor, Peter W. (2008),Topics in Differential Geometry,Graduate Studies in Mathematics, vol. 93, Providence: American Mathematical Society,ISBN 978-0-8218-2003-2
• Voitsekhovskii, M.I. (2001) [1994],"Fibre space",Encyclopedia of Mathematics,EMS Press

• Fiber Bundle, PlanetMath
• Rowland, Todd."Fiber Bundle".MathWorld.
• Making John Robinson's Symbolic Sculpture `Eternity'
• Sardanashvily, Gennadi, Fibre bundles, jet manifolds and Lagrangian theory. Lectures for theoreticians,arXiv:0908.1886

• Fiber bundles

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