Fiber Bundle - Wikipedia, The Free Encyclopedia

8/13/2019 Fiber Bundle - Wikipedia, The Free Encyclopedia

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A cylindrical hairbrush showing theintuition behind the term "fiberbundle". This hairbrush is like a fiber

bundle in which the base space is acylinder and the fibers (bristles) areline segments. The mapping : E Bwould take a point on any bristle andmap it to the point on the cylinderwhere the bristle attaches.

om Wikipedia, the free encyclopedia

mathematics,

d particularlypology, a fiber

undle (or, inritish English,bre bundle ) istuitively a spacehich locallyooks" like artain productace, but globallyay have a

fferentpologicalructure.pecifically, themilarity betweene fiber bundle E d a productace B F isfined using a continuous surjective map


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at in small regions of E behaves just like a projectionom corresponding regions of B F to B. The map ,lled the projection or submersion of the bundle, is

garded as part of the structure of the bundle. The spaceis known as the total space of the fiber bundle, B as these space , and F the fiber .

the trivial case, E is just B F , and the map is just theojection from the product space to the first factor. This islled a trivial bundle . Examples of non-trivial fiber

undles, that is, bundles twisted in the large, include thebius strip and Klein bottle, as well as nontrivialvering spaces. Fiber bundles such as the tangent bundlea manifold and more general vector bundles play an

mportant role in differential geometry and differentialpology, as do principal bundles.

appings which factor over the projection map are knownbundle maps , and the set of fiber bundles forms a

tegory with respect to such mappings. A bundle map

om the base space itself (with the identity mapping asojection) to E is called a section of E . Fiber bundles cangeneralized in a number of ways, the most common of

hich is requiring that the transition between the local


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vial patches should lie in a certain topological group,nown as the structure group , acting on the fiber F .

1 Formal definition2 Examples

2.1 Trivial bundle2.2 Mbius strip

2.3 Klein bottle2.4 Covering map2.5 Vector and principal bundles2.6 Sphere bundles2.7 Mapping tori2.8 Quotient spaces

3 Sections4 Structure groups and transition functions5 Bundle maps6 Differentiable fiber bundles7 Generalizations

8 See also9 Notes10 References11 External links


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fiber bundle consists of the data ( E , B, , F ), where E ,and F are topological spaces and : E B is a

ntinuous surjection satisfying a local triviality conditionutlined below. The space B is called the base space of theundle, E the total space , and F the fiber . The map islled the projection map (or bundle projection). We shallsume in what follows that the base space B is connected.

e require that for every x in E , there is an openighborhood U B of ( x) (which will be called avializing neighborhood) such that 1(U ) is

omeomorphic to the product space U F , in such a wayat agrees with the projection onto the first factor. Thatthe following diagram should commute:

here proj 1 : U F U is the natural projection and :


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1(U ) U F is a homeomorphism. The set of all {( U i,)} is called a local trivialization of the bundle.

hus for any p in B, the preimage 1({ p}) isomeomorphic to F (since proj 1-1({ p}) clearly is) and islled the fiber over p. Every fiber bundle : E B is an

pen map, since projections of products are open maps.herefore B carries the quotient topology determined bye map .

fiber bundle ( E , B, , F ) is often denoted

at, in analogy with a short exact sequence, indicateshich space is the fiber, total space and base space, as well

the map from total to base space.

smooth fiber bundle is a fiber bundle in the category of mooth manifolds. That is, E , B, and F are required to bemooth manifolds and all the functions above are required

be smooth maps.


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The Mbius strip is a nontrivialbundle over the circle.

ivial bundle

et E = B F and let : E B be the projection onto thest factor. Then E is a fiber bundle (of F ) over B. Here E not just locally a product but globally one. Any such

ber bundle is called a trivial bundle . Any fiber bundlever a contractible CW-complex is trivial.

Mbius strip

erhaps the simplestample of a nontrivialundle E is the Mbiusrip. It has the circle thatns lengthwise along thenter of the strip as ase B and a line segmentr the fiber F , so thebius strip is a bundlethe line segment over

e circle. Aighborhood U of a

oint x B is an arc; in the picture, this is the length of ne of the squares. The preimage in the picture is(somewhat twisted) slice of the strip four squares wide


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The Klein bottle

A torus.

d one long. The homeomorphism maps the preimageU to a slice of a cylinder: curved, but not twisted.

he corresponding trivial bundle B F would be alinder, but the Mbius strip has an overall "twist". Note

at this twist is visible only globally; locally the Mbiusrip and the cylinder are identical (making a single verticalt in either gives the same space).

lein bottle

similar nontrivial bundle is the Klein bottle which can beewed as a "twisted" circle bundle over another circle.he corresponding non-twisted (trivial) bundle is thetorus, S 1 S 1.


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immersed in three-dimensional space.

overing mapcovering space is a fiber bundle such that the bundleojection is a local homeomorphism. It follows that theber is a discrete space.

ector and principal bundles

special class of fiber bundles, called vector bundles , areose whose fibers are vector spaces (to qualify as a vector

undle the structure group of the bundle see below ust be a linear group). Important examples of vector

undles include the tangent bundle and cotangent bundlea smooth manifold. From any vector bundle, one cannstruct the frame bundle of bases which is a principal

undle (see below).

nother special class of fiber bundles, called principalundles , are bundles on whose fibers a free and transitivetion by a group G is given, so that each fiber is aincipal homogeneous space. The bundle is often


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ecified along with the group by referring to it as aincipal G-bundle. The group G is also the structure groupthe bundle. Given a representation of G on a vectorace V , a vector bundle with (G) Aut( V ) as a structureoup may be constructed, known as the associated

undle.

phere bundles

sphere bundle is a fiber bundle whose fiber is an

sphere. Given a vector bundle E with a metric (such ase tangent bundle to a Riemannian manifold) one cannstruct the associated unit sphere bundle , for which the

ber over a point x is the set of all unit vectors in E x.hen the vector bundle in question is the tangent bundleM ), the unit sphere bundle is known as the unit tangent

undle , and is denoted UT( M ).

sphere bundle is partially characterized by its Eulerass, which is a degree n+1 cohomology class in the totalace of the bundle. In the case n=1 the sphere bundle islled a circle bundle and the Euler class is equal to thest Chern class, which characterizes the topology of the

undle completely. For any n, given the Euler class of aundle, one can calculate its cohomology using a longact sequence called the Gysin sequence.


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Mapping tori

X is a topological space and f : X X is aomeomorphism then the mapping torus M f has a naturalructure of a fiber bundle over the circle with fiber X .apping tori of homeomorphisms of surfaces are of rticular importance in 3-manifold topology.

uotient spaces

G is a topological group and H is a closed subgroup, thennder some circumstances, the quotient space G / H gether with the quotient map : G G / H is a fiber

undle, whose fiber is the topological space H . Acessary and sufficient condition for ( G,G / H ,, H ) to formfiber bundle is that the mapping admit local cross-ctions (Steenrod & 1951 7).

he most general conditions under which the quotient mapill admit local cross-sections are not known, although if is a Lie group and H a closed subgroup (and thus a Liebgroup by Cartan's theorem), then the quotient map is a

ber bundle. One example of this is the Hopf fibration, S 2 which is a fiber bundle over the sphere S 2 whosetal space is S 3. From the perspective of Lie groups, S 3


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n be identified with the special unitary group SU(2). Theelian subgroup of diagonal matrices is isomorphic to thercle group U(1), and the quotient SU(2)/U(1) isffeomorphic to the sphere.

ore generally, if G is any topological group and H aosed subgroup which also happens to be a Lie group,en G G / H is a fiber bundle.

Main article: Section (fiber bundle)

section (or cross section ) of a fiber bundle is antinuous map f : B E such that ( f ( x))= x for all x in B.nce bundles do not in general have globally definedctions, one of the purposes of the theory is to accountr their existence. The obstruction to the existence of action can often be measured by a cohomology class,hich leads to the theory of characteristic classes ingebraic topology.

he most well-known example is the hairy ball theorem,here the Euler class is the obstruction to the tangentundle of the 2-sphere having a nowhere vanishingction.


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ften one would like to define sections only locallyspecially when global sections do not exist). A localction of a fiber bundle is a continuous map f : U E here U is an open set in B and ( f ( x))= x for all x in U . If

U , ) is a local trivialization chart then local sectionsways exist over U . Such sections are in 1-1rrespondence with continuous maps U F . Sectionsrm a sheaf.

ber bundles often come with a group of symmetrieshich describe the matching conditions betweenverlapping local trivialization charts. Specifically, let G be

topological group which acts continuously on the fiberace F on the left. We lose nothing if we require G to actfectively on F so that it may be thought of as a group of

omeomorphisms of F . A G-atlas for the bundle ( E , B, ,) is a local trivialization such that for any two overlappingarts ( U i, i) and ( U j, j) the function

given by


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here t ij : U i U j G is a continuous map called aansition function . Two G-atlases are equivalent if theirnion is also a G-atlas. A G-bundle is a fiber bundle with

equivalence class of G-atlases. The group G is callede structure group of the bundle; the analogous term in

hysics is gauge group.

the smooth category, a G-bundle is a smooth fiberundle where G is a Lie group and the correspondingtion on F is smooth and the transition functions are all

mooth maps.

he transition functions t ij satisfy the following conditions

1.2.3.

he third condition applies on triple overlaps U i U j U k d is called the cocycle condition (see echhomology). The importance of this is that the transitionnctions determine the fiber bundle (if one assumes theech cocycle condition).


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principal G-bundle is a G-bundle where the fiber F is aincipal homogeneous space for the left action of G itself quivalently, one can specify that the action of G on theber F is free and transitive). In this case, it is often aatter of convenience to identify F with G and so obtain a

ght) action of G on the principal bundle.

is useful to have notions of a mapping between two fiber

undles. Suppose that M and N are base spaces, and E : E M and F : F N are fiber bundles over M and N ,spectively. A bundle map (or bundle morphism )nsists of a pair of continuous [1] functions

ch that . That is, the followingagram commutes:


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or fiber bundles with structure group G and whose totalaces are (right) G-spaces (such as a principal bundle),

undle morphisms are also required to be G-equivariant on

e fibers. This means that is also-morphism from one G-space to another, i.e., for all and .

case the base spaces M and N coincide, then a bundleorphism over M from the fiber bundle E : E M to F : M is a map : E F such that . Thiseans that the bundle map : E F covers the identityM . That is, and the diagram commutes


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ssume that both E : E M and F : F M are definedver the same base space M . A bundle isomorphism is aundle map between E : E M and F : F M ch that and such that is also ameomorphism. [2]

the category of differentiable manifolds, fiber bundlesise naturally as submersions of one manifold to another.ot every (differentiable) submersion : M N from afferentiable manifold M to another differentiableanifold N gives rise to a differentiable fiber bundle. For

ne thing, the map must be surjective, and ( M , N ,) islled a fibered manifold. However, this necessary

ndition is not quite sufficient, and there are a variety of fficient conditions in common use.

M and N are compact and connected, then anybmersion f : M N gives rise to a fiber bundle in thense that there is a fiber space F diffeomorphic to each of

e fibers such that ( E , B,,F ) = ( M , N ,,F ) is a fiberundle. (Surjectivity of follows by the assumptionsready given in this case.) More generally, the assumptioncompactness can be relaxed if the submersion


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M N is assumed to be a surjective proper map,eaning that 1(K ) is compact for every compact subsetof N . Another sufficient condition, due to Ehresmann951), is that if : M N is a surjective submersion withand N differentiable manifolds such that the preimage

1{ x} is compact and connected for all x N , then mits a compatible fiber bundle structure (Michor 2008,7).

The notion of a bundle applies to many morecategories in mathematics, at the expense of appropriately modifying the local triviality condition.In topology, a fibration is a mapping : E B

which has certain homotopy-theoretic properties incommon with fiber bundles. Specifically, under mildtechnical assumptions a fiber bundle always has thehomotopy lifting property or homotopy coveringproperty (see Steenrod 1951, 11.7, for details). Thisis the defining property of a fibration.


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Covering mapFibrationGauge theoryHopf bundle

I-bundlePrincipal bundle

Pullback bundleUniversal bundleFibred manifoldVector bundle

Affine bundle

^ Depending on the category of spaces involved, thefunctions may be assumed to have properties other thancontinuity. For instance, in the category of differentiablemanifolds, the functions are assumed to be smooth. In thecategory of algebraic varieties, they are regularmorphisms.

1.

^ Or is, at least, invertible in the appropriate category;e.g., a diffeomorphism.

2.

Steenrod, Norman (1951), The Topology of Fibre Bundles , Princeton University Press,ISBN 0-691-08055-0Bleecker, David (1981), Gauge Theory and


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Variational Principles , Reading, Mass: Addison-Wesley publishing, ISBN 0-201-10096-7Ehresmann, C. "Les connexions infinitsimales dansun espace fibr diffrentiable". Colloque deTopologie (Espaces fibrs), Bruxelles, 1950 .

Georges Thone, Lige; Masson et Cie., Paris, 1951.pp. 2955.Husemller, Dale (1994), Fibre Bundles , SpringerVerlag, ISBN 0-387-94087-1Michor, Peter W. (2008), Topics in DifferentialGeometry , Graduate Studies in Mathematics, Vol.93, Providence: American Mathematical Society ( toappear ).Voitsekhovskii, M.I. (2001), "Fibre space"(http://www.encyclopediaofmath.org

/index.php?title=F/f040060), in Hazewinkel,Michiel, Encyclopedia of Mathematics , Springer,ISBN 978-1-55608-010-4

Fiber Bundle (http://planetmath.org/encyclopedia /FiberBundle.html), PlanetMathWeisstein, Eric W., "Fiber Bundle(http://mathworld.wolfram.com/FiberBundle.html)",

MathWorld .


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Making John Robinson's Symbolic Sculpture`Eternity' (http://www.popmath.org.uk/sculpmath

/pagesm/fibundle.html)Sardanashvily, G., Fibre bundles, jet manifolds andLagrangian theory. Lectures for theoreticians,arXiv:

0908.1886 (http://xxx.lanl.gov/abs/0908.1886)

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