World Library  
Flag as Inappropriate
Email this Article

Fiber bundle

Article Id: WHEBN0000276526
Reproduction Date:

Title: Fiber bundle  
Author: World Heritage Encyclopedia
Language: English
Subject: Vector bundle, BRST quantization, Teleparallelism, Affine bundle, Fibration
Collection: Algebraic Topology, Differential Topology, Fiber Bundles, Homotopy Theory
Publisher: World Heritage Encyclopedia

Fiber bundle

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 (bristles) are line segments. The mapping π:EB would take a point on any bristle and map it to its root on the cylinder.

In mathematics, and particularly topology, a fiber bundle (or, in British English, fibre bundle) is a space that is locally a product space, but globally may have a different topological structure. Specifically, the similarity between a space E and a product space B × F is defined using a continuous surjective map

\pi\colon E \to B

that in small regions of E behaves just like a projection from corresponding regions of B × F to B. The map π, called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space E is known as the total space of the fiber bundle, B as the base space, and F the fiber.

In the trivial case, E is just B × F, and the map π 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 spaces. Fiber bundles such as the tangent bundle of a manifold and more general vector bundles play an important role in differential geometry and differential topology, as do principal bundles.

Mappings between total spaces of fiber bundles that "commute" with the projection maps are known as bundle maps, and the class of fiber bundles forms a category with respect to such mappings. A bundle map from the base space itself (with the identity mapping as projection) to E is called a section of E. Fiber bundles can be specialized in a number of ways, the most common of which is requiring that the transitions between the local trivial patches lie in a certain topological group, known as the structure group, acting on the fiber F.


  • Formal definition 1
  • Examples 2
    • Trivial bundle 2.1
    • Möbius strip 2.2
    • Klein bottle 2.3
    • Covering map 2.4
    • Vector and principal bundles 2.5
    • Sphere bundles 2.6
    • Mapping tori 2.7
    • Quotient spaces 2.8
  • Sections 3
  • Structure groups and transition functions 4
  • Bundle maps 5
  • Differentiable fiber bundles 6
  • Generalizations 7
  • See also 8
  • Notes 9
  • References 10
  • External links 11

Formal definition

A fiber bundle is a structure (E, B, π, F), where E, B, and F are topological spaces and π : EB is a continuous surjection satisfying a local triviality condition outlined below. The space B is called the base space of the bundle, E the total space, and F the fiber. The map π is called the projection map (or bundle projection). We shall assume in what follows that the base space B is connected.

We require that for every x in E, there is an open neighborhood UB of π(x) (which will be called a trivializing neighborhood) such that there is a homeomorphism φ: π−1(U) → U × F (where U × F is the product space) in such a way that π agrees with the projection onto the first factor. That is, the following diagram should commute:

Local triviality condition

where proj1 : U × FU is the natural projection and φ : π−1(U) → U × F is a homeomorphism. The set of all {(Ui, φi)} is called a local trivialization of the bundle.

Thus for any p in B, the preimage π−1({p}) is homeomorphic to F (since proj1−1({p}) clearly is) and is called the fiber over p. Every fiber bundle π : EB is an open map, since projections of products are open maps. Therefore B carries the quotient topology determined by the map π.

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

F \longrightarrow E \ \xrightarrow{\, \ \pi \ } \ B

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


Trivial bundle

Let E = B × F and let π : EB be the projection onto the first factor. Then E is a fiber bundle (of F) over B. Here E is not just locally a product but globally one. Any such fiber bundle is called a trivial bundle. Any fiber bundle over a contractible CW-complex is trivial.

Möbius strip

The Möbius strip is a nontrivial bundle over the circle.

Perhaps the simplest example of a nontrivial bundle E is the Möbius strip. It has the circle that runs lengthwise along the center of the strip as a base B and a line segment for the fiber F, so the Möbius strip is a bundle of the line segment over the circle. A neighborhood U of a point xB is an arc; in the picture, this is the length of one of the squares. The preimage \pi^{-1}(U) in the picture is a (somewhat twisted) slice of the strip four squares wide and one long. The homeomorphism φ maps the preimage of U to a slice of a cylinder: curved, but not twisted.

The corresponding trivial bundle B × F would be a cylinder, but the Möbius strip has an overall "twist". Note that 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, S1 × S1.

The Klein bottle immersed in three-dimensional space.
A torus.

Covering map

A 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 bundles, are those whose fibers are vector spaces (to qualify as a vector bundle the structure group of the bundle — see below — must be a 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 bases which is a principal bundle (see below).

Another special class of fiber bundles, called principal bundles, are bundles on whose fibers a free and transitive action by a group G 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 G-bundle. The group G is also the structure group of the bundle. Given a representation ρ of G on a vector space V, a vector bundle with ρ(G)⊆Aut(V) as a structure group may be constructed, known as the associated bundle.

Sphere bundles

A sphere bundle is a fiber bundle whose fiber is an n-sphere. Given a vector bundle E with a 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 x is the set of all unit vectors in Ex. When the vector bundle in question is the tangent bundle T(M), the unit sphere bundle is known as the unit tangent bundle, and is denoted UT(M).

A sphere bundle is partially characterized by its Euler class, which is a degree n+1 cohomology class in the total space of the bundle. In the case n=1 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 n, 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 X is a topological space and f:X → X is a homeomorphism then the mapping torus Mf has a natural structure of a fiber bundle over the circle with fiber X. Mapping tori of homeomorphisms of surfaces are of particular importance in 3-manifold topology.

Quotient spaces

If G is a topological group and H is a closed subgroup, then under some circumstances, the quotient space G/H together with the quotient map π : G → G/H is a fiber bundle, whose fiber is the topological space H. A necessary and sufficient condition for (G,G/H,π,H) to form a fiber bundle is that the mapping π admit local cross-sections (Steenrod 1951, §7).

The most general conditions under which the quotient map will admit local cross-sections are not known, although if G is a Lie group and H a closed subgroup (and thus a Lie subgroup by Cartan's theorem), then the quotient map is a fiber bundle. One example of this is the Hopf fibration, S3 → S2 which is a fiber bundle over the sphere S2 whose total space is S3. From the perspective of Lie groups, S3 can be identified with the special unitary group SU(2). The abelian subgroup of diagonal matrices is isomorphic to the circle group U(1), and the quotient SU(2)/U(1) is diffeomorphic to the sphere.

More generally, if G is any topological group and H a closed subgroup which also happens to be a Lie group, then G → G/H is a fiber bundle.


A section (or cross section) of a fiber bundle π is a continuous map f : BE such that π(f(x))=x 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 to the existence of a section can often be measured by a cohomology class, which leads to the theory of characteristic classes 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 f : UE where U is an open set in B and π(f(x))=x for all x in U. If (U, φ) is a local trivialization chart then local sections always exist over U. Such sections are in 1-1 correspondence with continuous maps UF. Sections form a sheaf.

Structure groups and transition functions

Fiber bundles often come with a group of symmetries which describe the matching conditions between overlapping local trivialization charts. Specifically, let G be a topological group which acts continuously on the fiber space F on the left. We lose nothing if we require G to act effectively on F so that it may be thought of as a group of homeomorphisms of F. A G-atlas for the bundle (E, B, π, F) is a local trivialization such that for any two overlapping charts (Ui, φi) and (Uj, φj) the function

\varphi_i\varphi_j^{-1}\colon (U_i \cap U_j) \times F \to (U_i \cap U_j) \times F

is given by

\varphi_i\varphi_j^{-1}(x, \xi) = (x, t_{ij}(x)\xi)

where tij : UiUjG is a continuous map called a 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 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 tij satisfy the following conditions

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

The third condition applies on triple overlaps UiUjUk and is called the 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 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). 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

It is useful to have notions of a mapping between two fiber bundles. Suppose that M and N are base spaces, and πE : EM and πF : FN are fiber bundles over M and N, respectively. A bundle map (or bundle morphism) consists of a pair of continuous[1] functions

\varphi\colon E\to F,\quad f\colon M\to N

such that \pi_F\circ \varphi = f\circ\pi_E. That is, the following diagram commutes:

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

In case the base spaces M and N coincide, then a bundle morphism over M from the fiber bundle πE : EM to πF : FM is a map φ : EF such that \pi_E = \pi_F\circ\varphi. This means that the bundle map φ : EF covers the identity of M. That is, f\equiv {\mathrm {id}}_{M} and the diagram commutes

Assume that both πE : EM and πF : FM are defined over the same base space M. A bundle isomorphism is a bundle map (\varphi,f) between πE : EM and πF : FM such that f\equiv {\mathrm {id}}_{M} and such that φ is also a homeomorphism.[2]

Differentiable fiber bundles

In the category of differentiable manifolds, fiber bundles arise naturally as submersions of one manifold to another. Not every (differentiable) submersion ƒ : M → N 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 (M,N,ƒ) 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 and connected, then any submersion f : M → N gives rise to a fiber bundle in the sense that there is a fiber space F diffeomorphic to each of the fibers such that (E,B,π,F) = (M,N,ƒ,F) is a fiber bundle. (Surjectivity of ƒ follows by the assumptions already given in this case.) More generally, the assumption of compactness can be relaxed if the submersion ƒ : M → N is assumed to be a surjective proper map, meaning that ƒ−1(K) is compact for every compact subset K of N. Another sufficient condition, due to Ehresmann (1951), is that if ƒ : M → N is a surjective submersion with M and N differentiable manifolds such that the preimage ƒ−1{x} is compact and connected for all x ∈ N, then ƒ admits a compatible fiber bundle structure (Michor 2008, §17).


  • The notion of a 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 π : EB which has certain 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 Steenrod (1951, 11.7) for details). This is the defining property of a fibration.

See also


  1. ^ 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.
  2. ^ Or is, at least, invertible in the appropriate category; e.g., a diffeomorphism.


  • Bleecker, David (1981), Gauge Theory and Variational Principles, Reading, Mass: Addison-Wesley publishing,  
  • Husemöller, Dale (1994), Fibre Bundles, Springer Verlag,  
  • Michor, Peter W. (2008), Topics in Differential Geometry, ). to appear ( 
  • Voitsekhovskii, M.I. (2001), "Fibre space", in Hazewinkel, Michiel,  

External links

  • Fiber Bundle, PlanetMath
  • Rowland, Todd, "Fiber Bundle", MathWorld.
  • Making John Robinson's Symbolic Sculpture `Eternity'
  • Sardanashvily, G., Fibre bundles, jet manifolds and Lagrangian theory. Lectures for theoreticians,arXiv: 0908.1886
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.