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Fiber bundle
 
 
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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
Bristle

A bristle is a stiff hair or feather. Also used are synthetic materials such as nylon in items such as brooms and sweepers. Bristles are often used to make brushes for cleaning uses, as they are strongly abrasive; common examples include the toothbrush and toilet brush....
s) are line segments. The mapping p:E?B would take a point on any bristle and map it to the point on the cylinder where the bristle attaches.]]

In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, and particularly topology
Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
, a fiber bundle (or fibre bundle) is intuitively a space E which locally "looks" like a product space B × F, but globally may have a different topological structure.






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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
Bristle

A bristle is a stiff hair or feather. Also used are synthetic materials such as nylon in items such as brooms and sweepers. Bristles are often used to make brushes for cleaning uses, as they are strongly abrasive; common examples include the toothbrush and toilet brush....
s) are line segments. The mapping p:E?B would take a point on any bristle and map it to the point on the cylinder where the bristle attaches.]]

In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, and particularly topology
Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
, a fiber bundle (or fibre bundle) is intuitively a space E which locally "looks" like a product space B × F, but globally may have a different topological structure. Specifically, the similarity between the fiber bundle E and a product space B × F is defined using a continuous
Continuous function (topology)

In topology and related areas of mathematics a continuous function is a morphism between topological spaces. Intuitively, this is a function f where a set of points near f always contain the of a set of points near x....
 surjective
Surjective function

In mathematics, a function f is said to be surjective or onto, if its values span its whole codomain; that is, for every y in the codomain, there is at least one x in the domain such that f = y ....
 map
Map (mathematics)

In mathematics and related technical fields, the term map or mapping is often a synonym for Function . Thus, for example, a partial map is a partial function, and a total map is a total function....


that in small regions of E behaves just like a projection from corresponding regions of B × F to B. The map p, 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 p is just the projection from the product space to the first factor. This is called a trivial bundle. Examples of non-trivial fiber bundles, that is, bundles twisted in the large, include the Möbius strip
Möbius strip

The M?bius strip or M?bius band is a surface with only one side and only one boundary component. The M?bius strip has the mathematical property of being orientability....
 and Klein bottle
Klein bottle

In mathematics, the Klein bottle is a certain non-orientability surface, i.e., a surface with no distinct "inner" and "outer" sides. Other related non-orientable objects include the M?bius strip and the real projective plane....
, as well as nontrivial covering spaces. Fiber bundles such as the tangent bundle
Tangent bundle

In mathematics, the tangent bundle of a differentiable manifold M, denoted by T or just TM, is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector....
 of a manifold
Manifold

In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
 and more general vector bundle
Vector bundle

In mathematics, a vector bundle is a topology construction which makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together to form another space of the same kind as X , which is t...
s play an important role in differential geometry and differential topology
Differential topology

In mathematics, differential topology is the field dealing with differentiable function s on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds....
, as do principal bundle
Principal bundle

In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of a Cartesian product X × G of a space X with a group G....
s.

Mappings which factor over the projection map are known as bundle maps, and the set of fiber bundles forms a category
Category theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from set s and function s to objects linked in diagrams by morphisms or arrows....
 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
Section (fiber bundle)

In the mathematical field of topology, a section of a fiber bundle, π: E → B, over a topological space, B, is a continuous map, s : B → E, such that π=x for all x in B....
 of E. Fiber bundles can be generalized in a number of ways, the most common of which is requiring that the transition between the local trivial patches should lie in a certain topological group
Topological group

In mathematics, a topological group is a group G together with a topological space on G such that the group's binary operation and the group's inverse function are continuous function ....
, known as the structure group, acting on the fiber space F.

Formal definition


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

In topology and related branches of mathematics, a connected space is a topological space which cannot be represented as the disjoint union of two or more nonempty open subsets....
.

We require that for any x in B, there is an open neighborhood U of x (which will be called a trivializing neighborhood) such that p-1(U) is homeomorphic to the product space
Product topology

In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology....
 U × F, in such a way that p carries over to the projection onto the first factor. That is, the following diagram should commute
Commutative diagram

In mathematics, and especially in category theory a commutative diagram is a diagram of objects, also known as vertices, and morphisms, also known as arrows or edges, such that when selecting two objects any directed path through the diagram leads to the same result by composition....
:

where proj1 : U × F ? U is the natural projection and f : p-1(U) ? U × F is a homeomorphism. The set of all is called a local trivialization of the bundle.

For any x in B, the preimage p-1 is homeomorphic to F and is called the fiber over x. A fiber bundle (E, B, p, F) is often denoted
to indicate a short exact sequence of spaces. Note that every fiber bundle p : E ? B is an open map, since projections of products are open maps. Therefore B carries the quotient topology determined by the map p.

A smooth fiber bundle is a fiber bundle in the category
Category (mathematics)

In mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships. A category is composed of a collection of abstract "objects" of any kind, linked together by a collection of abstract "morphism" of any kind that have a few basic properties ....
 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.

Examples


Trivial bundle


Let E = B × F and let p : E ? B 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 fibre bundle over a contractible space
Contractible space

In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i.e. if it is homotopic to some constant map....
 is trivial.

Möbius strip


Perhaps the simplest example of a nontrivial bundle E is the Möbius strip
Möbius strip

The M?bius strip or M?bius band is a surface with only one side and only one boundary component. The M?bius strip has the mathematical property of being orientability....
. It has the circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
 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 x ? B is an arc; in the picture, this is the length of one of the squares. The preimage in the picture is a (somewhat twisted) slice of the strip four squares wide and one long. The homeomorphism f 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
Cylinder (geometry)

A cylinder is one of the most curvilinear basic geometric shapes: the surface formed by the points at a fixed distance from a given straight line, the axis of the 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
Klein bottle

In mathematics, the Klein bottle is a certain non-orientability surface, i.e., a surface with no distinct "inner" and "outer" sides. Other related non-orientable objects include the M?bius strip and the real projective plane....
 which can be viewed as a "twisted" circle bundle over another circle. The corresponding trivial bundle would be a torus
Torus

In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle, which does not touch the circle....
, S1 × S1.

in three-dimensional space.]] 


Covering map


A covering space
Covering map

File:PSTricks-Cubriente.pngIn mathematics, more specifically algebraic topology, a covering map is a continuous function surjective function p from a topological space, C, to a topological space, X, such that each point in X has a neighbourhood evenly covered by p....
 is a fiber bundle such that the bundle projection is a local homeomorphism
Local homeomorphism

In mathematics, more specifically topology, a local homeomorphism is intuitively a function, f, between topological spaces that preserves local structure....
. It follows in particular, that the fiber is a discrete space
Discrete space

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "Isolated point" from each other in a certain sense....
.

Vector and principal bundles


A special class of fiber bundles, called vector bundle
Vector bundle

In mathematics, a vector bundle is a topology construction which makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together to form another space of the same kind as X , which is t...
s, are those whose fibers are vector space
Vector space

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
s (to qualify as a vector bundle the structure group of the bundle — see below — must be a linear group
General linear group

In mathematics, the general linear group of degree n is the set of n×n invertible matrix, together with the operation of ordinary matrix multiplication....
). Important examples of vector bundles include the tangent bundle
Tangent bundle

In mathematics, the tangent bundle of a differentiable manifold M, denoted by T or just TM, is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector....
 and cotangent bundle
Cotangent bundle

In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold....
 of a smooth manifold. From any vector bundle, one can construct the frame bundle
Frame bundle

In mathematics, a frame bundle is a principal fiber bundle F associated to any vector bundle E. The fiber of F over a point x is the set of all ordered basis, or frames, for Ex....
 of bases which is a principal bundle (see below).

Another special class of fiber bundles, called principal bundle
Principal bundle

In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of a Cartesian product X × G of a space X with a group G....
s, are bundles on whose fibers a free and transitive action
Group action

In algebra and geometry, a group action is a way of describing symmetry of objects using group . The essential elements of the object are described by a Set and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformation of the set....
 by a group G is given, so that each fiber is a principal homogeneous space
Principal homogeneous space

In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G such that the stabilizer subgroup of any point is trivial....
. 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
Group representation

In the mathematics field of representation theory, group representations describe abstract group in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrix so that the group operation can be represented by matrix multiplication....
 ? of G on a vector space V, a vector bundle with ?(G)?Aut(V) as a structure group may be constructed, known as the associated bundle
Associated bundle

In mathematics, the theory of fiber bundles with a structure group allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from to , which are both topological spaces with a group action of ....
.

Sphere bundles


A sphere bundle is a fiber bundle whose fiber is an n-sphere
Hypersphere

In mathematics, an n-sphere is a generalization of the surface of an ordinary sphere to arbitrary dimension. For any natural number n, an n-sphere of radius r is defined as the set of points in -dimensional Euclidean space which are at distance r from a central point, where the radius r may be any positive real num...
. Given a vector bundle E with a metric
Metric tensor

In the mathematics field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of Vector in Euclidean space....
 (such as the tangent bundle to a Riemannian manifold
Riemannian manifold

In Riemannian geometry, a Riemannian manifold is a real differentiable manifold M in which each tangent space is equipped with an Inner product space g in a manner which varies smoothly from point to point....
) 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
Unit tangent bundle

In mathematics, the unit tangent bundle of a Finsler manifold , denoted by UT or simply UTM, is a fiber bundle over M given by the disjoint union...
, and is denoted UT(M).


A sphere bundle is partially characterized by its Euler class
Euler class

In mathematics, specifically in algebraic topology, the Euler class, named after Leonhard Euler, is a characteristic class of oriented, real vector bundles....
, which is a degree n+1 cohomology
Cohomology

In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex....
 class in the total space of the bundle. In the case n=1 the sphere bundle is called a circle bundle
Circle bundle

In mathematics, a circle bundle is a fiber bundle where the fiber is the circle , or, more precisely, a principal bundle. It is homotopically equivalent to a complex line bundle....
 and the Euler class is equal to the first Chern class
Chern class

In mathematics, in particular in algebraic topology and differential geometry and topology, the Chern classes are a particular type of characteristic class associated to complex vector bundles....
, 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
Gysin sequence

In the field of mathematics known as algebraic topology, the Gysin sequence is a long exact sequence which relates the cohomology classes of the base space, the fiber and the total space of a Fiber_bundle#Sphere_bundles....
.

Mapping tori

If X is a topological space
Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
 and f:X ? X is a homeomorphism
Homeomorphism

In the mathematics field of topology, a homeomorphism or topological isomorphism is a continuous function between two topological spaces....
 then the mapping torus
Mapping torus

In mathematics, the mapping torus in topology of a homeomorphism f of some topological space X to itself is a particular geometric construction with f....
 Mf has a natural structure of a fiber bundle over the circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
 with fiber X. Mapping tori of homeomorphisms of surfaces are of particular importance in 3-manifold topology
3-manifold

In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is usually made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds....
.

Quotient spaces

If G is a topological group
Topological group

In mathematics, a topological group is a group G together with a topological space on G such that the group's binary operation and the group's inverse function are continuous function ....
 and H is a closed subgroup, then under some circumstances, the quotient space
Quotient space

In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space....
 G/H together with the quotient map p : G ? G/H is a fiber bundle, whose fiber is the topological space H. A necessary and sufficient condition for (G,G/H,p,H) to form a fiber bundle is that the mapping p admit local cross-sections .

The most general conditions under which the quotient map will admit local cross-sections are not known, although if G is a Lie group
Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the Differential structure....
 and H a closed subgroup (and thus a Lie subgroup), 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
Special unitary group

In mathematics, the special unitary group of degree n, denoted SU, is the group of n×n unitary matrix Matrix with determinant 1....
 SU(2). The abelian subgroup of diagonal matrices is isomorphic to the circle group
Circle group

In mathematics, the circle group, denoted by T , is the multiplicative group of all complex numbers with absolute value 1, i.e., the unit circle in the complex plane....
 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.

Sections


A section
Section (fiber bundle)

In the mathematical field of topology, a section of a fiber bundle, π: E → B, over a topological space, B, is a continuous map, s : B → E, such that π=x for all x in B....
 (or cross section) of a fiber bundle is a continuous map f : B ? E such that p(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
Obstruction theory

In mathematics, obstruction theory is a name given to two different mathematical theory, both of which yield cohomological invariant ....
 to the existence of a section can often be measured by a cohomology class, which leads to the theory of characteristic class
Characteristic class

In mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X....
es in algebraic topology
Algebraic topology

Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant that classification theorem topological spaces up to homeomorphism....
.

The most well-known example is the hairy ball theorem
Hairy ball theorem

The hairy ball theorem of algebraic topology states that there is no nonvanishing continuous function tangent vector vector field on the sphere....
, where the Euler class
Euler class

In mathematics, specifically in algebraic topology, the Euler class, named after Leonhard Euler, is a characteristic class of oriented, real vector bundles....
 is the obstruction to the tangent bundle
Tangent bundle

In mathematics, the tangent bundle of a differentiable manifold M, denoted by T or just TM, is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector....
 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 : U ? E where U is an open set
Open set

In metric topology and related fields of mathematics, a Set U is called open if, intuitively speaking, starting from any point x in U one can move by a small amount in any direction and still be in the set U....
 in B and p(f(x))=x for all x in U. If (U, f) is a local trivialization chart then local sections always exist over U. Such sections are in 1-1 correspondence with continuous maps U ? F. Sections form a sheaf
Sheaf (mathematics)

In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original one....
.

Structure groups and transition functions


Fiber bundles often come with a group
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
 of symmetries which describe the matching conditions between overlapping local trivialization charts. Specifically, let G be a topological group
Topological group

In mathematics, a topological group is a group G together with a topological space on G such that the group's binary operation and the group's inverse function are continuous function ....
 which acts
Group action

In algebra and geometry, a group action is a way of describing symmetry of objects using group . The essential elements of the object are described by a Set and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformation of the set....
 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 homeomorphism
Homeomorphism

In the mathematics field of topology, a homeomorphism or topological isomorphism is a continuous function between two topological spaces....
s of F. A G-atlas
Atlas (topology)

In mathematics, particularly topology, an atlas describes how a manifold is equipped with a differential structure. Each piece is given by a chart ....
 for the bundle (E, B, p, F) is a local trivialization such that for any two overlapping charts (Ui, fi) and (Uj, fj) the function is given by where tij : Ui n Uj ? G 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
Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the Differential structure....
 and the corresponding action on F is smooth and the transition functions are all smooth maps.

The transition functions tij satisfy the following conditions
The third condition applies on triple overlaps Ui n Uj n Uk and is called the cocycle condition (see Cech cohomology
Cech cohomology

In mathematics, specifically algebraic topology, Cech cohomology is a cohomology theory based on the intersection properties of open set cover of a topological space....
). The importance of this is that the transition functions determine the fibre bundle (if one assumes the Cech cocycle condition).

A principal G-bundle
Principal bundle

In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of a Cartesian product X × G of a space X with a group G....
 is a G-bundle where the fiber F is a principal homogeneous space
Principal homogeneous space

In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G such that the stabilizer subgroup of any point is trivial....
 for the left action of G itself (equivalently, one can specify that the action of G on the fibre 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 two pair of base spaces, and pE : E ? M and pF : F ? N are fiber bundles over M and N, respectively. A bundle map
Bundle map

In mathematics, a bundle map is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common fiber bundle....
 (or bundle morphism) consists of a pair of continuous functions such that . That is, the following diagram commutes
Commutative diagram

In mathematics, and especially in category theory a commutative diagram is a diagram of objects, also known as vertices, and morphisms, also known as arrows or edges, such that when selecting two objects any directed path through the diagram leads to the same result by composition....
: For fiber bundles with structure group G (such as a principal bundle), bundle morphisms are required as well to be G-equivariant
Equivariant

In mathematics, an equivariant map is a function between two Set that commutes with the group action. Specifically, let G be a group and let X and Y be two associated group action....
 on the fibers.

In case the base spaces M and N coincide, then a bundle morphism over M from the fiber bundle pE : E ? M to pF : F ? M is a map f : E ? F such that . That is, the diagram commutes

A bundle isomorphism is a bundle map which is also a homeomorphism

Differentiable fiber bundles

In the category of differentiable manifold
Differentiable manifold

A differentiable manifold is a type of manifold that is locally similar enough to Euclidean space to allow one to do calculus. This article deals with differentiability in different contexts including: smooth function, k times differentiable, and holomorphic function....
s, fiber bundles arise naturally as submersions
Submersion (mathematics)

In mathematics, a submersion is a differentiable map between differentiable manifolds whose pushforward is everywhere surjective. Explicitly, f : M → N is a submersion if...
 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. 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,p,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
Proper map

In mathematics, a continuous function between topological spaces is called proper if inverse images of compact space are compact. In algebraic geometry, the analogous concept is called a proper morphism....
, meaning that ƒ-1(K) is compact for every compact subset K of N. Another sufficient condition, due to , is that if ƒ : M ? N is a surjective submersion
Submersion (mathematics)

In mathematics, a submersion is a differentiable map between differentiable manifolds whose pushforward is everywhere surjective. Explicitly, f : M → N is a submersion if...
 with M and N differentiable manifold
Differentiable manifold

A differentiable manifold is a type of manifold that is locally similar enough to Euclidean space to allow one to do calculus. This article deals with differentiability in different contexts including: smooth function, k times differentiable, and holomorphic function....
s such that the preimage ƒ-1 is compact and connected for all x ? N, then ƒ admits a compatible fiber bundle structure .

Generalizations

  • The notion of a bundle
    Bundle (mathematics)

    In mathematics, a bundle is a generalization of a fiber bundle dropping the condition of a local product structure. The requirement of a local product structure rests on the bundle having a topological space....
     applies to many more categories in mathematics, at the expense of relaxing the local triviality condition.
  • In topology, a fibration
    Fibration

    In mathematics, especially algebraic topology, a fibration is a continuous function satisfying the homotopy lifting property with respect to any space....
     is a mapping p : E ? B which is homotopic to a fiber bundle.


See also

  • Covering map
    Covering map

    File:PSTricks-Cubriente.pngIn mathematics, more specifically algebraic topology, a covering map is a continuous function surjective function p from a topological space, C, to a topological space, X, such that each point in X has a neighbourhood evenly covered by p....
  • Fibration
    Fibration

    In mathematics, especially algebraic topology, a fibration is a continuous function satisfying the homotopy lifting property with respect to any space....
  • Gauge theory
    Gauge theory

    In physics, gauge theory is a quantum field theory where the Lagrangian is invariant under certain transformations.The transformations form a Lie group which is referred to as the symmetry group or the gauge group of the theory....
  • Hopf bundle
    Hopf bundle

    In the mathematical field of topology, the Hopf fibration describes a 3-sphere in terms of circles and an ordinary sphere. Discovered by Heinz Hopf in 1931, it is an influential early example of a fiber bundle....
  • Principal bundle
    Principal bundle

    In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of a Cartesian product X × G of a space X with a group G....
  • Pullback bundle
    Pullback bundle

    In mathematics, a pullback bundle or induced bundle is a useful construction in the theory of fiber bundles. Given a fiber bundle π : E → B and a continuous f : B′ → B one can define a "pullback" of E by f as a bundle f*E over B′....
  • Universal bundle
    Universal bundle

    In mathematics, the universal bundle in the theory of fiber bundles with structure group a given topological group G, is a specific bundle over a classifying space BG, such that every bundle with the given structure group G over M is a pullback bundle by means of a continuous map...
  • Vector bundle
    Vector bundle

    In mathematics, a vector bundle is a topology construction which makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together to form another space of the same kind as X , which is t...


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