Fiber bundle

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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, and particularly topology, a fiber bundle (or, in
Commonwealth English The use of the English language in current and former member countries of the Commonwealth of Nations was largely inherited from British colonisation, with some exceptions. English serves as the medium of inter-Commonwealth relations. Many ...
: fibre bundle) is a space that is a product space, but may have a different
topological structure In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
. Specifically, the similarity between a space $E$ and a product space $B \times F$ is defined using a continuous surjective map, $\pi : E \to B,$ that in small regions of $E$ behaves just like a projection from corresponding regions of $B \times F$ to $B.$ The map $\pi,$ 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 $B \times F,$ and the map $\pi$ 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 In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and Au ...
and
Klein bottle In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is ...
, as well as nontrivial
covering space A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete sp ...
s. Fiber bundles, such as the
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
of a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
and other more general
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every ...
s, play an important role in
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
and
differential topology In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
, as do principal bundles. Mappings between total spaces of fiber bundles that "commute" with the projection maps are known as
bundle map In mathematics, a bundle map (or bundle morphism) 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 base space. The ...
s, and the class of fiber bundles forms a
category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce ...
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, Sectioning or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sign ...
of $E.$ Fiber bundles can be specialized in a number of ways, the most common of which is requiring that the transition maps between the local trivial patches lie in a certain
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two s ...
, known as the structure group, acting on the fiber $F$.

# History

In topology, the terms ''fiber'' (German: ''Faser'') and ''fiber space'' (''gefaserter Raum'') appeared for the first time in a paper by
Herbert Seifert Herbert Karl Johannes Seifert (; 27 May 1897, Bernstadt – 1 October 1996, Heidelberg) was a German mathematician known for his work in topology. Biography Seifert was born in Bernstadt auf dem Eigen, but soon moved to Bautzen, where he attend ...
in 1933, 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 under the name sphere space, but in 1940 Whitney changed the name to sphere bundle. The theory of fibered spaces, of which
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every ...
s, principal bundles, topological
fibration The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in postnikov-systems or obstruction theory. In this article, all map ...
s and
fibered manifold In differential geometry, in the category of differentiable manifolds, a fibered manifold is a surjective submersion \pi : E \to B\, that is, a surjective differentiable mapping such that at each point y \in U the tangent mapping T_y \pi : T_ E ...
s are a special case, is attributed to Seifert,
Heinz Hopf Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of topology and geometry. Early life and education Hopf was born in Gräbschen, Germany (now , part of Wrocław, Poland), the son of Elizabeth ( ...
,
Jacques Feldbau Jacques Feldbau was a French mathematician, born on 22 October 1914 in Strasbourg, of an Alsatian Jewish traditionalist family. He died on 22 April 1945 at the ''Ganacker'' Camp, annex of the concentration camp of Flossenbürg in Germany. As a ...
, Whitney,
Norman Steenrod Norman Earl Steenrod (April 22, 1910October 14, 1971) was an American mathematician most widely known for his contributions to the field of algebraic topology. Life He was born in Dayton, Ohio, and educated at Miami University and University o ...
,
Charles Ehresmann Charles Ehresmann (19 April 1905 – 22 September 1979) was a German-born French mathematician who worked in differential topology and category theory. He was an early member of the Bourbaki group, and is known for his work on the differentia ...
,
Jean-Pierre Serre Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
, 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. Whitney came to the general definition of a fiber bundle from his study of a more particular notion of a sphere bundle, that is a fiber bundle whose fiber is a sphere of arbitrary dimension.

# Formal definition

A fiber bundle is a structure $\left(E,\, B,\, \pi,\, F\right),$ where $E, B,$ and $F$ are topological spaces and $\pi : E \to B$ is a continuous
surjection In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
satisfying a ''local triviality'' condition outlined below. The space $B$ is called the of the bundle, $E$ the , and $F$ the . The map $\pi$ is called the (or ). We shall assume in what follows that the base space $B$ is connected. We require that for every $x \in B$, there is an open
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural ...
$U \subseteq B$ of $x$ (which will be called a trivializing neighborhood) such that there is a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomo ...
$\varphi : \pi^\left(U\right) \to U \times F$ (where $\pi^\left(U\right)$ is given the
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induce ...
, and $U \times F$ is the product space) in such a way that $\pi$ agrees with the projection onto the first factor. That is, the following diagram should commute: Local triviality condition, 230px, center where $\operatorname_1 : U \times F \to U$ is the natural projection and $\varphi : \pi^\left(U\right) \to U \times F$ is a homeomorphism. The set of all $\left\$ is called a of the bundle. Thus for any $p \in B$, the
preimage In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
$\pi^\left(\\right)$ is homeomorphic to $F$ (since this is true of $\operatorname_1^\left(\\right)$) and is called the fiber over $p.$ Every fiber bundle $\pi : E \to B$ is an
open map In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y. Likewise ...
, since projections of products are open maps. Therefore $B$ carries the
quotient topology In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient t ...
determined by the map $\pi.$ A fiber bundle $\left(E,\, B,\, \pi,\, F\right)$ is often denoted 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 is a fiber bundle in the
category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce ...
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 map In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
s.

# Examples

## Trivial bundle

Let $E = B \times F$ and let $\pi : E \to B$ be the projection onto the first factor. Then $\pi$ 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 . Any fiber bundle over a
contractible 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. Intuitively, a contractible space is one that can be continuously shrunk to a point within tha ...
CW-complex is trivial.

## Nontrivial bundles

### Möbius strip

Perhaps the simplest example of a nontrivial bundle $E$ is the
Möbius strip In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and Au ...
. It has the circle that runs lengthwise along the center of the strip as a base $B$ and a
line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
for the fiber $F$, so the Möbius strip is a bundle of the line segment over the circle. A
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural ...
$U$ of $\pi\left(x\right) \in B$ (where $x \in E$) is an arc; in the picture, this is the length of one of the squares. The
preimage In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
$\pi^\left(U\right)$ 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 $U$). A homeomorphism ($\varphi$ in ) exists that maps the preimage of $U$ (the trivializing neighborhood) to a slice of a cylinder: curved, but not twisted. This pair locally trivializes the strip. The corresponding trivial bundle $B\times F$ would be a
cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an inf ...
, 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 In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is ...
, which can be viewed as a "twisted" circle bundle over another circle. The corresponding non-twisted (trivial) bundle is the 2- torus, $S^1 \times S^1$.

## Covering map

A
covering space A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete sp ...
is a fiber bundle such that the bundle projection is a
local homeomorphism In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure. If f : X \to Y is a local homeomorphism, X is said to be an ...
. It follows that the fiber is a discrete space.

## Vector and principal bundles

A special class of fiber bundles, called
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every ...
s, 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 In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
and
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. It may be described also as the dual bundle to the tangent bundle. This ...
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 In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non-em ...
. 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 Representation may refer to: Law and politics *Representation (politics), political activities undertaken by elected representatives, as well as other theories ** Representative democracy, type of democracy in which elected officials represent a ...
$\rho$ of $G$ on a vector space $V$, a vector bundle with $\rho\left(G\right) \subseteq \text\left(V\right)$ as a structure group may be constructed, known as the
associated bundle In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces with ...
.

## Sphere bundles

A sphere bundle is a fiber bundle whose fiber is an ''n''-sphere. Given a vector bundle $E$ with a
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...
(such as the tangent bundle to a
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space '' ...
) one can construct the associated unit sphere bundle, for which the fiber over a point $x$ is the set of all unit vectors in $E_x$. When the vector bundle in question is the tangent bundle $TM$, the unit sphere bundle is known as the unit tangent bundle. 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 In mathematics, a circle bundle is a fiber bundle where the fiber is the circle S^1. Oriented circle bundles are also known as principal ''U''(1)-bundles. In physics, circle bundles are the natural geometric setting for electromagnetism. A circl ...
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 An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the contex ...
called the Gysin sequence.

## Mapping tori

If $X$ is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
and $f : X \to X$ is a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomo ...
then the mapping torus $M_f$ 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 In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two s ...
and $H$ is a closed subgroup, then under some circumstances, the quotient space $G/H$ together with the quotient map $\pi : G \to G/H$ is a fiber bundle, whose fiber is the topological space $H$. A necessary and sufficient condition for ($G,\, G/H,\, \pi,\, H$) to form a fiber bundle is that the mapping $\pi$ admits 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 In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the add ...
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 In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz Ho ...
, $S^3 \to S^2$, which is a fiber bundle over the sphere $S^2$ whose total space is $S^3$. From the perspective of Lie groups, $S^3$ can be identified with the
special unitary group In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the speci ...
$SU\left(2\right)$. The abelian subgroup of diagonal matrices is isomorphic to the
circle group In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers. \mathbb T = \. ...
$U\left(1\right)$, and the quotient $SU\left(2\right)/U\left(1\right)$ is
diffeomorphic In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given tw ...
to the sphere. More generally, if $G$ is any topological group and $H$ a closed subgroup that also happens to be a Lie group, then $G \to G/H$ is a fiber bundle.

# Sections

A (or cross section) of a fiber bundle $\pi$ is a continuous map $f : B \to E$ such that $\pi\left(f\left(x\right)\right) = x$ 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 Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
. The most well-known example is the
hairy ball theorem The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) states that there is no nonvanishing continuous tangent vector field on even-dimensional ''n''-spheres. For the ordinary sphere, or 2‑sphere, i ...
, where the Euler class is the obstruction to the
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
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 $f : U \to E$ where ''U'' is an
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are s ...
in ''B'' and $\pi\left(f\left(x\right)\right) = x$ for all ''x'' in ''U''. If $\left(U,\, \varphi\right)$ is a local trivialization chart then local sections always exist over ''U''. Such sections are in 1-1 correspondence with continuous maps $U \to F$. Sections form a sheaf.

# Structure groups and transition functions

Fiber bundles often come with a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic id ...
of symmetries that describe the matching conditions between overlapping local trivialization charts. Specifically, let ''G'' be a
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two s ...
that acts continuously on the fiber space ''F'' on the left. We lose nothing if we require ''G'' to act faithfully on ''F'' so that it may be thought of as a group of
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomo ...
s of ''F''. A ''G''-
atlas An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth. Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geogra ...
for the bundle $\left(E, B, \pi, F\right)$ is a set of local trivialization charts $\$ such that for any $\varphi_i,\varphi_j$ for the overlapping charts $\left(U_i,\, \varphi_i\right)$ and $\left(U_j,\, \varphi_j\right)$ the function $\varphi_i\varphi_j^ : \left(U_i \cap U_j\right) \times F \to \left(U_i \cap U_j\right) \times F$ is given by $\varphi_i\varphi_j^(x,\, \xi) = \left(x,\, t_(x)\xi\right)$ where $t_ : U_i \cap U_j \to G$ is a continuous map called a . 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 of the bundle; the analogous term in physics is
gauge group In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie grou ...
. In the smooth category, a ''G''-bundle is a smooth fiber bundle where ''G'' is a
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the add ...
and the corresponding action on ''F'' is smooth and the transition functions are all smooth maps. The transition functions $t_$ satisfy the following conditions # $t_\left(x\right) = 1\,$ # $t_\left(x\right) = t_\left(x\right)^\,$ # $t_\left(x\right) = t_\left(x\right)t_\left(x\right).\,$ The third condition applies on triple overlaps ''Ui'' ∩ ''Uj'' ∩ ''Uk'' 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 In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non-em ...
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. 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

It is useful to have notions of a mapping between two fiber bundles. Suppose that ''M'' and ''N'' are base spaces, and $\pi_E : E \to M$ and $\pi_F : F \to N$ are fiber bundles over ''M'' and ''N'', respectively. A or consists of a pair of continuous functions $\varphi : E \to F,\quad f : M \to N$ such that $\pi_F\circ \varphi = f \circ \pi_E.$ That is, the following diagram is
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
: 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 In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, ...
on the fibers. This means that $\varphi : E \to F$ is also ''G''-morphism from one ''G''-space to another, that is, $\varphi\left(xs\right) = \varphi\left(x\right)s$ 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 $\pi_E : E \to M$ to $\pi_F : F \to M$ is a map $\varphi : E \to F$ such that $\pi_E = \pi_F \circ \varphi.$ This means that the bundle map $\varphi : E \to F$ covers the identity of ''M''. That is, $f \equiv \mathrm_$ and the following diagram commutes: Assume that both $\pi_E : E \to M$ and $\pi_F : F \to M$ are defined over the same base space ''M''. A bundle isomorphism is a bundle map $\left(\varphi,\, f\right)$ between $\pi_E : E \to M$ and $\pi_F : F \to M$ such that $f \equiv \mathrm_M$ and such that $\varphi$ is also a homeomorphism. Or is, at least, invertible in the appropriate category; e.g., a diffeomorphism.

# Differentiable fiber bundles

In the category of
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One m ...
s, fiber bundles arise naturally as submersions of one manifold to another. Not every (differentiable) submersion $f : M \to N$ 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 $\left(M, N, f\right)$ is called a
fibered manifold In differential geometry, in the category of differentiable manifolds, a fibered manifold is a surjective submersion \pi : E \to B\, that is, a surjective differentiable mapping such that at each point y \in U the tangent mapping T_y \pi : T_ E ...
. 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 Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
and connected, then any submersion $f : M \to N$ gives rise to a fiber bundle in the sense that there is a fiber space ''F'' diffeomorphic to each of the fibers such that $\left(E, B, \pi, F\right) = \left(M, N, f, F\right)$ is a fiber bundle. (Surjectivity of $f$ follows by the assumptions already given in this case.) More generally, the assumption of compactness can be relaxed if the submersion $f : M \to N$ is assumed to be a surjective
proper map In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism. Definition There are several competing defini ...
, meaning that $f^\left(K\right)$ is compact for every compact subset ''K'' of ''N''. Another sufficient condition, due to , is that if $f : M \to N$ is a surjective submersion with ''M'' and ''N''
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One m ...
s such that the preimage $f^\$ is compact and connected for all $x \in N,$ then $f$ admits a compatible fiber bundle structure .

# Generalizations

* 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 In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non-em ...
and
torsor (algebraic geometry) In algebraic geometry, a torsor or a principal bundle is an analog of a principal bundle in algebraic topology. Because there are few open sets in Zariski topology, it is more common to consider torsors in étale topology or some other flat topolo ...
. * In topology, a
fibration The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in postnikov-systems or obstruction theory. In this article, all map ...
is a mapping $\pi : E \to B$ that 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 for details). This is the defining property of a fibration. * A section of a fiber bundle is a "function whose output range is continuously dependent on the input." This property is formally captured in the notion of
dependent type In computer science and logic, a dependent type is a type whose definition depends on a value. It is an overlapping feature of type theory and type systems. In intuitionistic type theory, dependent types are used to encode logic's quantifiers lik ...
.

* Affine bundle * Algebra bundle * Characteristic class * Covering map * Equivariant bundle *
Fibered manifold In differential geometry, in the category of differentiable manifolds, a fibered manifold is a surjective submersion \pi : E \to B\, that is, a surjective differentiable mapping such that at each point y \in U the tangent mapping T_y \pi : T_ E ...
*
Fibration The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in postnikov-systems or obstruction theory. In this article, all map ...
*
Gauge theory In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie grou ...
* Hopf bundle * I-bundle * Natural bundle * Principal bundle * Projective bundle *
Pullback bundle In mathematics, a pullback bundle or induced bundle is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle and a continuous map one can define a "pullback" of by as a bundle over . The fiber of over a point in ...
* Quasifibration * Universal bundle *
Vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every ...

* * * * * * *