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
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, 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
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consi ...
that is a
product space, but may have a different
topological structure. Specifically, the similarity between a space
and a product space
is defined using a
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
surjective
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 o ...
map,
that in small regions of
behaves just like a projection from corresponding regions of
to
The map
called the projection or
submersion of the bundle, is regarded as part of the structure of the bundle. The space
is known as the total space of the fiber bundle,
as the base space, and
the fiber.
In the ''trivial'' case,
is just
and the map
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 A ...
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 a ...
, as well as nontrivial
covering spaces. Fiber bundles, such as the
tangent bundle 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 n ...
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 p ...
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 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
is called a
section of
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 st ...
, known as the structure group, acting on the fiber
.
History
In
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, 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 atten ...
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 p ...
s,
principal bundles, topological
fibrations and
fibered manifolds 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 Eliza ...
,
Jacques Feldbau, Whitney,
Norman Steenrod,
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 differential ...
,
Jean-Pierre Serre, 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
where
and
are
topological spaces
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 point ...
and
is a
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
surjection satisfying a ''local triviality'' condition outlined below. The space
is called the of the bundle,
the , and
the . The map
is called the (or ). We shall assume in what follows that the base space
is
connected.
We require that for every
, 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 area, ...
of
(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 isom ...
(where
is given the
subspace topology, and
is the product space) in such a way that
agrees with the projection onto the first factor. That is, the following diagram should
commute
Commute, commutation or commutative may refer to:
* Commuting, the process of travelling between a place of residence and a place of work
Mathematics
* Commutative property, a property of a mathematical operation whose result is insensitive to th ...
:
Local triviality condition, 230px, center
where
is the natural projection and
is a homeomorphism. The set of all
is called a of the bundle.
Thus for any
, the
preimage is homeomorphic to
(since this is true of
) and is called the fiber over
Every fiber bundle
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
carries the
quotient topology determined by the map
A fiber bundle
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 of
smooth manifolds. That is,
and
are required to be smooth manifolds and all the functions above are required to be
smooth maps.
Examples
Trivial bundle
Let
and let
be the projection onto the first factor. Then
is a fiber bundle (of
) over
Here
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 th ...
CW-complex is trivial.
Nontrivial bundles
Möbius strip
Perhaps the simplest example of a nontrivial bundle
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 A ...
. It has the
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
that runs lengthwise along the center of the strip as a base
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 i ...
for the fiber
, 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 area, ...
of
(where
) 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 (i.e. all the points that project to
).
A homeomorphism (
in ) exists that maps the preimage of
(the trivializing neighborhood) to a slice of a cylinder: curved, but not twisted. This pair locally trivializes the strip. The corresponding trivial bundle
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 ...
, 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 a ...
, which can be viewed as a "twisted" circle bundle over another circle. The corresponding non-twisted (trivial) bundle is the 2-
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does n ...
,
.
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
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest top ...
.
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 p ...
s, are those whose fibers are
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s (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
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. Th ...
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
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
-bundle. The group
is also the structure group of the bundle. Given a
representation of
on a vector space
, a vector bundle with
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
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 mathe ...
(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
is the set of all unit vectors in
. When the vector bundle in question is the tangent bundle
, the unit sphere bundle is known as the
unit tangent bundle
In Riemannian geometry, the unit tangent bundle of a Riemannian manifold (''M'', ''g''), denoted by T1''M'', UT(''M'') or simply UT''M'', is the unit sphere bundle for the tangent bundle T(''M''). It is a fiber bundle over ''M'' whose fiber at each ...
.
A sphere bundle is partially characterized by its
Euler class
In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle ...
, which is a degree
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be view ...
class in the total space of the bundle. In the case
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
, given the Euler class of a bundle, one can calculate its cohomology using a
long exact sequence called the
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 sphere bundle. The Gysin sequence is a useful tool for ...
.
Mapping tori
If
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 poin ...
and
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 isom ...
then the
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''. Take the cartesian product of ''X'' with a closed interval ''I'', and glue the bound ...
has a natural structure of a fiber bundle over the
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
with fiber
Mapping tori of homeomorphisms of surfaces are of particular importance in
3-manifold topology.
Quotient spaces
If
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 st ...
and
is a
closed subgroup
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 ...
, then under some circumstances, the
quotient space together with the quotient map
is a fiber bundle, whose fiber is the topological space
. A necessary and sufficient condition for (
) to form a fiber bundle is that the mapping
admits
local cross-sections .
The most general conditions under which the quotient map will admit local cross-sections are not known, although if
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 addi ...
and
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 ...
,
, which is a fiber bundle over the sphere
whose total space is
. From the perspective of Lie groups,
can be identified with the
special unitary group . The abelian subgroup of diagonal matrices is isomorphic to the
circle group , and the quotient
is
diffeomorphic to the sphere.
More generally, if
is any topological group and
a closed subgroup that also happens to be a Lie group, then
is a fiber bundle.
Sections
A (or cross section) of a fiber bundle
is a continuous map
such that
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, where the
Euler class
In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle ...
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
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 su ...
in ''B'' and
for all ''x'' in ''U''. If
is a local trivialization chart then local sections always exist over ''U''. Such sections are in 1-1 correspondence with continuous maps
. Sections form a
sheaf.
Structure groups and transition functions
Fiber bundles often come with a
group 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 st ...
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 isom ...
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
is a set of local trivialization charts
such that for any
for the overlapping charts
and
the function
is given by
where
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 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 addi ...
and the corresponding action on ''F'' is smooth and the transition functions are all smooth maps.
The transition functions
satisfy the following conditions
#
#
#
The third condition applies on triple overlaps ''U
i'' ∩ ''U
j'' ∩ ''U
k'' and is called the
cocycle
In mathematics a cocycle is a closed cochain. Cocycles are used in algebraic topology to express obstructions (for example, to integrating a differential equation on a closed manifold). They are likewise used in group cohomology. In autonomous ...
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, 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
and
are fiber bundles over ''M'' and ''N'', respectively. A or consists of a pair of continuous functions
such that
That is, the following diagram is
commutative:
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
is also ''G''-morphism from one ''G''-space to another, that is,
for all
and
In case the base spaces ''M'' and ''N'' coincide, then a bundle morphism over ''M'' from the fiber bundle
to
is a map
such that
This means that the bundle map
covers the identity of ''M''. That is,
and the following diagram commutes:
Assume that both
and
are defined over the same base space ''M''. A bundle isomorphism is a bundle map
between
and
such that
and such that
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 ma ...
s, fiber bundles arise naturally as
submersions of one manifold to another. Not every (differentiable) submersion
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
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
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
gives rise to a fiber bundle in the sense that there is a fiber space ''F'' diffeomorphic to each of the fibers such that
is a fiber bundle. (Surjectivity of
follows by the assumptions already given in this case.) More generally, the assumption of compactness can be relaxed if the submersion
is assumed to be a surjective
proper map, meaning that
is compact for every compact subset ''K'' of ''N''. Another sufficient condition, due to , is that if
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 ma ...
s such that the preimage
is compact and connected for all
then
admits a compatible fiber bundle structure .
Generalizations
* The notion of a
bundle
Bundle or Bundling may refer to:
* Bundling (packaging), the process of using straps to bundle up items
Biology
* Bundle of His, a collection of heart muscle cells specialized for electrical conduction
* Bundle of Kent, an extra conduction path ...
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 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 topo ...
.
* In topology, a
fibration is a mapping
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 quantifier ...
.
See also
*
Affine bundle
In mathematics, an affine bundle is a fiber bundle whose typical fiber, fibers, trivialization morphisms and transition functions are affine.. (page 60)
Formal definition
Let \overline\pi:\overline Y\to X be a vector bundle with a typical fiber ...
*
Algebra bundle In mathematics, an algebra bundle is a fiber bundle whose fibers are algebras and local trivializations respect the algebra structure. It follows that the transition functions are algebra isomorphisms. Since algebras are also vector spaces, e ...
*
Characteristic class
*
Covering map
*
Equivariant bundle In geometry and topology, given a group ''G'', an equivariant bundle is a fiber bundle such that the total space and the base spaces are both ''G''-spaces and the projection map \pi between them is equivariant: \pi \circ g = g \circ \pi with som ...
*
Fibered manifold
*
Fibration
*
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
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 Hop ...
*
I-bundle
In mathematics, an I-bundle is a fiber bundle whose fiber is an interval and whose base is a manifold. Any kind of interval, open, closed, semi-open, semi-closed, open-bounded, compact, even rays, can be the fiber. An I-bundle is said to be ...
*
Natural bundle In mathematics, a natural bundle is any fiber bundle associated to the ''s''-frame bundle F^s(M) for some s \geq 1. It turns out that its transition functions depend functionally on local changes of coordinates in the base manifold M together with ...
*
Principal bundle
*
Projective bundle
In mathematics, a projective bundle is a fiber bundle whose fibers are projective spaces.
By definition, a scheme ''X'' over a Noetherian scheme ''S'' is a P''n''-bundle if it is locally a projective ''n''-space; i.e., X \times_S U \simeq \math ...
*
Pullback bundle
*
Quasifibration
In algebraic topology, a quasifibration is a generalisation of fibre bundles and fibrations introduced by Albrecht Dold and René Thom. Roughly speaking, it is a continuous map ''p'': ''E'' → ''B'' having the same behaviour as a fibration regar ...
*
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 p ...
Notes
References
*
*
*
*
*
*
*
External links
Fiber Bundle PlanetMath
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Sardanashvily, Gennadi, Fibre bundles, jet manifolds and Lagrangian theory. Lectures for theoreticians,
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