Principal bundle
Encyclopedia
In mathematics
, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product
X × G of a space X with a group
G. In the same way as with the Cartesian product, a principal bundle P is equipped with
Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of (x,e). Likewise, there is not generally a projection onto G generalizing the projection onto the second factor, X × G → G which exists for the Cartesian product. They may also have a complicated topology
, which prevents them from being realized as a product space even if a number of arbitrary choices are made to try to define such a structure by defining it on smaller pieces of the space.
A common example of a principal bundle is the frame bundle
FE of a vector bundle
E, which consists of all ordered bases of the vector space attached to each point. The group G in this case is the general linear group
, which acts in the usual way on ordered bases. Since there is no preferred way to choose an ordered basis of a vector space, a frame bundle lacks a canonical choice of identity cross-section.
Principal bundles have important applications in topology
and differential geometry. They have also found application in physics
where they form part of the foundational framework of gauge theories
. Principal bundles provide a unifying framework for the theory of fiber bundles in the sense that all fiber bundles with structure group G determine a unique principal G-bundle from which the original bundle can be reconstructed.
, is a fiber bundle
π : P → X together with a continuous right action
P × G → P such that G preserves the fibers of P and acts freely and transitively on them. This implies that the fiber of the bundle is homeomorphic to the group G itself. Frequently, one requires the base space X to be Hausdorff
and possibly paracompact.
Since the group action preserves the fibers of π : P → X and acts transitively, it follows that the orbits of the G-action are precisely these fibers and the orbit space P/G is homeomorphic to the base space X. Because the action is free, the fibers have the structure of G-torsors. A G-torsor is a space which is homeomorphic to G but lacks a group structure since there is no preferred choice of an identity element
.
An equivalent definition of a principal G-bundle is as a G-bundle π : P → X with fiber G where the structure group acts on the fiber by left multiplication. Since right multiplication by G on the fiber commutes with the action of the structure group, there exists an invariant notion of right multiplication by G on P. The fibers of π then become right G-torsors for this action.
The definitions above are for arbitrary topological spaces. One can also define principal G-bundles in the category
of smooth manifolds. Here π : P → X is required to be a smooth map between smooth manifolds, G is required to be a Lie group
, and the corresponding action on P should be smooth.
of a smooth manifold M, often denoted FM or GL(M). Here the fiber over a point x in M is the set of all frames (i.e. ordered bases) for the tangent space
TxM. The general linear group
GL(n,R) acts freely and transitively on these frames. These fibers can be glued together in a natural way so as to obtain a principal GL(n,R)-bundle over M.
Variations on the above example include the orthonormal frame bundle of a Riemannian manifold
. Here the frames are required to be orthonormal with respect to the metric
. The structure group is the orthogonal group
O(n). The example also works for bundles other than the tangent bundle; if E is any vector bundle of rank k over M, then the bundle of frames of E is a principal GL(k,R)-bundle, sometimes denoted F(E).
A normal (regular) covering space p : C → X is a principal bundle where the structure group
acts on the fibres of p via the monodromy action. In particular, the universal cover of X is a principal bundle over X with structure group 
(since the universal cover is simply connected and thus 
is trivial).
Let G be a Lie group and let H be a closed subgroup (not necessarily normal
). Then G is a principal H-bundle over the (left) coset space G/H. Here the action of H on G is just right multiplication. The fibers are the left cosets of H (in this case there is a distinguished fiber, the one containing the identity, which is naturally isomorphic to H).
Consider the projection π: S1 → S1 given by z ↦ z2. This principal Z2-bundle is the associated bundle
of the Möbius strip
. Besides the trivial bundle, this is the only principal Z2-bundle over S1.
Projective space
s provide some more interesting examples of principal bundles. Recall that the n-sphere
Sn is a two-fold covering space of real projective space
RPn. The natural action of O(1) on Sn gives it the structure of a principal O(1)-bundle over RPn. Likewise, S2n+1 is a principal U(1)-bundle over complex projective space
CPn and S4n+3 is a principal Sp(1)-bundle over quaternionic projective space
HPn. We then have a series of principal bundles for each positive n:
Here S(V) denotes the unit sphere in V (equipped with the Euclidean metric). For all of these examples the n = 1 cases give the so-called Hopf bundle
s.
The same is not true for other fiber bundles. For instance, Vector bundle
s always have a zero section whether they are trivial or not and sphere bundles may admit many global sections without being trivial.
The same fact applies to local trivializations of principal bundles. Let π : P → X be a principal G-bundle. An open set
U in X admits a local trivialization if and only if there exists a local section on U. Given a local trivialization
one can define an associated local section 
by
where e is the identity
in G. Conversely, given a section s one defines a trivialization Φ by
The simple transitivity of the G action on the fibers of P guarantees that this map is a bijection
, it is also a homeomorphism
. The local trivializations defined by local sections are G-equivariant
in the following sense. If we write
in the form 
then the map 
satisfies
Equivariant trivializations therefore preserve the G-torsor structure of the fibers. In terms of the associated local section s the map φ is given by
The local version of the cross section theorem then states that the equivariant local trivializations of a principal bundle are in one-to-one correspondence with local sections.
Given an equivariant local trivialization ({Ui}, {Φi}) of P, we have local sections si on each Ui. On overlaps these must be related by the action of the structure group G. In fact, the relationship is provided by the transition function
s
For any x in Ui ∩ Uj we have
on P so that the orbit space P/G is diffeomorphic to the base space X. It turns out that these properties completely characterize smooth principal bundles. That is, if P is a smooth manifold, G a Lie group and μ : P × G → P a smooth, free, and proper right action then
whose fibers are homeomorphic to the coset space 
. If the new bundle admits a global section, then one says that the section is a reduction of the structure group from G to H . The reason for this name is that the (fiberwise) inverse image of the values of this section form a subbundle of P which is a principal H-bundle. If H is the identity, then a section of P itself is a reduction of the structure group to the identity. Reductions of the structure group do not in general exist.
Many topological questions about the structure of a manifold or the structure of bundles over it that are associated to a principal G-bundle may be rephrased as questions about the admissibility of the reduction of the structure group (from G to H). For example:
Also note: an n-dimensional manifold admits n vector fields that are linearly independent at each point if and only if its frame bundle
admits a global section. In this case, the manifold is called parallelizable.
with fibre V, as the quotient of the product P×V by the diagonal action of G. This is a special case of the associated bundle
construction, and E is called an associated vector bundle to P. If the representation of G on V is faithful
, so that G is a subgroup of the general linear group GL(V), then E is a G-bundle and P provides a reduction of structure group of the frame bundle of E from GL(V) to G. This is the sense in which principal bundles provide an abstract formulation of the theory of frame bundles.
space EG, i.e. a topological space for which all its homotopy group
s are trivial by a free action of G. The classifying space has the property that any G principal bundle over a paracompact manifold B is isomorphic to a pullback
of the principal bundle
. In fact, more is true, as the set of isomorphism classes of principal G bundles over the base B identifies with the set of homotopy classes of maps B → BG.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
X × G of a space X with a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
G. In the same way as with the Cartesian product, a principal bundle P is equipped with
- An actionGroup actionIn algebra and geometry, a group action is a way of describing symmetries of objects using groups. 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 transformations of the set...
of G on P, analogous to (x,g)h = (x, gh) for a product space. - A projection onto X. For a product space, this is just the projection onto the first factor, (x,g) → x.
Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of (x,e). Likewise, there is not generally a projection onto G generalizing the projection onto the second factor, X × G → G which exists for the Cartesian product. They may also have a complicated topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, which prevents them from being realized as a product space even if a number of arbitrary choices are made to try to define such a structure by defining it on smaller pieces of the space.
A common example of a principal bundle is 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 bases, or frames, for Ex...
FE of a vector 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 : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
E, which consists of all ordered bases of the vector space attached to each point. The group G in this case is the general linear group
General linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
, which acts in the usual way on ordered bases. Since there is no preferred way to choose an ordered basis of a vector space, a frame bundle lacks a canonical choice of identity cross-section.
Principal bundles have important applications in topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
and differential geometry. They have also found application in physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
where they form part of the foundational framework of gauge theories
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
. Principal bundles provide a unifying framework for the theory of fiber bundles in the sense that all fiber bundles with structure group G determine a unique principal G-bundle from which the original bundle can be reconstructed.
Formal definition
A principal G-bundle, where G denotes any topological groupTopological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
, is a fiber bundle
Fiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
π : P → X together with a continuous right action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. 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 transformations of the set...
P × G → P such that G preserves the fibers of P and acts freely and transitively on them. This implies that the fiber of the bundle is homeomorphic to the group G itself. Frequently, one requires the base space X to be Hausdorff
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
and possibly paracompact.
Since the group action preserves the fibers of π : P → X and acts transitively, it follows that the orbits of the G-action are precisely these fibers and the orbit space P/G is homeomorphic to the base space X. Because the action is free, the fibers have the structure of G-torsors. A G-torsor is a space which is homeomorphic to G but lacks a group structure since there is no preferred choice of an identity element
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
.
An equivalent definition of a principal G-bundle is as a G-bundle π : P → X with fiber G where the structure group acts on the fiber by left multiplication. Since right multiplication by G on the fiber commutes with the action of the structure group, there exists an invariant notion of right multiplication by G on P. The fibers of π then become right G-torsors for this action.
The definitions above are for arbitrary topological spaces. One can also define principal G-bundles in the category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
of smooth manifolds. Here π : P → X is required to be a smooth map between smooth manifolds, G is required to be 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 smooth structure...
, and the corresponding action on P should be smooth.
Examples
The prototypical example of a smooth principal bundle is the frame bundleFrame 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 bases, or frames, for Ex...
of a smooth manifold M, often denoted FM or GL(M). Here the fiber over a point x in M is the set of all frames (i.e. ordered bases) for the tangent space
Tangent space
In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
TxM. The general linear group
General linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
GL(n,R) acts freely and transitively on these frames. These fibers can be glued together in a natural way so as to obtain a principal GL(n,R)-bundle over M.
Variations on the above example include the orthonormal frame bundle of a Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
. Here the frames are required to be orthonormal with respect to the metric
Metric tensor
In the mathematical 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 vectors in Euclidean...
. The structure group is the orthogonal group
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
O(n). The example also works for bundles other than the tangent bundle; if E is any vector bundle of rank k over M, then the bundle of frames of E is a principal GL(k,R)-bundle, sometimes denoted F(E).
A normal (regular) covering space p : C → X is a principal bundle where the structure group
acts on the fibres of p via the monodromy action. In particular, the universal cover of X is a principal bundle over X with structure group (since the universal cover is simply connected and thus is trivial).Let G be a Lie group and let H be a closed subgroup (not necessarily normal
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
). Then G is a principal H-bundle over the (left) coset space G/H. Here the action of H on G is just right multiplication. The fibers are the left cosets of H (in this case there is a distinguished fiber, the one containing the identity, which is naturally isomorphic to H).
Consider the projection π: S1 → S1 given by z ↦ z2. This principal Z2-bundle is the associated bundle
Associated bundle
In mathematics, the theory of fiber bundles with a structure group G 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 a group action of G...
of the Möbius strip
Möbius strip
The Möbius strip or Möbius band is a surface with only one side and only one boundary component. The Möbius strip has the mathematical property of being non-orientable. It can be realized as a ruled surface...
. Besides the trivial bundle, this is the only principal Z2-bundle over S1.
Projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
s provide some more interesting examples of principal bundles. Recall that the n-sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
Sn is a two-fold covering space of real projective space
Real projective space
In mathematics, real projective space, or RPn, is the topological space of lines through 0 in Rn+1. It is a compact, smooth manifold of dimension n, and a special case of a Grassmannian.-Construction:...
RPn. The natural action of O(1) on Sn gives it the structure of a principal O(1)-bundle over RPn. Likewise, S2n+1 is a principal U(1)-bundle over complex projective space
Complex projective space
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines...
CPn and S4n+3 is a principal Sp(1)-bundle over quaternionic projective space
Quaternionic projective space
In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates lie in the ring of quaternions H. Quaternionic projective space of dimension n is usually denoted byand is a closed manifold of dimension 4n...
HPn. We then have a series of principal bundles for each positive n:
Here S(V) denotes the unit sphere in V (equipped with the Euclidean metric). For all of these examples the n = 1 cases give the so-called 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...
s.
Trivializations and cross sections
One of the most important questions regarding any fiber bundle is whether or not it is trivial, i.e. isomorphic to a product bundle. For principal bundles there is a convenient characterization of triviality:- Proposition. A principal bundle is trivial if and only if it admits a global cross section.
The same is not true for other fiber bundles. For instance, Vector 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 : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
s always have a zero section whether they are trivial or not and sphere bundles may admit many global sections without being trivial.
The same fact applies to local trivializations of principal bundles. Let π : P → X be a principal G-bundle. An open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
U in X admits a local trivialization if and only if there exists a local section on U. Given a local trivialization
one can define an associated local section bywhere e is the identity
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
in G. Conversely, given a section s one defines a trivialization Φ by
The simple transitivity of the G action on the fibers of P guarantees that this map is a bijection
Bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
, it is also a homeomorphism
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
. The local trivializations defined by local sections are G-equivariant
Equivariant
In mathematics, an equivariant map is a function between two sets that commutes with the action of a group. Specifically, let G be a group and let X and Y be two associated G-sets. A function f : X → Y is said to be equivariant iffor all g ∈ G and all x in X...
in the following sense. If we write
in the form then the map satisfiesEquivariant trivializations therefore preserve the G-torsor structure of the fibers. In terms of the associated local section s the map φ is given by
The local version of the cross section theorem then states that the equivariant local trivializations of a principal bundle are in one-to-one correspondence with local sections.
Given an equivariant local trivialization ({Ui}, {Φi}) of P, we have local sections si on each Ui. On overlaps these must be related by the action of the structure group G. In fact, the relationship is provided by the transition function
Transition function
In mathematics, a transition function has several different meanings:* In topology, a transition function is a homeomorphism from one coordinate chart to another...
s
For any x in Ui ∩ Uj we have
Characterization of smooth principal bundles
If π : P → X is a smooth principal G-bundle then G acts freely and properlyProper map
In mathematics, a continuous 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 :...
on P so that the orbit space P/G is diffeomorphic to the base space X. It turns out that these properties completely characterize smooth principal bundles. That is, if P is a smooth manifold, G a Lie group and μ : P × G → P a smooth, free, and proper right action then
- P/G is a smooth manifold,
- the natural projection π : P → P/G is a smooth submersionSubmersion (mathematics)In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology...
, and - P is a smooth principal G-bundle over P/G.
Reduction of the structure group
Given a subgroup H of G one may consider the bundle whose fibers are homeomorphic to the coset space . If the new bundle admits a global section, then one says that the section is a reduction of the structure group from G to H . The reason for this name is that the (fiberwise) inverse image of the values of this section form a subbundle of P which is a principal H-bundle. If H is the identity, then a section of P itself is a reduction of the structure group to the identity. Reductions of the structure group do not in general exist.Many topological questions about the structure of a manifold or the structure of bundles over it that are associated to a principal G-bundle may be rephrased as questions about the admissibility of the reduction of the structure group (from G to H). For example:
- A 2n-dimensional real manifold admits an almost-complex structure if the frame bundleFrame bundleIn 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 bases, or frames, for Ex...
on the manifold, whose fibers are
, can be reduced to the group 
.
- An n-dimensional real manifold admits a k-plane field if the frame bundle can be reduced to the structure group
.
- A manifold is orientable if and only if its frame bundle can be reduced to the special orthogonal group,
.
- A manifold has spin structureSpin structureIn differential geometry, a spin structure on an orientable Riemannian manifold \,allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry....
if and only if its frame bundle can be further reduced from
to 
the Spin group, which maps to 
as a double cover.
Also note: an n-dimensional manifold admits n vector fields that are linearly independent at each point if and only if its 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 bases, or frames, for Ex...
admits a global section. In this case, the manifold is called parallelizable.
Associated vector bundles and frames
If P is a principal G-bundle and V is a linear representation of G, then one can construct a vector bundle with fibre V, as the quotient of the product P×V by the diagonal action of G. This is a special case of the associated bundleAssociated bundle
In mathematics, the theory of fiber bundles with a structure group G 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 a group action of G...
construction, and E is called an associated vector bundle to P. If the representation of G on V is faithful
Faithful representation
In mathematics, especially in the area of abstract algebra known as representation theory, a faithful representation ρ of a group G on a vector space V is a linear representation in which different elements g of G are represented by distinct linear mappings ρ.In more abstract language, this means...
, so that G is a subgroup of the general linear group GL(V), then E is a G-bundle and P provides a reduction of structure group of the frame bundle of E from GL(V) to G. This is the sense in which principal bundles provide an abstract formulation of the theory of frame bundles.
Classification of principal bundles
Any topological group G admits a classifying space BG: the quotient of some weakly contractibleWeakly contractible
In mathematics, a topological space is said to be weakly contractible if all of its homotopy groups are trivial.-Property:It follows from Whitehead's Theorem that if a CW-complex is weakly contractible then it is contractible.-Example:...
space EG, i.e. a topological space for which all its homotopy group
Homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
s are trivial by a free action of G. The classifying space has the property that any G principal bundle over a paracompact manifold B is isomorphic to a pullback
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 map f : B′ → B one can define a "pullback" of E by f as a bundle f*E over B′...
of the principal bundle
. In fact, more is true, as the set of isomorphism classes of principal G bundles over the base B identifies with the set of homotopy classes of maps B → BG.See also
- Associated bundleAssociated bundleIn mathematics, the theory of fiber bundles with a structure group G 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 a group action of G...
- Vector bundleVector bundleIn mathematics, a vector bundle is a topological construction that 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...
- G-structureG-structureIn differential geometry, a G-structure on an n-manifold M, for a given structure group G, is a G-subbundle of the tangent frame bundle FM of M....
- Gauge theoryGauge theoryIn physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
- Connection (principal bundle)Connection (principal bundle)In mathematics, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points...