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 ...
, a principal bundle is a mathematical object that formalizes some of the essential features of the
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\t ...
of a space
with a
group . In the same way as with the Cartesian product, a principal bundle
is equipped with
# An
action of
on
, analogous to
for a
product space.
# A projection onto
. For a product space, this is just the projection onto the first factor,
.
Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of
. Likewise, there is not generally a projection onto
generalizing the projection onto the second factor,
that exists for the Cartesian product. They may also have a complicated
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 ...
that 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
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts nat ...
of a
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 ...
, which consists of all ordered
bases of the vector space attached to each point. The group
in this case, is the
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
, which acts on the right
in the usual way: by
changes of basis. Since there is no natural 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
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 ...
and
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 mathematical
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 ...
. They have also found application in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
where they form part of the foundational framework of physical
gauge theories.
Formal definition
A principal
-bundle, where
denotes any
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 ...
, is a
fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...
together with 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 ...
right action such that
preserves the fibers of
(i.e. if
then
for all
) and acts
freely and
transitively (meaning each fiber is a
G-torsor) on them in such a way that for each
and
, the map
sending
to
is a homeomorphism. In particular each fiber of the bundle is homeomorphic to the group
itself. Frequently, one requires the base space
to be
Hausdorff and possibly
paracompact
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal ...
.
Since the group action preserves the fibers of
and acts transitively, it follows that the
orbits of the
-action are precisely these fibers and the orbit space
is
homeomorphic
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 isomor ...
to the base space
. Because the action is free and transitive, the fibers have the structure of G-torsors. A
-torsor is a space that is homeomorphic to
but lacks a group structure since there is no preferred choice of an
identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
.
An equivalent definition of a principal
-bundle is as a
-bundle
with fiber
where the structure group acts on the fiber by left multiplication. Since right multiplication by
on the fiber commutes with the action of the structure group, there exists an invariant notion of right multiplication by
on
. The fibers of
then become right
-torsors for this action.
The definitions above are for arbitrary topological spaces. One can also define principal
-bundles 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. Here
is required to be a
smooth map between smooth manifolds,
is required to be 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
should be smooth.
Examples
Trivial bundle and sections
Over an open ball
, or
, with induced coordinates
, any principal
-bundle is isomorphic to a trivial bundle
and a smooth section
is equivalently given by a (smooth) function
since
for some smooth function. For example, if
, the Lie group of
unitary matrices, then a section can be constructed by considering four real-valued functions
and applying them to the parameterization
This same procedure by taking a parameterization of a collection of matrices defining a Lie group and by considering the set of functions from a patch to
and inserting them into the parameterization.
Other examples
* The prototypical example of a smooth principal bundle is the
frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts nat ...
of a smooth manifold
, often denoted
or
. Here the fiber over a point
is the set of all frames (i.e. ordered bases) for the
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
. The
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
acts freely and transitively on these frames. These fibers can be glued together in a natural way so as to obtain a principal
-bundle over
.
* Variations on the above example include the
orthonormal frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts n ...
of 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 ...
. Here the frames are required to be
orthonormal
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of ...
with respect to the
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 ...
. The structure group is the
orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
. The example also works for bundles other than the tangent bundle; if
is any vector bundle of rank
over
, then the bundle of frames of
is a principal
-bundle, sometimes denoted
.
* A normal (regular)
covering space is a principal bundle where the structure group
:
: acts on the fibres of
via the
monodromy action. In particular, the
universal cover 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 spa ...
of
is a principal bundle over
with structure group
(since the universal cover is simply connected and thus
is trivial).
* Let
be a Lie group and let
be a closed subgroup (not necessarily
normal). Then
is a principal
-bundle over the (left)
coset space . Here the action of
on
is just right multiplication. The fibers are the left cosets of
(in this case there is a distinguished fiber, the one containing the identity, which is naturally isomorphic to
).
* Consider the projection
given by
. This principal
-bundle is the
associated bundle of 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 ...
. Besides the trivial bundle, this is the only principal
-bundle over
.
*
Projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
s provide some more interesting examples of principal bundles. Recall that the
-
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
is a two-fold covering space of
real projective space . The natural action of
on
gives it the structure of a principal
-bundle over
. Likewise,
is a principal
-bundle over
complex projective space and
is a principal
-bundle over
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 \mathbb. Quaternionic projective space of dimension ''n'' ...
. We then have a series of principal bundles for each positive
:
:
:
:
: Here
denotes the unit sphere in
(equipped with the Euclidean metric). For all of these examples the
cases give the so-called
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 ...
s.
Basic properties
Trivializations and cross sections
One of the most important questions regarding any fiber bundle is whether or not it is
trivial
Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense.
Latin Etymology
The ancient Romans used the word ''triviae'' to describe where one road split or fork ...
, ''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
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 sig ...
.''
The same is not true for other fiber bundles. For instance,
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 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 be a principal -bundle. 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 admits a local trivialization if and only if there exists a local section on . Given a local trivialization
:
one can define an associated local section
:
where is the
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), an ...
in . Conversely, given a section one defines a trivialization by
:
The simple transitivity of the action on the fibers of guarantees that this map is a
bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
, it is also 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 ...
. The local trivializations defined by local sections are -
equivariant in the following sense. If we write
:
in the form
:
then the map
:
satisfies
:
Equivariant trivializations therefore preserve the -torsor structure of the fibers. In terms of the associated local section the map 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 of , we have local sections on each . On overlaps these must be related by the action of the structure group . In fact, the relationship is provided by the
transition functions
:
By gluing the local trivializations together using these transition functions, one may reconstruct the original principal bundle. This is an example of the
fiber bundle construction theorem.
For any we have
:
Characterization of smooth principal bundles
If
is a smooth principal
-bundle then
acts freely and
properly on
so that the orbit space
is
diffeomorphic to the base space
. It turns out that these properties completely characterize smooth principal bundles. That is, if
is a smooth manifold,
a Lie group and
a smooth, free, and proper right action then
*
is a smooth manifold,
*the natural projection
is a smooth
submersion, and
*
is a smooth principal
-bundle over
.
Use of the notion
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
to
. The reason for this name is that the (fiberwise) inverse image of the values of this section form a subbundle of
that is a principal
-bundle. If
is the identity, then a section of
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
-bundle may be rephrased as questions about the admissibility of the reduction of the structure group (from
to
). For example:
* A
-dimensional real manifold admits an
almost-complex structure
In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not comple ...
if the
frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts nat ...
on the manifold, whose fibers are
, can be reduced to the group
.
* An
-dimensional real manifold admits a
-plane field if the frame bundle can be reduced to the structure group
.
* A manifold is
orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
if and only if its frame bundle can be reduced to the
special orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
,
.
* A manifold has
spin structure if and only if its frame bundle can be further reduced from
to
the
Spin group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As a ...
, which maps to
as a double cover.
Also note: an
-dimensional manifold admits
vector fields that are linearly independent at each point if and only if its
frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts nat ...
admits a global section. In this case, the manifold is called
parallelizable.
Associated vector bundles and frames
If
is a principal
-bundle and
is a
linear representation of
, then one can construct a vector bundle
with fibre
, as the quotient of the product
×
by the diagonal action of
. This is a special case of the
associated bundle construction, and
is called an
associated vector 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 ...
to
. If the representation of
on
is
faithful, so that
is a subgroup of the general linear group GL(
), then
is a
-bundle and
provides a reduction of structure group of the frame bundle of
from
to
. 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 admits a classifying space : the quotient by the action of of some
weakly contractible space, ''e.g.'', a topological space with vanishing
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, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...
s. The classifying space has the property that any principal bundle over a
paracompact
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal ...
manifold ''B'' is isomorphic to a
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: ...
of the principal bundle .
[, Theorem 2] In fact, more is true, as the set of isomorphism classes of principal bundles over the base identifies with the set of homotopy classes of maps .
See also
*
Associated 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 ...
*
G-structure
In differential geometry, a ''G''-structure on an ''n''-manifold ''M'', for a given structure group ''G'', is a principal ''G''- subbundle of the tangent frame bundle F''M'' (or GL(''M'')) of ''M''.
The notion of ''G''-structures includes vari ...
*
Reduction of the structure group
*
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 ...
*
Connection (principal bundle)
In mathematics, and especially differential geometry and gauge theory, 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. A principal ''G''-conn ...
*
G-fibration
References
Sources
*
*
*
*
*
{{DEFAULTSORT:Principal Bundle
Differential geometry
Fiber bundles
Group actions (mathematics)