Principal bundle on arvau

Wikipedia / Principal bundle

In the mathematical area of topology, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product of a topological space with a group is equipped with

An action for a product space; the group action is conventionally a right action).
.

that exists for the Cartesian product. It may also have a complicated topology that prevents it from being realized as a product space.

-torsors. Since this bundle looks like a helix, there is no canonical choice of identity cross-section, which should look like a circle.

Another example of a principal bundle is the frame bundle of a vector bundle, which consists of all ordered bases in this case, is the general linear group, 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 and differential geometry and mathematical gauge theory. They have also found application in physics where they form part of the foundational framework of physical gauge theories. Important cases are principal U(1)-bundles and principal SU(2)-bundles.

Formal definition

denotes any topological group, is a fiber bundle together with a continuous right action) and acts freely and transitively (meaning each fiber is a G-torsor to be Hausdorff and possibly paracompact.

and acts transitively, it follows that the orbits is homeomorphic but lacks a group structure since there is no preferred choice of an identity element.

-torsors for this action.

-bundles in the category of smooth manifolds is required to be a smooth map is required to be a Lie group should be smooth.

Examples

Trivial bundle and sections

-bundle is isomorphic to a trivial bundle since unitary matrices, then a section can be constructed by considering four real-valued functionsand applying them to the parameterization

and inserting them into the parameterization.

Other examples

The prototypical example of a smooth principal bundle is the frame bundle is the set of all frames (i.e. ordered bases) for the tangent space. The general linear group.
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.
A normal (regular) covering space is a principal bundle where the structure group
via the monodromy action. Intuitively, G is the group of loops in base space that are unliftable. In particular, the universal cover is trivial).
be a closed subgroup (not necessarily normal-bundle over the (left) coset space).
. One is the trivial bundle. The other is the associated bundle of the Möbius strip.
Projective spaces-sphere is a two-fold covering space of real projective space-bundle over complex projective space-bundle over quaternionic projective space is the group of unit quaternions, isomorphic to the 3-sphere, , and :
cases give the so-called Hopf bundles.

Basic properties

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 section.

The same is not true in general for other fiber bundles. For instance, vector bundles always have a zero section whether they are trivial or not and sphere bundles may admit many global sections without being trivial (such as the Klein bottle).

The same fact applies to local trivializations of principal bundles. Let be a principal -bundle. An open set 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 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, it is also a homeomorphism. 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

acts freely and properly is diffeomorphic a smooth, free, and proper right action then

is a smooth manifold,
is a smooth submersion, and
.

Use of the notion

Reduction of the structure group

of whose fibers are homeomorphic to the coset space. If the new bundle admits a global section, then one says that the section is a itself is a reduction of the structure group to the identity. Reductions of the structure group do not in general exist.

). For example:

-dimensional real manifold admits an almost-complex structure if the frame bundle.
.
A manifold is orientable if and only if its frame bundle can be reduced to the special orthogonal group.
A manifold has spin structure the Spin group as a double cover.

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.

Associated vector bundles and frames

is a linear representation. This is a special case of the associated bundle is called an associated vector bundle is faithful. 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 groups. The classifying space has the property that any 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 bundles over the base identifies with the set of homotopy classes of maps .