Principal bundle

In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product formula_1 of a space formula_2 with a group formula_3. In the same way as with the Cartesian product, a principal bundle formula_4 is equipped with
Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of formula_10. Likewise, there is not generally a projection onto formula_3 generalizing the projection onto the second factor, formula_12 that exists for the Cartesian product. They may also have a complicated topology 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 formula_13 of a vector bundle formula_14, which consists of all ordered bases of the vector space attached to each point. The group formula_15 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. 
Formal definition.
A principal formula_3-bundle, where formula_3 denotes any topological group, is a fiber bundle formula_18 together with a continuous right action formula_19 such that formula_3 preserves the fibers of formula_4 (i.e. if formula_22 then formula_23 for all formula_24) and acts freely and transitively (i.e. regularly) on them in such a way that for each formula_25 and formula_22, the map formula_27 sending formula_28 to formula_29 is a homeomorphism. In particular each fiber of the bundle is homeomorphic to the group formula_3 itself. Frequently, one requires the base space formula_2 to be Hausdorff and possibly paracompact.
Since the group action preserves the fibers of formula_18 and acts transitively, it follows that the orbits of the formula_3-action are precisely these fibers and the orbit space formula_34 is homeomorphic to the base space formula_2. Because the action is free, the fibers have the structure of G-torsors. A formula_3-torsor is a space that is homeomorphic to formula_3 but lacks a group structure since there is no preferred choice of an identity element.
An equivalent definition of a principal formula_3-bundle is as a formula_3-bundle formula_18 with fiber formula_3 where the structure group acts on the fiber by left multiplication. Since right multiplication by formula_3 on the fiber commutes with the action of the structure group, there exists an invariant notion of right multiplication by formula_3 on formula_4. The fibers of formula_45 then become right formula_3-torsors for this action.
The definitions above are for arbitrary topological spaces. One can also define principal formula_3-bundles in the category of smooth manifolds. Here formula_18 is required to be a smooth map between smooth manifolds, formula_3 is required to be a Lie group, and the corresponding action on formula_4 should be smooth.
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:
The same is not true 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.
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
For any we have
Characterization of smooth principal bundles.
If formula_117 is a smooth principal formula_3-bundle then formula_3 acts freely and properly on formula_4 so that the orbit space formula_34 is diffeomorphic to the base space formula_2. It turns out that these properties completely characterize smooth principal bundles. That is, if formula_4 is a smooth manifold, formula_3 a Lie group and formula_125 a smooth, free, and proper right action then
Use of the notion.
Reduction of the structure group.
Given a subgroup H of G one may consider the bundle formula_131 whose fibers are homeomorphic to the coset space formula_77. If the new bundle admits a global section, then one says that the section is a reduction of the structure group from formula_3 to formula_74 . The reason for this name is that the (fiberwise) inverse image of the values of this section form a subbundle of formula_4 that is a principal formula_74-bundle. If formula_74 is the identity, then a section of formula_4 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 formula_3-bundle may be rephrased as questions about the admissibility of the reduction of the structure group (from formula_3 to formula_74). For example:
Also note: an formula_87-dimensional manifold admits formula_87 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.
If formula_4 is a principal formula_3-bundle and formula_105 is a linear representation of formula_3, then one can construct a vector bundle formula_158 with fibre formula_105, as the quotient of the product formula_4×formula_105 by the diagonal action of formula_3. This is a special case of the associated bundle construction, and formula_14 is called an associated vector bundle to formula_4. If the representation of formula_3 on formula_105 is faithful, so that formula_3 is a subgroup of the general linear group GL(formula_105), then formula_14 is a formula_3-bundle and formula_4 provides a reduction of structure group of the frame bundle of formula_14 from formula_173 to formula_3. 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 , "i.e." 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 .