In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product of a space with a group. In the same way as with the Cartesian product, a principal bundle is equipped with
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 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 of a vector bundle, which consists of all ordered bases of the vector space attached to each point. The group 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 -bundle, where denotes any topological group, is a fiber bundle together with a continuousright action such that preserves the fibers of and acts freely and transitively 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. 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 to the base space. Because the action is free, 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. 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 of smooth manifolds. Here is required to be a smooth map between smooth manifolds, is required to be a Lie group, and the corresponding action on should be smooth.
Examples
The prototypical example of a smooth principal bundle is the frame bundle of a smooth manifold, often denoted or. Here the fiber over a point is the set of all frames for the tangent space. The general linear group 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.
Let be a Lie group and let be a closed subgroup. Then is a principal -bundle over the coset space. Here the action of on is just right multiplication. The fibers are the left cosets of .
Consider the projection given by. This principal -bundle is the associated bundle of the Möbius strip. Besides the trivial bundle, this is the only principal -bundle over.
Projective spaces provide some more interesting examples of principal bundles. Recall that the -sphere 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. We then have a series of principal bundles for each positive :
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 π : P → X is a smooth principal G-bundle then G acts freely and properly 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 submersion, and
P is a smooth principal G-bundle over P/G.
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 G to H. The reason for this name is that the inverse image of the values of this section form a subbundle of P that 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. For example:
A 2n-dimensional real manifold admits an almost-complex structure if the frame bundle 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 has spin structure 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 admits a global section. In this case, the manifold is called parallelizable.
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 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, then E is a G-bundle and P provides a reduction of structure group of the frame bundle of E from GL 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 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 manifoldB 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.
