In mathematics, the theory of fiber bundles with a structure group (a topological group) allows an operation of creating an associated bundle, which are both topological spaces with a group action, the transition functions of the fiber (i.e., the cocycle as a new fiber bundle having the same transition functions, but possibly a different fiber.

Construction

acts, to the associated principal bundle, via the principal bundle. Details in terms of data for an open covering are given as a case of descent.

This section is organized as follows. We first introduce the general procedure for producing an associated bundle, with specified fiber, from a given fiber bundle. This then specializes to the case when the specified fiber is a principal homogeneous space for the left action of the group on itself, yielding the associated principal bundle. If, in addition, a right action is given on the fiber of the principal bundle, we describe how to construct any associated bundle by means of a fiber product construction.

Associated bundles in general

be a fiber bundle over a topological space. By definition, there is a left action (as a transformation group. Suppose furthermore that this action is faithful. There is a local trivialization consisting of an open cover, and a collection of fiber mapssuch that the transition maps. Then the bundle associated satisfy the cocycle condition. Hence, by the existence part of the fiber bundle construction theorem as claimed.

Principal G-bundle associated with a G-bundle

has a free and transitive becomes a principal bundle in the usual sense.

-equivariant bundle isomorphism. This is also called a gauge transformation. This allows us to speak of the. One may then, as in the next section, go the other way around and derive any fiber bundle by using a fiber product.

G-bundle associated with a principal G-bundle

be a principal G-bundle. Given a faithful left action.

Define a right via

Take the quotient. Note that

. This is well-defined.

In category theory, this is the coequalizer is the coequalizer of these maps.

Example

Consider the Möbius strip, the cyclic group in each case.

to itself directly at one end, and with the twist over at the other end. The associated bundle. That is, for each G-bundle, the only essential part of it is the principal G-bundle associated with it, which encodes all of the transition data. Since their associated principal G-bundle are isomorphic, all these constructions of the Möbius strip are essentially the same construction.

Extension of the structure group

, one per coset.

is the inclusion function.

The extension always exists, is functorial, and unique up to isomorphism.

is the inclusion function.

.

This construction is functorial and has a universal property is unique up to unique isomorphism.

Reduction of the structure group

Reduction of the structure group asks whether there is an inverse to the extension.

up to isomorphism. In other words, we ask to identify the image of the associated bundle mapping (which is actually a functor).

Unlike the case of extension, reduction is not always possible. A goal of obstruction theory is to explain when and how reduction may be impossible.

Examples

again.

Given a smooth manifold, its frame bundle is a principal general linear group. A Riemannian metric is a reduction of the structure group to the orthogonal group. An orientation is a reduction to the special linear group.

Given a smooth manifold, its tangent bundle, or more generally, a vector bundle-bundle, and we can perform similar reductions. An almost complex structure as a Whitney sum. Extra integrability conditions are necessary for it to make it a "complex" structure, not merely "almost complex".

A distribution (i.e. a flagWhen the distribution is integrable, Frobenius theorem applies, producing a foliation.

See also

• Spinor bundle

References

Books