# Associated bundle

In mathematics, the theory of fiber bundles with a structure group [itex]G[itex] (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from [itex]F_1[itex] to [itex]F_2[itex], which are both topological spaces with a group action of [itex]G[itex].

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## An example

A simple case comes with the Möbius band, for which [itex]G[itex] is a cyclic group of order 2. We can take as [itex]F[itex] any of: the real number line [itex]\mathbb{R}[itex], the interval [itex][-1,\ 1][itex], the real number line less the point 0, or the two-point set [itex]\{-1,\ 1\}[itex]. The action of [itex]G[itex] on these (the non-identity element acting as [itex]x\ \rightarrow\ -x[itex] in each case) is comparable, in an intuitive sense. We could say that more formally in terms of gluing two rectangles [itex][-1,\ 1] \times I[itex] and [itex][-1,\ 1] \times J[itex] together: what we really need is the data to identify [itex][-1,\ 1][itex] to itself directly at one end, and with the twist over at the other end. This data can be written down as a patching function, with values in G. The associated bundle construction is just the observation that this data does just as well for [itex]\{-1,\ 1\}[itex] as for [itex][-1,\ 1][itex].

## Construction

In general it is enough to explain the transition from a bundle with fiber [itex]F[itex], on which [itex]G[itex] acts, to the principal bundle (namely the bundle where the fiber is [itex]G[itex], considered to act by translation on itself). For then we can go from from [itex]F_1[itex] to [itex]F_2[itex], via the principal bundle. Details in terms of data for an open covering are given as a case of descent.

### Fiber bundle associated to a principal bundle

Let π : PX be a principal G-bundle and let ρ : G → Homeo(F) be an continuous left action of G on a space F (in the smooth category, we should have a smooth action on a smooth manifold). Without loss of generality, we can take this action to be effective (ker(ρ) = 1).

Define a right action of G on P × F via

[itex](p,f)\cdot g = (p\cdot g, \rho(g^{-1})f)[itex]

We then identify by this action to obtain the space E = P ×ρ F = (P × F)/G. Denote the equivalence class of (p,f) by [p,f]. Note that

[itex][p\cdot g,f] = [p,\rho(g)f] \mbox{ for all } g\in G.[itex]

Define a projection map πρ : EX by πρ([p,f]) = π(p). Note that this is well-defined.

Then πρ : EX is a fiber bundle with fiber F and structure group G. The transition functions are given by ρ(tij) where tij are the transition functions of the principal bundle P.

## Relation with subgroups

One very useful case is to take a subgroup [itex]H[itex] of [itex]G[itex]. Then an [itex]H[itex]-bundle has an associated [itex]G[itex]-bundle: this is trite for bundles, but looking at their sections it is essentially the induced representation construction, in a different light. This does suggest there will be some adjoint functors involved.

## Complexifying a real vector bundle

One application is to complexifying a real vector bundle (as required to define Pontryagin classes, for example). If we have a real vector bundle [itex]V[itex], and want to create the associated bundle with complex vector space fibers, we should take [itex]H=GL_n(\mathbb{R})[itex] and [itex]G=GL_n(\mathbb{C})[itex] in that schematic.

## Reduction of structure group

The companion concept to associated bundles is the reduction of the structure group of a [itex]G[itex]-bundle [itex]B[itex]. We ask whether there is an [itex]H[itex]-bundle [itex]C[itex], such that the associated [itex]G[itex]-bundle is [itex]B[itex], up to isomorphism. More concretely, this asks whether the transition data for [itex]B[itex] can consistently be written with values in [itex]H[itex]. In other words, we ask to identify the image of the associated bundle mapping (which is actually a functor).

## Examples of reduction of group

Examples for vector bundles include: the introduction of a metric (equivalently, reduction to an orthogonal group from [itex]GL_n[itex]); and the existence of complex structure on a real bundle (from [itex]GL_{2n}(\mathbb{R})[itex] to [itex]GL_n(\mathbb{C})[itex].)

Another important case is the reduction from [itex]GL_{n}(\mathbb{R})[itex] to [itex]GL_k(\mathbb{R}) \times GL_{n-k}(\mathbb{R})[itex], the latter sitting inside as block matrices. A reduction here is a consistent way of taking complementary [itex]k[itex]- and [itex]n-k[itex]-dimensional subspaces; in other words, finding a decomposition of a vector bundle [itex]V[itex] as a Whitney sum (direct sum) of sub-bundles of the specified fiber dimensions.

One can also express the condition for a foliation to be defined as a reduction of the tangent bundle to a block matrix subgroup - but here the reduction is only a necessary condition, there being an integrability condition so that the Frobenius theorem applies.

## Spinor bundles

The language of associated bundles is helpful in expressing the meaning of spinor bundles.

Here the two groups SO and Spin are involved (for a fixed choice of signature [itex](p,\ q)[itex]), the former having a faithful matrix representation of dimension [itex]n\ =\ p\ +\ q[itex], but the latter acting (in general) only faithfully in a higher dimension, on a space of spinors. Spin is a double cover of SO, so that the latter is a quotient of the former. That does mean that transition data with values in Spin give rise to transition data for SO, automatically: passing to a quotient group simply loses information.

Therefore a Spin-bundle always gives rise to an associated bundle with fibers [itex]\mathbb{R}^n[itex], since Spin acts on [itex]\mathbb{R}^n[itex], via its quotient SO. Conversely, there is a lifting problem for SO-bundles: there is a consistency question on the transition data, in passing to a Spin-bundle. The existence of such a spin structure is extra information on a real vector bundle. es:Fibrado asociado

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