Associated bundle

From Academic Kids

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

Contents

An example

A simple case comes with the Möbius band, for which <math>G<math> is a cyclic group of order 2. We can take as <math>F<math> any of: the real number line <math>\mathbb{R}<math>, the interval <math>[-1,\ 1]<math>, the real number line less the point 0, or the two-point set <math>\{-1,\ 1\}<math>. The action of <math>G<math> on these (the non-identity element acting as <math>x\ \rightarrow\ -x<math> in each case) is comparable, in an intuitive sense. We could say that more formally in terms of gluing two rectangles <math>[-1,\ 1] \times I<math> and <math>[-1,\ 1] \times J<math> together: what we really need is the data to identify <math>[-1,\ 1]<math> 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 <math>\{-1,\ 1\}<math> as for <math>[-1,\ 1]<math>.

Construction

In general it is enough to explain the transition from a bundle with fiber <math>F<math>, on which <math>G<math> acts, to the principal bundle (namely the bundle where the fiber is <math>G<math>, considered to act by translation on itself). For then we can go from from <math>F_1<math> to <math>F_2<math>, 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

<math>(p,f)\cdot g = (p\cdot g, \rho(g^{-1})f)<math>

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

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

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 <math>H<math> of <math>G<math>. Then an <math>H<math>-bundle has an associated <math>G<math>-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 <math>V<math>, and want to create the associated bundle with complex vector space fibers, we should take <math>H=GL_n(\mathbb{R})<math> and <math>G=GL_n(\mathbb{C})<math> in that schematic.

Reduction of structure group

The companion concept to associated bundles is the reduction of the structure group of a <math>G<math>-bundle <math>B<math>. We ask whether there is an <math>H<math>-bundle <math>C<math>, such that the associated <math>G<math>-bundle is <math>B<math>, up to isomorphism. More concretely, this asks whether the transition data for <math>B<math> can consistently be written with values in <math>H<math>. 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 <math>GL_n<math>); and the existence of complex structure on a real bundle (from <math>GL_{2n}(\mathbb{R})<math> to <math>GL_n(\mathbb{C})<math>.)

Another important case is the reduction from <math>GL_{n}(\mathbb{R})<math> to <math>GL_k(\mathbb{R}) \times GL_{n-k}(\mathbb{R})<math>, the latter sitting inside as block matrices. A reduction here is a consistent way of taking complementary <math>k<math>- and <math>n-k<math>-dimensional subspaces; in other words, finding a decomposition of a vector bundle <math>V<math> 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 <math>(p,\ q)<math>), the former having a faithful matrix representation of dimension <math>n\ =\ p\ +\ q<math>, 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 <math>\mathbb{R}^n<math>, since Spin acts on <math>\mathbb{R}^n<math>, 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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