Associated bundle

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

=An example=

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

=Construction=

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

==Fiber bundle associated to a principal bundle==

Let π : P → X be a 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 :(p,f)cdot g = (pcdot g, ho(g^{-1})f) We then quotient space 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 :[pcdot g,f] = [p, ho(g)f] mbox{ for all } gin G. Define a projection map πρ : E → X by πρ([ p , f ]) = π( p ). Note that this is well-defined.

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

=Relation with subgroups=

One very useful case is to take a subgroup H of G. Then an H-bundle has an associated G-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 V, and want to create the associated bundle with complex vector space fibers, we should take H=GL_n(mathbb{R}) and G=GL_n(mathbb{C}) in that schematic.

=Reduction of structure group=

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

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