Table of contents |

2 General case 3 Relation with subgroups 4 Complexifying a real vector bundle 5 Reduction of structure group 6 Examples of reduction of group 7 Spinor bundles |

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 **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* -> -*x* in each case) is comparable, in an intuitive sense. We could say that more formally in terms of gluing two rectangles [-1,1]x*I* and [-1,1]x*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].

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.

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.

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}(**R**) and *G* = *GL*_{n}(**C**), in that schematic.

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 for vector bundles include: the introduction of a *metric* (equivalently, reduction to an orthogonal group from *GL*_{n}); and the existence of complex structure on a real bundle (from *GL*_{2n}(**R**) to *GL*_{n}(**C**)).

Another important case is the reduction from *GL*_{n}(**R**) to *GL*_{k}(**R**)x*GL*_{n-k}(**R**), the latter sitting inside as block matrices. 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.

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 (*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 **R**^{n}, since *Spin* acts on **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.