Associated bundle

This section is organized as follows. We first introduce the general procedure for producing an associated bundle, with specified fibre, from a given fibre bundle. This then specializes to the case when the specified fibre 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 fibre of the principal bundle, we describe how to construct any associated bundle by means of a fibre product construction.[1]

Let π : EX be a fiber bundle over a topological space X with structure group G and typical fibre F. By definition, there is a left action of G (as a transformation group) on the fibre F. Suppose furthermore that this action is effective.[2] There is a local trivialization of the bundle E consisting of an open cover Ui of X, and a collection of fibre maps

such that the transition maps are given by elements of G. More precisely, there are continuous functions gij : (UiUj) → G such that

Now let F′ be a specified topological space, equipped with a continuous left action of G. Then the bundle associated with E with fibre F′ is a bundle E′ with a local trivialization subordinate to the cover Ui whose transition functions are given by

where the G-valued functions gij(u) are the same as those obtained from the local trivialization of the original bundle E.

This definition clearly respects the cocycle condition on the transition functions, since in each case they are given by the same system of G-valued functions. (Using another local trivialization, and passing to a common refinement if necessary, the gij transform via the same coboundary.) Hence, by the fiber bundle construction theorem, this produces a fibre bundle E′ with fibre F′ as claimed.

As before, suppose that E is a fibre bundle with structure group G. In the special case when G has a free and transitive left action on F′, so that F′ is a principal homogeneous space for the left action of G on itself, then the associated bundle E′ is called the principal G-bundle associated with the fibre bundle E. If, moreover, the new fibre F′ is identified with G (so that F′ inherits a right action of G as well as a left action), then the right action of G on F′ induces a right action of G on E′. With this choice of identification, E′ becomes a principal bundle in the usual sense. Note that, although there is no canonical way to specify a right action on a principal homogeneous space for G, any two such actions will yield principal bundles which have the same underlying fibre bundle with structure group G (since this comes from the left action of G), and isomorphic as G-spaces in the sense that there is a G-equivariant isomorphism of bundles relating the two.

In this way, a principal G-bundle equipped with a right action is often thought of as part of the data specifying a fibre bundle with structure group G, since to a fibre bundle one may construct the principal bundle via the associated bundle construction. One may then, as in the next section, go the other way around and derive any fibre bundle by using a fibre product.

Let π : PX be a principal G-bundle and let ρ : G → Homeo(F) be a 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.

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

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.

Examples for vector bundles include: the introduction of a metric resulting in reduction of the structure group from a general linear group GL(n) to an orthogonal group O(n); and the existence of complex structure on a real bundle resulting in reduction of the structure group from real general linear group GL(2n,R) to complex general linear group GL(n,C).

Another important case is finding a decomposition of a vector bundle V of rank n as a Whitney sum (direct sum) of sub-bundles of rank k and n-k, resulting in reduction of the structure group from GL(n,R) to GL(k,R) × GL(n-k,R).

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.