Pullback bundle

In mathematics, a pullback bundle or induced bundle is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle Template:Math and a continuous map Template:Math one can define a "pullback" of Template:Math by Template:Math as a bundle Template:Math over Template:Math. The fiber of Template:Math over a point Template:Math in Template:Math is just the fiber of Template:Math over Template:Math. Thus Template:Math is the disjoint union of all these fibers equipped with a suitable topology.

Formal definition

Let Template:Math be a fiber bundle with abstract fiber Template:Math and let Template:Math be a continuous map. Define the pullback bundle by

$f^{*}E=\{(b^{\prime },e)\in B^{\prime }\times E\mid f(b^{\prime })=\pi (e)\}\subseteq B^{\prime }\times E$

and equip it with the subspace topology and the projection map Template:Math given by the projection onto the first factor, i.e.,

${\pi }^{\prime }(b^{\prime },e)=b^{\prime }.$

The projection onto the second factor gives a map

$h:f^{*}E\to E$

such that the following diagram commutes:

$\begin{array}{ccc}{f}^{\ast }E& \stackrel{h}{⟶}& E\\ {\pi }^{\prime }\downarrow & & \downarrow \pi \\ B^{\prime }& \stackrel{f}{⟶}& B\end{array}$

If Template:Math is a local trivialization of Template:Math then Template:Math is a local trivialization of Template:Math where

$\psi (b^{\prime },e)=(b^{\prime },{\text{proj}}_2(\phi (e))).$

It then follows that Template:Math is a fiber bundle over Template:Math with fiber Template:Math. The bundle Template:Math is called the pullback of E by Template:Math or the bundle induced by Template:Math. The map Template:Math is then a bundle morphism covering Template:Math.

Properties

Any section Template:Math of Template:Math over Template:Math induces a section of Template:Math, called the pullback section Template:Math, simply by defining

$f^{*}s(b^{\prime }):=(b^{\prime },s(f(b^{\prime })))$ for all $b^{\prime }\in B^{\prime }$.

If the bundle Template:Math has structure group Template:Math with transition functions Template:Math (with respect to a family of local trivializations Template:Math) then the pullback bundle Template:Math also has structure group Template:Math. The transition functions in Template:Math are given by

$f^{*}{t}_{ij}=t_{ij}\circ f.$

If Template:Math is a vector bundle or principal bundle then so is the pullback Template:Math. In the case of a principal bundle the right action of Template:Math on Template:Math is given by

$(x,e)\cdot g=(x,e\cdot g)$

It then follows that the map Template:Math covering Template:Math is equivariant and so defines a morphism of principal bundles.

In the language of category theory, the pullback bundle construction is an example of the more general categorical pullback. As such it satisfies the corresponding universal property.

The construction of the pullback bundle can be carried out in subcategories of the category of topological spaces, such as the category of smooth manifolds. The latter construction is useful in differential geometry and topology.

Bundles and sheaves

Bundles may also be described by their sheaves of sections. The pullback of bundles then corresponds to the inverse image of sheaves, which is a contravariant functor. A sheaf, however, is more naturally a covariant object, since it has a pushforward, called the direct image of a sheaf. The tension and interplay between bundles and sheaves, or inverse and direct image, can be advantageous in many areas of geometry. However, the direct image of a sheaf of sections of a bundle is not in general the sheaf of sections of some direct image bundle, so that although the notion of a 'pushforward of a bundle' is defined in some contexts (for example, the pushforward by a diffeomorphism), in general it is better understood in the category of sheaves, because the objects it creates cannot in general be bundles.

Sources

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Further reading

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