In mathematics, a pullback bundle or induced bundle is the fiber bundle and a continuous map is the disjoint union of all these fibers equipped with a suitable topology.

Formal definition

be a continuous map. Define the pullback bundle

and equip it with the subspace topology and the projection map

such that the following diagram commutes

is a local trivialization

is called the pullback of E by or the bundle induced by is then a bundle morphism.

Properties

Any section, called the pullback section, simply by defining

1. .

has structure group are given by

is a vector bundle or principal bundle. In the case of a principal bundle the right action is given by

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.

References

Sources

Further reading