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 and a continuous map one can define a "pullback" of by as a bundle over. The fiber of over a point in is just the fiber of over. Thus is the disjoint union of all these fibers equipped with a suitable topology.

Formal definition

Let be a fiber bundle with abstract fiber and let be a continuous map. Define the pullback bundle by
and equip it with the subspace topology and the projection map given by the projection onto the first factor, i.e.,
The projection onto the second factor gives a map
such that the following diagram commutes:
If is a local trivialization of then is a local trivialization of where
It then follows that is a fiber bundle over with fiber. The bundle is called the pullback of E by or the bundle induced by . The map is then a bundle morphism covering.

Properties

Any section of over induces a section of, called the pullback section, simply by defining
If the bundle has structure group with transition functions and sometimes disconnected space, but always several copies of the circle.

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, in general it is better understood in the category of sheaves, because the objects it creates cannot in general be bundles.