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 (with respect to a family of local trivializations ) then the pullback bundle also has structure group . The transition functions in are given by
If is a vector bundle or principal bundle then so is the pullback . In the case of a principal bundle the right action of on is given by
It then follows that the map covering 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.
Examples: It is illuminating to consider the pullback of the degree 2 map from the circle to itself over the degree or map from the circle to itself. In such examples one sometimes gets a connected (e.g. choosing degree ) and sometimes disconnected space (degree ), 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 (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.