Pullback Bundle - Properties

Properties

Any section s of E over B induces a section of f*E, called the pullback section f*s, simply by defining .

If the bundle E → B has structure group G with transition functions t_ij (with respect to a family of local trivializations {(U_i, φ_i)} ) then the pullback bundle f*E also has structure group G. The transition functions in f*E are given by

If E → B is a vector bundle or principal bundle then so is the pullback f*E. In the case of a principal bundle the right action of G on f*E is given by

It then follows that the map 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 3 or 4 map from the circle to itself. In such examples one sometimes gets a connected and sometimes disconnected space, but always several copies of the circle.