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 π : E → B and a continuous map f : B′ → B one can define a "pullback" of E by f as a bundle fE over B′. The fiber of fE over a point b′ in B′ is just the fiber of E over f(b′). Thus fE is the disjoint union of all these fibers equipped with a suitable topology. Let π : E → B be a fiber bundle with abstract fiber F and let f : B′ → B be a continuous map. Define the pullback bundle by and equip it with the subspace topology and the projection map π′ : fE → B′ 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 (U, φ) is a local trivialization of E then (f−1U, ψ) is a local trivialization of fE where It then follows that fE is a fiber bundle over B′ with fiber F. The bundle fE is called the pullback of E by f or the bundle induced by f. The map h is then a bundle morphism covering f. Any section s of E over B induces a section of fE, called the pullback section fs, simply by defining for all . If the bundle E → B has structure group G with transition functions tij (with respect to a family of local trivializations {(Ui, φi)}) then the pullback bundle fE also has structure group G. The transition functions in fE are given by If E → B is a vector bundle or principal bundle then so is the pullback fE. In the case of a principal bundle the right action of G on f*E is given by It then follows that the map h covering f is equivariant and so defines a morphism of principal bundles. In the language of , 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.