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 f^*E over B′. The fiber of f^*E over a point b′ in B′ is just the fiber of E over f(b′). Thus f^*E 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′ : f^*E → 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 f^*E where

It then follows that f^*E is a fiber bundle over B′ with fiber F. The bundle f^*E is called the pullback of E by f or the bundle induced by f. The map g is then a bundle morphism covering f.

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