Pullback Bundle - Formal Definition

Formal Definition

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 is then a bundle morphism covering f.