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 : EB 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*EB 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

This page was last edited on 7 June 2017, at 21:48.
Reference: https://en.wikipedia.org/wiki/Pullback_bundle under CC BY-SA license.

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