pullback bundle

If $\pi :E\to B$ is a bundle and $f:B^{\prime }\to B$ is an arbitrary continuousmap, then there exists a pullback, or induced, bundle $f^{*}(\pi ):E^{\prime }\to B^{\prime }$, where

$$E^{\prime }=\{(e,b)\in E\times B^{\prime }|f(b)=\pi (e)\},$$

and $f^{*}(\pi )$ is the restriction of the projection mapto $B^{\prime }$. There is a natural bundle map from $f^{*}(\pi )$ to $\pi$ with the map $B^{\prime }\to B$ givenby $f$, and the map $\phi :E^{\prime }\to E$ given by the restriction of projection.

If $\pi$ is locally trivial, a principal $G$-bundle, or a fiber bundle, then $f^{*}(\pi )$ is as well.The pullback satisfies the following universal property:

(i.e. given a diagram with the solid arrows, a map satisfying the dashed arrow exists).