# pullback bundle

If $\pi:E\to B$ is a bundle and $f:B^{\prime}\to B$ is an arbitrary continuous map, 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 map to $B^{\prime}$. There is a natural bundle map from $f^{*}(\pi)$ to $\pi$ with the map $B^{\prime}\to B$ given by $f$, and the map $\varphi:E^{\prime}\to E$ 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:

 $\xymatrix{&\ar[ddl]X\ar[ddr]\ar@{-->}[d]&\\ &\ar[dl]^{f^{*}{\pi}}E^{\prime}\ar[dr]_{\varphi}&\\ B^{\prime}\ar[dr]^{f}&&E\ar[dl]_{\pi}\\ &B&}$

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

Title pullback bundle PullbackBundle 2013-03-22 13:17:19 2013-03-22 13:17:19 bwebste (988) bwebste (988) 7 bwebste (988) Definition msc 55R10 induced bundle