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# 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:

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

Synonym:

induced bundle

Type of Math Object:

Definition

Major Section:

Reference

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