# 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 $\phi :{E}^{\prime}\to E$ 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:

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

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

Title | pullback bundle |
---|---|

Canonical name | PullbackBundle |

Date of creation | 2013-03-22 13:17:19 |

Last modified on | 2013-03-22 13:17:19 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 7 |

Author | bwebste (988) |

Entry type | Definition |

Classification | msc 55R10 |

Synonym | induced bundle |