| Version 2 |
Version 1 |
| If $\pi:E\to B$ is a bundle and $f:B'\to B$ is an arbitrary continuous |
If $\pi:E\to B$ is a bundle and $f:B'\to B$ is an arbitrary continuous |
| map, then there exists a pullback, or induced, bundle $f^*(\pi):E'\to B'$, where |
map, then there exists a pullback, or induced, bundle $f^*(\pi):E'\to B'$, where |
| $$E'=\{(e,b)\in E\times B'| f(b)=\pi(e)\},$$ and $f^*(\pi)$ is the restriction of the projection map |
$$E'=\{(e,b)\in E\times B'| f(b)=\pi(e)\},$$ and $f^*(\pi)$ is the restriction of the projection map |
| to $B'$. There is a natural bundle map from $f^*(\pi)$ to $\pi$ with the map $B'\to B$ given |
to $B'$. There is a natural bundle map from $f^*(\pi)$ to $\pi$ with the map $B'\to B$ given |
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by $f$, and the map $\vp:E'\to E$ given by the restriction of projection.
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by $f$, and the map $E'\to E$ given by the restriction of projection.
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| If $\pi$ is locally trivial, a principal $G$-bundle, or a fiber bundle, then $f^*(\pi)$ is as well. |
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: |
The pullback satisfies the following universal property: |
| $$\xymatrix{ &\ar[ddl]X\ar[ddr]\ar@{--}[d]&\\ |
$$\xymatrix{ &\ar[ddl]X\ar[ddr]\ar{--}[d]&\\&\ar[dl]E'\ar[dr]&\\B'\ar[dr]& &E\ar[dl]\\ &B&}$$ |
| &\ar[dl]^{f^*{\pi}}E'\ar[dr]^\vp&\\ |
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| B'\ar[dr]^f& &E\ar[dl]^\pi\\ |
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| &B&}$$ |
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| (i.e. given a diagram with the solid arrows, a map satisfying the dashed arrow exists). |
(i.e. given a diagram with the solid arrows, a map satisfying the dashed arrow exists). |
