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pullback bundle
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(Definition)
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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 $$E'=\{(e,b)\in E\times B'| f(b)=\pi(e)\},$$ 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 by $f$ , and the map $\vp:E'\to E$ 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).
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"pullback bundle" is owned by bwebste.
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(view preamble | get metadata)
| Other names: |
induced bundle |
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Cross-references: arrow, solid, diagram, universal property, fiber bundle, projection, map, bundle map, projection map, restriction, induced, pullback, continuous map
There is 1 reference to this entry.
This is version 4 of pullback bundle, born on 2002-12-18, modified 2003-08-22.
Object id is 3775, canonical name is PullbackBundle.
Accessed 5122 times total.
Classification:
| AMS MSC: | 55R10 (Algebraic topology :: Fiber spaces and bundles :: Fiber bundles) |
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Pending Errata and Addenda
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![$\displaystyle \xymatrix{ &\ar[ddl]X\ar[ddr]\ar@{-->}[d] &\ &\ar[dl]^{f^*{\pi}}E'\ar[dr]_\varphi &\ B'\ar[dr]^f & &E\ar[dl]_\pi\ &B & }$](http://images.planetmath.org:8080/cache/objects/3775/js/img1.png "$\displaystyle \xymatrix{ &\ar[ddl]X\ar[ddr]\ar@{-->}[d] &\ &\ar[dl]^{f^*{\pi}}E'\ar[dr]_\varphi &\ B'\ar[dr]^f & &E\ar[dl]_\pi\ &B & }$")