PlanetMath: pullback bundle

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pullback bundle

(Definition)

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

$\displaystyle E'=\{(e,b)\in E\times B'\vert f(b)=\pi(e)\},$

and $f^*(\pi)$ is the restriction of the projection map to

. There is a natural bundle map from $f^*(\pi)$ to $\pi$ with the map $B'\to B$ given by

, and the map $\varphi :E'\to E$ given by the restriction of projection.

If $\pi$ is locally trivial, a principal -bundle, or a fiber bundle, then $f^*(\pi)$ is as well. The pullback satisfies the following universal property:

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ &\ar[ddl]X\ar[ddr]\ar@{-->}[d] &... ...^*{\pi}}E'\ar[dr]_\varphi &\ B'\ar[dr]^f & &E\ar[dl]_\pi\ &B & } } \end{xy}$

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

"pullback bundle" is owned by bwebste.

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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 4138 times total.

Classification:

AMS MSC:

55R10 (Algebraic topology :: Fiber spaces and bundles :: Fiber bundles)

Pending Errata and Addenda

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