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  "body": "\n## Summary\n\nLet $P \\to B$ be a [[principal bundle|8E37E74XVP]] with group $G$, and let $G$ act on a space $F$.\nThe associated bundle $P \\times_G F$ is a [[fiber bundle|CX4M8K61AB]] over $B$ with fiber $F$.\nIt is built by gluing copies of $F$ along $B$ with the same twisting as $P$.\nIn this way one principal bundle produces many bundles: vector bundles, sphere bundles, bundles of frames and so on.\nFor example, the tangent bundle of a manifold is associated to its frame bundle through the action of $GL_n(\\R)$ on $\\R^n$.\n\n## Explanations\n\n### L1 human\n\nA principal bundle records only how the fibers are twisted, by elements of a group, as one moves around the base.\nAn associated bundle uses this twisting recipe on another fiber, any space on which the group acts.\nFor example, the circle double cover $S^1 \\to S^1$, $z \\mapsto z^2$, records a single flip as one goes around once.\nApplying this flip to a line, by $t \\mapsto -t$, gives the Möbius band.\n",
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