In Dan Freed's notes pp.4 beneath (12.24) he comments that, let $P \to M$ be a principal $G$-bundle and $E = (P \times F)/G = P \times_G F$ its associated fiber bundle. The fibers of $E \to M$ are identified with $F$ only up to the action of $G$. The principal bundle controls this uncertainty. More precisely, each point $p \in P_x$ gives an identification of the fiber $E_x$ with $F$. In that sense points of a principal bundle are generalized bases for all associated fiber bundles, and it is the principal bundle which controls the geometry and topology. He adds that this is the geometric view of fiber bundles advocated by Steenrod.

I do not quite get his idea in particular how the point $p \in P_x$ serve as generalized bases for all associated fiber bundles thus a combination of them fixes an associated fiber bundle thereby determines the topology and geometry of that bundle, and why this point of view could be useful.

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    $\begingroup$ The term "generalized basis" is by analogy with the case of vector bundles, where one way of describing the principal $GL_n$-bundle associated to an $n$-dimensional vector bundle is that it is the bundle of bases in the usual linear algebra sense. This point of view is useful because it reduces the study of fiber bundles to the study of principal bundles for many purposes. $\endgroup$ – Qiaochu Yuan Sep 14 '16 at 20:39
  • $\begingroup$ @QiaochuYuan: many thanks for the answer! Would you mind elaborating a bit more to make it an official answer? I think it would be helpful to write down explicitly an associated fiber bundle expressed in terms of a combination of the "generalized basis". $\endgroup$ – PhysicsMath Sep 15 '16 at 18:40

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