We know that a principal bundle can induce a fiber bundle as follows: if $F$ is a space which admits a $G$-action then a principal $G$-bundle $p: E \to B$ induces a fiber bundle $p: E \times_G F \to B$ with fiber $F$ where $E \times_G F = (E \times F)/G$ denotes the orbit space and the $G$-action on $E \times F$ is defined ``diagonally'' as $(e,f)g = (eg,fg)$. By doing so we are effectively changing the fiber from $G$ to $F$ for the base space $B$. This is because the local trivialization for the induced fiber bundle is $\psi: p^{-1}(U) \to U \times (G \times_G F)$ (please check whether this is correct) and $G \times_G F$ is naturally identified with $F$.

My question is, could this fiber changing trick be generalized to other fiber bundles? Say for example you are given a non-principle bundle with a nasty fiber $F$ and you want to change the fiber to a nicer one $F'$?

Thank you very much for your attention and help!

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    $\begingroup$ Locally, $E\approx B\times G$ so locally $E\times_GF\approx B\times (G\times_G F)$ . $\endgroup$ – Steven Landsburg Apr 16 '16 at 23:38
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    $\begingroup$ One would do well to go back to Norman Steenrod's classical book "The topology of fiber bundles'', where it is made clear how to construct bundles of either sort in terms of coordinate transformations defined solely in terms of the base space and the group G. No cellular structure needed. See Theorem 3.3, page 16. $\endgroup$ – Peter May Apr 16 '16 at 23:47
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    $\begingroup$ First, choose a trivialization of $E \to B$, then use the same gluing laws with $G$ replaced by $F$. $\endgroup$ – S. Carnahan Apr 17 '16 at 6:58
  • $\begingroup$ Thank you very much for your all comments and answers especially Prof May (I could have never imagined having the fortune to meet famous mathematicians here!). This simple question is in fact a "prelude" to a much harder (hopefully legitimate!) question (refer to the revised version above). Please let me know how you think of it! $\endgroup$ – PhysicsMath Apr 18 '16 at 17:03

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