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Let $p:E\to B$ be a fiber bundle and $f:B'\to B$ a map. Under what conditions does a lift $f':B'\to E$ exist? In the context of covering spaces, I remember a necessary and sufficient condition is that

$f_{\#}(\pi_1(B'))\subseteq p_{\#}(\pi_1(E)) $

where $p_{\#}:\pi_1(E)\to \pi_1(B)$ and $f_{\#}:\pi_1(B')\to \pi_1(B)$ are the induced morphisms at the level of the fundamental groups. I am guessing that since covering spaces are a particular case of fiber bundles then at least this condition has to be satisfied, but is it also enough?

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Define the pullback of the fiber bundle as $f^*E = \{(x,e) \in B' \times E \mid f(x)=p(e)\}$. This is a fiber bundle over $B'$, and a section of this fiber bundle is the same thing as a lift of the map $f$.

Whether or not fiber bundles admit sections is not so easily stated as the answer for covering spaces. The answer to this question goes by the name of "obstruction theory", and is complicated enough I'm not going to describe it here. One source I like for this is Davis & Kirk's algebraic topology book, but if I remember Hatcher talks briefly about it.

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