Hello, can somebody help with the following question that I have thought over for quite some time, to no avail?


Suppose f: X--->Y is a universal cover and g: Y--->Z a fiber bundle, where X, Y and Z are manifolds. Is the composition gof: X--->Z necessarily a fiber bundle?


THanks!



composition of covering map and bundle projection
It is better to just assume that $f$ is a covering space. By shrinking $Z$ we may assume that that $Z$ is a ball and that $Y=Z\times F$. As $X\to Y$ is a covering space and $Z$ is simply-connected there is a covering space $X'\to F$ such that $X\to Y$ is isomorphic to $Z\times X'\to Z\times F$ which gives what you want.


**Addendum**: The reason that $X'$ exists is that if $h\colon T\to T'$ is a homotopy equivalence (and possibly $T$ and $T'$ fulfil some local niceness conditions which certainly are fulfilled in the case at hand), then pullback along $h$ induces an equivalence between the category of covering spaces of $T$ and that of $T'$. This is true irregardless on whether $T$ and $T'$ are connected or not. 


