TITLE: Is the composition of two bundle projections necessarily a bundle projection?
QUESTION [9 upvotes]: That is, if $f: X \rightarrow Y$ and $g:Y \rightarrow Z$ are bundle projections, is $g \circ f: X \rightarrow Z$ a bundle projection? Assume $X$, $Y$ and $Z$ are manifolds.
Here is what I know. The answer is affirmative when (1) $f$ is a covering map and $g$ is bundle projection; (2) $f$ is a bundle projection and $g$ is a covering map of finite degree.
What can we say about the most general situation? 
Thanks.

REPLY [8 votes]: In the category of finite dimensional manifolds, I proved this in a very long paper (p. 23): the composition of smooth fiber bundle maps is a smooth fiber bundle map. Spanier's counterexample is not a finite dimensional manifold.