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$Z$aremanifolds.
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?