I was wondering, is the composition of fiber bundles a fiber bundle? (No smoothness imposed-just topological). I found this https://mathoverflow.net/questions/74611/is-the-composition-of-two-bundle-projections-necessarily-a-bundle-projection, I was wondering about a proof of item (b) in the case the maps are fiber bundles
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https://math.stackexchange.com/questions/146976/composition-of-covering-maps may be of interest. – Rob Arthan Nov 06 '23 at 23:45
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I don't understand the difference between your question and the question in the link. What is item (b) (in Georges Elencwajg's answer) for fiber bundles? – ronno Nov 07 '23 at 14:28
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@ronno I am asking about fiber bundles. Item b is point b in Georges answer – monoidaltransform Nov 13 '23 at 00:07
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But the counterexample in the covering spaces case is also a counterexample for fiber bundles. Are you asking if the composition of fiber bundles is a fiber bundle assuming the base has a universal cover? – ronno Nov 13 '23 at 07:28
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@ronno yes. That's what i'm asking – monoidaltransform Nov 13 '23 at 20:37