When is a diffeomorphism a bundle map?

Question

Let $F\rightarrow E_0 \rightarrow B$ and $F\rightarrow E_1\rightarrow B$ be two smooth fiber bundles. Suppose $E_0$ and $E_1$ are diffeomorphic.

What are the obstructions for $E_0$ and $E_1$ to be diffeomorphic via a bundle map?

To be more specific, I am interestid in the following cases:

Search on "bundles diffeomorphic total spaces". It is common for a fiber bundle to fiber in many distinct ways. In rare cases a complete classification is possible, e.g. see arxiv.org/abs/math/0004147. — Igor Belegradek, Commented Nov 2, 2020 at 12:52
At the level of generality you ask the question, there is no uniform answer. But in many cases (dimension restrictions, types of bundles, etc) there are useful tools. You might want to narrow down your question, or you risk asking for a summary of all smooth manifold theory, in which case your question might be closed. — Ryan Budney, Commented Nov 2, 2020 at 13:43
@IgorBelegradek Maybe I'm missing something obvious, but I don't see how Crowley and Escher's article adresses the question of whether the diffeomorphism is via bundle maps... — Kafka91, Commented Nov 2, 2020 at 14:07
Corollary 1.6 of the Crowley and Escher's article shows that there are many non-isomorphic $S^3$ bundles over $S^4$ with diffeomorphic total spaces. There are many examples of this type. Like Ryan says, if you specify which bundles you care about the problem may become more tractable. As stated the question is too broad. — Igor Belegradek, Commented Nov 2, 2020 at 16:05

Kafka91 · Accepted Answer · 2020-11-02 14:01:49Z

3

I actually found a partial answer to the case of vector bundles over spheres (that is, under certain dimensional restrictions): https://www.sciencedirect.com/science/article/pii/S0040938399000397

answered Nov 2, 2020 at 13:53

Kafka91

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