I would like to better understand the relationship between different notions of orientable sphere bundle. Let me say that a locally trivial fiber bundle $\pi\colon E\to M$ with fiber $S^n$ and structure group $G$ is a
topological sphere bundle if $G=\mathrm{Homeo}^+(S^n)$,
smooth sphere bundle if $G=\mathrm{Diffeo}^+(S^n)$,
linear sphere bundle if $G=\mathrm{Gl}^+(n,\mathbb{R})$.
It is known that there exist smooth sphere bundles that are not equivalent (as smooth sphere bundles) to linear sphere bundles (see e.g. the following posts here on MO):
Examples of sphere bundlesExamples of sphere bundles
Is it true that all sphere bundles are boundaries of disk bundles?Is it true that all sphere bundles are boundaries of disk bundles?
For example, in an answer to the second question linked above R. Budney shows that there are smooth sphere bundles over $S^2$ which are not linear. Here comes my first question:
- Is it possible to construct a smooth sphere bundle over $S^1$ which is not smoothly equivalent to a linear one? One could take an exotic diffeomorphism $f\colon S^n\to S^n$, and consider the mapping cone of $f$, which is a smooth sphere bundle. Is this bundle linear?
The second question is about topological bundles:
- Do there exist a topological sphere bundle which is not topologically isomorphic to a linear one (or even to a smooth one)?
