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 bundles](https://mathoverflow.net/questions/116003/examples-of-sphere-bundles)


[Is it true that all sphere bundles are boundaries of disk bundles?](https://mathoverflow.net/questions/74756/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:


1. 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:


2. Do there exist a topological sphere bundle which is not topologically isomorphic to a linear one (or even to a smooth one)?



Two questions about sphere bundles
Recall that linear, smooth, and topological $S^k$-bundles over a finite complex $X$ are classified by the sets of homotopy classes $[X, BO\_{k+1}]$, $[X, B\mathrm{Diff}(S^k)]$ and $[X, B\mathrm{Homeo}(S^k)]$. There are obvious maps between the sets, and we are interested in their cokernels.


For example, if $X=S^1$, then the smooth nonlinear bundles are elements of the cokernel of $\pi\_1(BO\_{k+1})\to \pi\_1(B\mathrm{Diff}(S^k))$. By the exact homotopy sequence the cokernel is isomorphic to $\pi\_0(\mathrm{Diff}(S^k)/O\_{k+1})$. The space $\mathrm{Diff}(S^k)/O\_{k+1}$ is simply $\mathrm{Diff}(D^k, \partial)$, the diffeomorphisms of the disk that are the identity on the boundary. The latter space is classically studied, and by results of Smale and Cerf the group $\pi\_0(\mathrm{Diff}(D^k, \partial))$ is isomorphic to the group of homotopy $(k+1)$-spheres, and hence it is often nonzero. Thus there are lots of smooth non-linear bundles over a circle.


EDIT: *the crossed out text below is nonsense, and I retract it. What one actually needs is that $\mathrm{Diff}(S^k)\to \mathrm{Homeo}(S^k)$ is not $\pi\_i$-surjective for some $i$, and at the moment I do not know any $k, i$ for which this is true.*



Non-smoothable sphere bundles over spheres also exist because the space $\mathrm{Homeo}(S^k)/\mathrm{Diff}(S^k)$ is not contractible, for otherwise the inclusion $\mathrm{Diff}(S^k)\to \mathrm{Homeo}(S^k)$ would be a $\pi\_i$-isomorphism for all $i$, which it is not for some $k$ (this is due to Alexander's trick, see page 3 of the reference 2 below).

Some references to online sources which you may find useful, and which contains references to the above claims):


1. A. Hatcher's survey: [http://www.math.cornell.edu/~hatcher/Papers/Diff%28M%292012.pdf](http://www.math.cornell.edu/%7Ehatcher/Papers/Diff%28M%292012.pdf)
2. P.L. Antonelli, D. Burghelea,
P.J. Kahn, <http://www.sciencedirect.com/science/article/pii/0040938372900213>
3. D. Crowley, T. Schick: <https://arxiv.org/abs/1204.6474>


