Does there exist a topological (not necessarily trivial) vector bundle $\pi: E\to M$, with $\dim(M)=1$ and $\dim\big(\pi^{-1}(\{x\})\big) =3$, that has no $C^1$ (or $C^2,C^\infty$) structure?
I know that every topological manifold up to dimension 3 has a unique smooth structure and there are 4-manifolds with no smooth structure. So the question here is that can such a non-smooth 4-manifold be a rank 3 base 1 vector bundle? I am guessing that the local triviality should forbid the existence of such bundles but I do not know for sure.
$\textbf{Addition:}$ It would also be sufficient to prove that a general topological vector bundle $\pi:E\to M$ is automatically a $C^k$(or $C^\infty$) bundle if the base manifold $M$ is $C^k$(or $C^\infty$). In other words, to prove that it would be possible to construct an atlas of $E$ with bundle charts that are not only continuous but also has the same regularity of the charts of $M$.
Any references are warmly welcome. Thanks.