How much is known about exotic $C^k$-manifolds? For example,

- Is it known whether there are $C^k$-differentiable manifolds that are homeomorphic, but not $C^k$ diffeomorphic?
- More generally, is it known whether there are $C^k$-differentiable manifolds that are $C^l$ diffeomorphic, but not $C^k$ diffeomorphic for $l < k$?
- Do any of the famous examples for $k=\infty$ (such as $\mathbb R^4$ or Milnor-Kervaire's exotic spheres) give examples for $k < \infty$?