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

  1. Is it known whether there are $C^k$-differentiable manifolds that are homeomorphic, but not $C^k$ diffeomorphic?
  2. More generally, is it known whether there are $C^k$-differentiable manifolds that are $C^l$ diffeomorphic, but not $C^k$ diffeomorphic for $l < k$?
  3. Do any of the famous examples for $k=\infty$ (such as $\mathbb R^4$ or Milnor-Kervaire's exotic spheres) give examples for $k < \infty$?
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There are no such exotic manifolds. Whitney proved in

Whitney, Hassler Differentiable manifolds. Ann. of Math. (2) 37 (1936), no. 3, 645–680.

that every $C^1$-differential structure can be uniquely smoothed to a $C^{\infty}$ differential structure. This was improved by Morrey and Grauert to show that in fact it can be uniquely smoothed to a real analytic structure; see

H. Grauert, On Levi's problem and the imbedding of real-analytic manifolds, Ann. of Math. 68 (1958), 460-472.

C. B. Morrey, The analytic embedding of abstract real-analytic manifolds, Ann. of Math. 68 (1958), 159-201.

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