TITLE: Exotic $C^k$ manifolds
QUESTION [7 upvotes]: 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 $k < \infty$?

REPLY [19 votes]: 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.