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I would like to know if there are examples of topological manifolds that not admit a smooth atlas, or more generally a differentiable structure of class $C^k$ for $k\geq1$. In the book "Introduction to Manifolds - Loring W. Tu"" the author says that an example of this type is dued to M. Kervaire.

Vincenzo Zaccaro
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    http://math.stackexchange.com/questions/408221/the-easiest-non-smoothable-manifold – Travis Willse Mar 05 '16 at 22:18
  • For the sake of completeness, all topological manifolds of dimension at most $3$ admit a unique smooth stiffening. –  Mar 05 '16 at 22:33
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    Your second question is actually not more general. A manifold that admits a $C^1$ structure admits an essentially unique compatible $C^\infty$ (even $C^\omega$, analytic) structure. –  Mar 05 '16 at 22:49

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