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Kervaire proved in [M. Kervaire , A manifold which does not admit any differentiable structure. Comment. Math. Helv. 34 (1960), pp. 257–270.] that there are topological manifolds which cannot be smoothed. On the other hand, every $C^k$ manifold with $k>0$ can be uniquely smoothed to a $C^\infty$ manifold, by a theorem of Whitney. Finally, |
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Kervaire proved in the 60s [M. Kervaire , A manifold which does not admit any differentiable structure. Comment. Math. Helv. 34 (1960), pp. 257–270.] that there are topological manifolds which cannot be smoothed. On the other hand, every $C^k$ manifold with $k>0$ can be uniquely smoothed to a $C^\infty$ manifold, by a theorem of Whitney. Finally, if I recall correctly not every $C^k$ manifold had a compatible analytic structure... |
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Kervaire proved in the 60s that there are topological manifolds which cannot be smoothed. On the other hand, every $C^k$ manifold with $k>0$ can be uniquely smoothed to a $C^\infty$ manifold, by a theorem of Whitney. |
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