Give me an example of a topological manifold which is not a smooth manifold.

Question

I was just wondering whether someone could give me an example of a topological manifold which is not a smooth manifold. In particular, I want an example of a topological manifold where no differentiable structure could be given.

I have thought a lot but in vain. If someone could put some light it would be of great help.

Answer 1

Theorem. Any topological manifold of dimension $2$ or $3$ can be endowed with a real analytic manifold structure. Furthermore, two such structures which are homeomorphic are $C^\omega$-diffeomorphic.

Proof. See Geometric topology in dimension $2$ and $3$ by E. Moise. $\Box$

Therefore, an example must have dimension at least $4$, such a construction is not easy, see for example the article A manifold which does not admit any differentiable structure by M. Kervaire.

Answer 2

You can see the example of Kervaire The "Easiest" non-smoothable manifold