9

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.

2 Answers2

14

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.

C. Falcon
  • 19,553
  • 2
    What about dimension 1? [Honest question.] – Randall Mar 14 '18 at 17:55
  • 4
    @Randall Any connected topological manifold (without boundary) of dimension $1$ is either homeomorphic to the real line or the unit circle and both of them can be endowed with a real analytic manifold structure, so the theorem also holds true in dimension $1$! I didn't mention it because it is a less deep result, but this is a good question! :) – C. Falcon Mar 14 '18 at 18:37
3

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

Tsemo Aristide
  • 89,587
  • 1
    As a topological space this is homeomorphic to the real line, so can be "given a differentiable structure" that respects the topology. I think the question is unclear. – Ethan Bolker Mar 14 '18 at 17:25