I read in the lecture notes of my course on manifolds (undergraduate) a little side-note that stated that every manifold with dimension less then 4 can be equipped with a differentiable structure, but without proof.

I was interested in this statement, so I started to look on the internet and in the library for a proof, but I couldn't find it anywhere!

At first I tried to prove it myself, but then some websites stated that is was a very difficult proof.

Could you help me with the proof? I honestly have to say I don't have a clue where to start. Maybe first embed it in euclidean space?

If you also think the proof is way to difficult for an undergraduate, could you then please help me find a place where I can read it?

Thank you very much,



1 Answer 1


This theorem is due to Rado in dimension 2 and Moise in dimension 3. You can find its proof in Moise "Geometric topology in dimensions 2 and 3" and in Ahlfors and Sario "Riemann surfaces" in 2d case. Both proofs are likely to be too hard for an undergraduate. Thre are other proofs but they are still hard. You can try to prove this by yourself in 1d case and see if you can give a rigorous proof, this is a good exercise.

  • $\begingroup$ I thank you! I'll go and find the books $\endgroup$ Commented Oct 29, 2013 at 14:18

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