One of the remark in my lecture notes said:

In dimension $\leq 3$, every topological manifold has a unique smooth structure (up to diffeomorphism.)

I don't quite understand what is a structure in a manifold. Can you give me some examples, say what is a smooth structure of $S^2=$ surface of unit sphere in $ \mathbb{R^3}$

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    $\begingroup$ Are you sure your lecture notes don't describe what a smooth structure is prior to this statement? It seems out of place otherwise. $\endgroup$ – Michael Albanese Mar 30 '15 at 4:17
  • $\begingroup$ @MichaelAlbanese Smooth structure on a manifold is an equivalence class of $C^\infty$-atlases if I recall correctly. $\endgroup$ – SamC Mar 30 '15 at 4:24
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    $\begingroup$ Your statement can be interpreted as: If $M_1$, $M_2$ are two smooth manifold of dimension $\le 3$ so that $M_1$ and $M_2$ are homeomorphic as topological space. Then they are diffeomorphic. This is not true when the dimension is $\ge 4$. (I guess the easiest example was given by Milnor in dimension 7, where he constructed some $S^7$ with different smooth structures) $\endgroup$ – user99914 Mar 30 '15 at 5:11
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    $\begingroup$ @Mnifldz: Your 1 dimensional toplogical manifold is homeomorphic to $\mathbb R$, thus CAN be given a smooth structure (that of $\mathbb R$). The statement is that That smooth structure is the only one you can put on it. If I understand correctly, it is also true that all topological manifold of dimension $\le 3$ can be given a smooth structure (someone correct me if I am wrong). $\endgroup$ – user99914 Mar 30 '15 at 5:45
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    $\begingroup$ If $ f: W\to \mathbb R$ is a homeomorphism, then we can define a smooth structure on $W$, so that $f$ is diffeomorphic ($W$ is your wave). @Mnifldz $\endgroup$ – user99914 Mar 30 '15 at 6:32

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