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The solution to number 5 of this released exam seems rather sophisticated to me. I would have said:

The dimension of $S^1 \times R$ is $3$ as the $\dim(S^1 \times R) = \dim(S^1) + \dim(R) = 2 + 1 = 3$ but $\dim(R^2) = 2$, as they do not have the same dimension, there cannot exist a bijection between the two spaces thus there can be no diffeomorphism between the two.

I felt relativively confident about my answer originally, but the answer seems so long that I have doubts that I am correct. Of course, she says there are multiple ways to show this, but the way she presents is rather involved and makes me concerned that I have ommitted necessary rigour (as I have done before).

Furthermore, if I did do it incorrectly, is there a way to correct my reasoning to make it rigourous?

Dair
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  • $S^1 \times \mathbb{R}$ is the cylinder and has dimension 2, not 3. $S^1$ has dimension 1 (it locally looks like the real line). – ODF Dec 11 '16 at 20:29
  • @ODF: I thought that the circle required 2 parameterizations and therefore was dim 2, as the minimal number of parameterizations was 2? – Dair Dec 11 '16 at 20:30
  • The dimension of a (connected, smooth) manifold $X$ is defined to be $k$, where each point of $X$ has an open neighbourhood $U$ diffeomorphic to an open subset of $\mathbb{R}^k$. – ODF Dec 11 '16 at 20:33
  • @ODF Ok, thanks! Out of curiosity, is there a way to utilize the fact that $S^1$ requires two parameterizations to prove this? – Dair Dec 11 '16 at 20:35
  • The reason $S^1$ requires two paramaterisations is because it is not homeomorphic to $\mathbb{R}$. There are ways using this to prove the cylinder and plane aren't diffeomorphic (or even homeomorphic), for example using homology ($H_1(S^1 \times \mathbb{R}) = \mathbb{Z} \neq 0 = H_1(\mathbb{R}^2)$), but I can't think of a way not using algebraic topology off the top of my head. – ODF Dec 11 '16 at 20:44
  • @ODF The circle is compact; $\Bbb R$ is not. $S^1 \times \Bbb R$ is not diffeomorphic to $\Bbb R^2$ because the former has two ends, and the latter one. –  Dec 11 '16 at 22:25

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