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Define $M= \mathbb{R}^2/\sim$ where $(x,y)\sim(x',y')$ if $x-x'=2n$ for some integer $n$ and $y = (-1)^n y'$.

Then how can I give a differentiable sturucture on $M$? Is there a general technique for this?

(Currently I'm reading Lee's SM. I hope you give me some ref. pages too.)

Arthur
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le4m
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1 Answers1

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There is a unique differential structure on $M$ such that the quotient map $p:\mathbb R^2\to M$ is locally a diffeomorphism. You can construct charts using the fact that $p$ is a covering map.

More generally, if $M$ is a smooth manifold and $p:M\to N$ is a covering, then $N$ has a unique smooth structure such that $p$ is locally a diffeomorphism.

This is a good exercise!

Mariano Suárez-Álvarez
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  • Could you help me with the explicit construction part using $p$'s covering property? – le4m Apr 02 '15 at 07:15
  • I could but I think you should do it yourself. I suggest you review the definition of what a covering map is if you do not have that fresh and then try to do this. As I wrote, this is a very good exercise. – Mariano Suárez-Álvarez Apr 02 '15 at 07:21
  • Okay. But just one more question (I'm very new to this DG and I'm not reading the book in order). Do I have to use the covering map $\epsilon:\mathbb{R} \to \mathbb{S}^1$ to construct an atlas on $M$? – le4m Apr 02 '15 at 07:38
  • No. ${}{}{}{}{}$ – Mariano Suárez-Álvarez Apr 02 '15 at 07:54