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Provide the structure of a differentiable manifold on the surface of the cube:

$\{x\in\mathbb{R}^{n+1}\colon \max\{|x_1|,\dotso,|x_{n+1}|\}=1\}$

Hello,

I have a problem with this task. I do not know how to solve this task and I am not able to come up with an appropriate approach.

I know, that the sphere has a differentiable structure. I know that it is possible to give a homeomorphism between the cube and the sphere.

Would that be enough to solve this problem?

Thanks in advance.

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Cornman
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  • Yes. Worth noting that this throws away some geometry. – Ethan Bolker Apr 23 '17 at 14:52
  • Thank you for your fast answer. So it is enough to give this homeomorphism, or would there be more to show? Since I know the maps of the sphere, I get the maps on the cube by concatenation with the homeomorphism between the cube and sphere, right? – Cornman Apr 23 '17 at 14:54
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    I think that's the whole story, unless there's more in the context that generated the question. When you're done you can answer your own question here. – Ethan Bolker Apr 23 '17 at 14:57
  • No, this is the whole question. When I am done, I write an answer. This might take a few hours. Thank you very much. – Cornman Apr 23 '17 at 15:02

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