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## Main Question or Discussion Point

Using the Mayer–Vietoris sequence, how can we calculate the

De Rham cohomology of the Kelin bottle?

De Rham cohomology of the Kelin bottle?

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- #1

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Using the Mayer–Vietoris sequence, how can we calculate the

De Rham cohomology of the Kelin bottle?

De Rham cohomology of the Kelin bottle?

- #2

morphism

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First step would be to find a good open cover. Have you done so yet?

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First step would be to find a good open cover. Have you done so yet?

Yes, I used tow cylinders to cover it. The result I got this way shows that Klein bottle has the same cohomology as the torus. I am not sure whether this is correct.

- #4

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Once you figure out what they are (and compute their cohomology, which is pretty straightforward), you'll have the answer.

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You are right. But still they have the sam cohomology as the cylinder, which is the same as the cohomology of the circle. In this way, I got the result that the Klein bottle and the torus have the same cohomology. I do not know if this is correct, but I think I am doing the right thing.

Once you figure out what they are (and compute their cohomology, which is pretty straightforward), you'll have the answer.

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Why do they have the same cohomology as the cylinder? That's certainly not true.

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They are two Mobius strips, right? So they have the same cohomology as the circle.Why do they have the same cohomology as the cylinder? That's certainly not true.

- #8

morphism

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So either you haven't computed the cohomology of the torus correctly, or you're messing up the MV argument.

- #9

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Yeah, I did mess up with MV. I got it now. Thanks guys.

So either you haven't computed the cohomology of the torus correctly, or you're messing up the MV argument.

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