How to compute the cohomology of the Klein bottle

  • Thread starter TFT
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  • #1
TFT
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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?
 

Answers and Replies

  • #2
morphism
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First step would be to find a good open cover. Have you done so yet?
 
  • #3
TFT
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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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I don't think those are cylinders ;)

Once you figure out what they are (and compute their cohomology, which is pretty straightforward), you'll have the answer.
 
  • #5
TFT
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I don't think those are cylinders ;)

Once you figure out what they are (and compute their cohomology, which is pretty straightforward), you'll have the answer.
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.
 
  • #6
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Why do they have the same cohomology as the cylinder? That's certainly not true.
 
  • #7
TFT
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Why do they have the same cohomology as the cylinder? That's certainly not true.
They are two Mobius strips, right? So they have the same cohomology as the circle.
 
  • #8
morphism
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That's the open cover I had in mind. Those two mobius strips and their intersection (another mobius strip) have the cohomology of the circle. In fact, it turns out that the Klein bottle does as well.

So either you haven't computed the cohomology of the torus correctly, or you're messing up the MV argument.
 
  • #9
TFT
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That's the open cover I had in mind. Those two mobius strips and their intersection (another mobius strip) have the cohomology of the circle. In fact, it turns out that the Klein bottle does as well.

So either you haven't computed the cohomology of the torus correctly, or you're messing up the MV argument.
Yeah, I did mess up with MV. I got it now. Thanks guys.
 

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