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How to compute the cohomology of the Klein bottle

  1. Oct 11, 2009 #1

    TFT

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    Using the Mayer–Vietoris sequence, how can we calculate the
    De Rham cohomology of the Kelin bottle?
     
  2. jcsd
  3. Oct 12, 2009 #2

    morphism

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    First step would be to find a good open cover. Have you done so yet?
     
  4. Oct 13, 2009 #3

    TFT

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    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.
     
  5. Oct 13, 2009 #4
    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.
     
  6. Oct 14, 2009 #5

    TFT

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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.
     
  7. Oct 14, 2009 #6
    Why do they have the same cohomology as the cylinder? That's certainly not true.
     
  8. Oct 14, 2009 #7

    TFT

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    They are two Mobius strips, right? So they have the same cohomology as the circle.
     
  9. Oct 14, 2009 #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.
     
  10. Oct 15, 2009 #9

    TFT

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