How to compute the cohomology of the Klein bottle

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Discussion Overview

The discussion revolves around computing the De Rham cohomology of the Klein bottle using the Mayer–Vietoris sequence. Participants explore the appropriate open cover for the Klein bottle and its implications for cohomological results, comparing it to the torus and other surfaces.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests using the Mayer–Vietoris sequence to calculate the De Rham cohomology of the Klein bottle.
  • Another participant emphasizes the importance of finding a good open cover, mentioning the use of two cylinders.
  • A participant claims that the Klein bottle has the same cohomology as the torus based on their calculations.
  • Some participants challenge the assertion that the open cover consists of cylinders, suggesting they are actually Mobius strips.
  • There is a discussion about the cohomology of Mobius strips and their relationship to the cohomology of the circle.
  • One participant acknowledges a mistake in their application of the Mayer–Vietoris argument after further clarification from others.

Areas of Agreement / Disagreement

Participants express differing views on the nature of the open cover and the resulting cohomological conclusions. There is no consensus on the correctness of the initial claims regarding the cohomology of the Klein bottle and its comparison to the torus.

Contextual Notes

Participants reference the need to compute the cohomology of the torus and the implications of their findings, but there are unresolved questions about the accuracy of these computations and the definitions used in the discussion.

TFT
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Using the Mayer–Vietoris sequence, how can we calculate the
De Rham cohomology of the Kelin bottle?
 
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First step would be to find a good open cover. Have you done so yet?
 
morphism said:
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.
 
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.
 
zhentil said:
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.
 
Why do they have the same cohomology as the cylinder? That's certainly not true.
 
zhentil said:
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.
 
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.
 
morphism said:
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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