[Math] De Rham cohomology group of the Klein Bottle

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I need to compute all the cohomology rings of the Klein Bottle.

I want to apply the Mayer-Vietoris sequence. Here I'm using the same good open cover suggested by the Wikipedia page 🙂
enter image description here

It's reasonable to think that the blue set is a Moebius band and each connected component of the intersection as well. What it's not clear to me is why the red set is also a Moebius band.

Any help?

Best Answer

If you cut open the square vertically in the middle and connect the two vertical lines, you would get the same picture, only the red part now is connected and the blue part is separated.

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