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I wanted to explain to me, or give me a reference of how to calculate the cohomology groups of the complex and real, torus $\mathbb{T}^2$. I want to use this as an example in a seminar that I will present to my teacher.

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You can use Poincaré duality. The torus is a closed oriented manifold so that the $k$-cohomology group is isomorphic to the $n-k$-th homology group: $H^k (\mathbb T^2) \cong H_{n-k} (\mathbb T^2)$. Then just compute the simplicial homology of the torus.

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I feel Poincare duality is overkill in this situation. It's simple enough to give the torus a simplicial/$\Delta$/cell structure and compute the corresponding cohomology directly. –  Miha Habič Dec 12 '12 at 16:55
    
@MihaHabič I am not sure one can fill 2 hours of seminar lecture with computations of cohomology groups. But what I wrote is probably overkill. I wrote it because I have never computed cohomology groups but I know how to do simplicial homology for the torus. : ) –  Rudy the Reindeer Dec 12 '12 at 16:58

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