Co-homology Groups of the Torus

Question

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.

Answer 1

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.