Computing the homology groups of the torus or a cell complex

I've found this table of homology groups of the tori $T^n$.

My question is: How did they compute these? More generally: what's the "recipe" to compute the homology group of say, a cell complex?

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You might want to look at Allen Hatcher's book "Algebraic Topology," which is available at his webpage for free download. –  Grumpy Parsnip May 25 '11 at 12:03
I'm actually reading that but not sequentially. I did have a look but couldn't find where he does it. Can someone point me to it please? –  Matt N. May 25 '11 at 12:07
The section on cellular homology, starting p137 is relevant. In particular example 2.36 on p141 does the computation for surfaces of genus g, including the torus as a special case. –  Grumpy Parsnip May 25 '11 at 12:16
You can use the Kunneth formula: en.wikipedia.org/wiki/K%C3%BCnneth_theorem –  Qiaochu Yuan May 25 '11 at 12:17
I think I have to look into "Cellular homology". –  Matt N. May 25 '11 at 12:17
In a simple case, such as the 2-torus, it's very straightforward to compute the simplicial homology from a simplicial (or Delta) complex. You should really sit down and do this for - at least - the torus, the circle, and $\mathbb{R}P^2$, because it's a great way to start to get a handle on what homology is all about. You'll soon realise that it's easy to calculate the homology of a given simplicial complex... but also long-winded.