# 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? – Rudy the Reindeer 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". – Rudy the Reindeer 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.

For cell complexes, we can use cellular homology, which is a much more powerful way to find the homology groups for CW complexes. But often that's not enough (with the information we've got, or, we don't want to spend hours computing), and we have to use tools such as excision, the Mayer-Vietoris sequence, and the Kunneth formula to compute the homology of a space by considering its geometry, "nicer" subspaces, and the like.

Everything I've talked about here (apart from Kunneth) is in Chapter 2 of Hatcher's book, which I can't recommend enough if you're just starting to learn about homology. It's a very lucid exposition.

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Cellular homology will always allow you to compute the homology groups of a cell complex as long as you know everything about the cell structure. It may not be the most efficient way of course. – Grumpy Parsnip May 25 '11 at 12:28