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?
Many thanks for your help!
 A: 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.
