Right. and .

First of all, a 3-torus is a 3-dimensional manifold so it is not a surface.The first homology group would be the abelianization of this group... but how do you compute higher homology groups of this surface?

To compute a homology groups of a 3-torus, the easiest way I think is to use a CW complex and compute a cellular homology group.

The CW structure of 3-torus consists of one 3-cell, three 2-cells, three-1 cells, and one 0-cell.

The cellular chain complexes for a 3-torus are as follows:

Note that all the celluar boundary maps of the above are zero. I leave it to you to verify this.

Now, the (cellular) homology group of is

( a single connected component),

,

,

for k > 3.