Thread: Homology Groups of T3

Find the fundamental group and the homology groups of the 3-dimensional torus, T3 = S1 X S1 X S1.

The fundamental group preserves products, so it's Z X Z X Z right? The first homology group would be the abelianization of this group... but how do you compute higher homology groups of this surface? I came across a method called the Mayor-Vietoris sequence, but I'm not sure how to apply it here.

Originally Posted by joeyjoejoe

Find the fundamental group and the homology groups of the 3-dimensional torus, T3 = S1 X S1 X S1.

The fundamental group preserves products, so it's Z X Z X Z right?

Right. and .

The first homology group would be the abelianization of this group... but how do you compute higher homology groups of this surface?

First of all, a 3-torus is a 3-dimensional manifold so it is not a 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.