I'm trying to compute the homology of the 3-torus $T^3=S^1 \times S^1 \times S^1$. Trying to use the typical construction the 2-torus $T^2$ as a starting point, I identified pairs of opposite faces on a cube as shown below:
This gives a single 0-cell, three 1-cells, three 2-cells, and a single 3-cell. Still using my $T^2$ computations as a guide, I've gotten that the cellular chain complex for this space is
$0 \to \mathbb{Z} \overset{d_3} \longrightarrow \mathbb{Z}^3 \overset{d_2}\longrightarrow \mathbb{Z}^3 \overset{d_1}\longrightarrow \mathbb{Z} \to 0$
Since there is only one 0-cell, I get that $d_1=0$. I'm having trouble understanding how to compute $d_2$ and $d_3$. Could anyone shed some light on this?
For reference, this is problem IV.10.1 in Bredon's "Topology and Geometry".