# Homology of the 3-Torus

• Mar 29th 2009, 06:42 PM
robeuler
Homology of the 3-Torus
I'm learning the basics of homology theory. I'm trying to give the 3-Torus a CW structure and then compute its homology.

I think I understand the approach to giving the 2-torus a CW complex. Then you have a diagram with sides identified to work from. Can I create a 3-d box with side identifications and do something similar? Even if I can, that approach is unsatisfying, as generalizing to the n-torus would be difficult.
• Mar 30th 2009, 02:32 AM
vsywod
Quote:

Originally Posted by robeuler
I think I understand the approach to giving the 2-torus a CW complex. Then you have a diagram with sides identified to work from. Can I create a 3-d box with side identifications and do something similar? Even if I can, that approach is unsatisfying, as generalizing to the n-torus would be difficult.

I think you can. Giving the 2-torus a CW structure is based on the observation that $\displaystyle T^2=S^1\times S^1 = \mathbb{R}/\mathbb{Z} \times \mathbb{R}/\mathbb{Z}$.

Now we can add another factor to get the 3-torus:
$\displaystyle T^3=S^1\times S^1 \times S^1= \mathbb{R}/\mathbb{Z} \times \mathbb{R}/\mathbb{Z} \times \mathbb{R}/\mathbb{Z}$.
This shows how to identify sides on a cube to get a 3-torus. And that should lead to a CW-structure on $\displaystyle T^3$.

Further generalizations for higher dimensions should be analogous.