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Math Help - Homology of the 3-Torus

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by robeuler View Post
    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 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:
    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 T^3.

    Further generalizations for higher dimensions should be analogous.
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