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Triangulation of Torus, Algorithms for Calculating Simplicial Homology

  1. Apr 24, 2010 #1
    Hi, everyone:

    A couple of points, please:

    1) I am reviewing last semester's Simplicial Homology. I was able to do a triangulation
    of the torus T2=S1xS1 , and I was able to do
    a triangulation of T2 , although the best I could do was use 18 triangles.
    (the triangulation checked out: 18 triangles/faces, 8 vertices, 10 edges)

    I tried to do a triangulation with 12 triangles/faces , 6 vertices, but the edges did not
    add up ; I need 18, but could only come up with 15.
    Anyone know of a smaller triangulation of the Torus (i.e., with fewer than 18 triangles).?. Anyone know how to find the Euler number of general surfaces (like
    the Mobius band) , without using Homology (e.g., Betti numbers.).? . I am
    trapped in the catch-22 : to find the (simplicial )Homology, I need to find
    a triangulation. But to check if the triangulation is a valid one, I need t find
    the Euler number, which I only know how to find using the homology group.


    2) Is there an algorithm to find the Simplicial Homology of a space X , once X has
    been triangulated, and we know the chain groups of X.?. It seems to come
    down to some basic calculations, and some basic linear algebra (row-reduction).

    Thanks For any Suggestions/Ideas.
     
  2. jcsd
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