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Topology question

  1. Sep 30, 2012 #1
    1. The problem statement, all variables and given/known data
    Find the fundamental group of [itex]T^{n}[/itex], the torus with n holes, by finding the planar representation of [itex]T^{n}[/itex].

    2. Relevant equations
    I'm just having a hard time finding the planar representation of [itex]T^{n}[/itex]. I can't picture it.


    3. The attempt at a solution
    I can see how the picture attached rolls up into a torus. It rolls into a cylinder and then curls around into a doughnut. I am just having a hard time seeing how I can get a torus with 2 holes, 3 holes etc.

    Thanks!

    P.S. this isn't a very rigorous class. If I just understand how/why then the teacher is happy. He's not much of a proof guy. He's happy with geometric/picture arguments.
     

    Attached Files:

  2. jcsd
  3. Sep 30, 2012 #2

    jgens

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    Can you visualize the double torus? You should also be thinking in 3-dimensional space and not the plane.
     
  4. Sep 30, 2012 #3
    I can visualize it in 3-D but I'm having a hard time imagining cutting it down into an identification space
     
  5. Sep 30, 2012 #4

    jgens

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    Find a space homotopy equivalent to the double torus whose fundamental group you can compute. The 3-dimensional picture will help with this.

    You can also do this by decomposing the double torus into a cell complex, as you suggested above, but I have always found that computation less obvious.
     
    Last edited: Sep 30, 2012
  6. Sep 30, 2012 #5
    Do you mean something like a sphere with 2 handles?

    I could see the sphere with 1 handle has a group isomorphic to [itex]Z^{2}[/itex]:
    G=<[itex]g_{1}[/itex],[itex]g_{2}[/itex]|[itex]g_{1}[/itex]*[itex]g_{1}[/itex]^-1,[itex]g_{2}[/itex]*[itex]g_{2}[/itex]^-1> ??

    So maybe the sphere with 2 handles has a group isomorphic to [itex]Z^{4}[/itex] ??
     
  7. Sep 30, 2012 #6

    jgens

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    If you know how to compute the fundamental group of the sphere with 2 handles, then that works. The double torus is also homotopy equivalent to the wedge sum of two tori, which is (in my opinion) the easiest way to compute the fundamental group.
     
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