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Topology question

  • #1

Homework Statement


Find the fundamental group of [itex]T^{n}[/itex], the torus with n holes, by finding the planar representation of [itex]T^{n}[/itex].

Homework Equations


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


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.
 

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Answers and Replies

  • #2
jgens
Gold Member
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Can you visualize the double torus? You should also be thinking in 3-dimensional space and not the plane.
 
  • #3
I can visualize it in 3-D but I'm having a hard time imagining cutting it down into an identification space
 
  • #4
jgens
Gold Member
1,580
49
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:
  • #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] ??
 
  • #6
jgens
Gold Member
1,580
49
Do you mean something like a sphere with 2 handles?
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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