Finding Planar Representation of Torus with n Holes

[stephenkeiths]

Homework Statement

Find the fundamental group of $T^{n}$, the torus with n holes, by finding the planar representation of $T^{n}$.

Homework Equations

I'm just having a hard time finding the planar representation of $T^{n}$. 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.

 

[jgens]

Can you visualize the double torus? You should also be thinking in 3-dimensional space and not the plane.

 

[stephenkeiths]

I can visualize it in 3-D but I'm having a hard time imagining cutting it down into an identification space

 

[jgens]

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.

 

[stephenkeiths]

Do you mean something like a sphere with 2 handles?

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

So maybe the sphere with 2 handles has a group isomorphic to $Z^{4}$ ??

 

[jgens]

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.