The fundamental Group of the Torus is abelian

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This video illustrates the proof of the Theorem in the title. The proof goes like this: Consider a rectangle. Then the path going up the left side of the rectangle and then along the top is homeomorphic to the path going first along the bottom and then up the right side. Gluing the rectancle to make a torus, this shows that going first around through the hole and then along the outside is homeomorphic to going first along the outside and then through the hole. Since these two path generate the fundamental group of the torus this proves that this group is abelan. q.e.d. Remark: This is a very special property. Many topological spaces have nonabelian fundamental groups. This video was produces for a topology seminar at the Leibniz Universitaet Hannover. http://www-ifm.math.uni-hannover.de/~fugru/?topologie_teil1

Channel: Howto & Style
Uploaded: November 8, 2006 at 10:08 am
Author: bothmer

Length: 0:01:33
Rating: 4.63
Views: 17,668

Tags: topology math fundamental group torus abelian path homotopy bothmer fugru cg

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Video Comments:
sorrysonofa (Thursday 27th of November 2008 06:27:33 PM)
I thought the torus' fundamental group was the free product of Z with it self... and so is noncommutative. is it the direct product?
kolomgorov (Saturday 1st of November 2008 11:27:57 PM)
Much easier to show that the torus is homeomorphic to S^1*S^1 and then remember that the fundamental group is a topological property. I like these videos though, I'm so bad at picturing this stuff.
greenphantom (Friday 5th of September 2008 04:07:14 AM)
It's Z+Z.
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LoboLoko007 (Sunday 8th of June 2008 05:53:42 PM)
someone please explain to me what is this video telling? looks cool
sc316ru (Wednesday 2nd of July 2008 09:53:50 PM)
It's telling what it shows. That the one loop on the "donught" can be deformed into the other loop. This is important in math because these two loops represent important things. One can talk about simlilar loops on other spacs, like a figure 8, and one cannot deform the two loops into eachother (notice on the first slide how the loops moved through the "middle" which is not there in the figure 8!)
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Genial. Anschaulicher geht es kaum.