The fundamental Group of the Torus is abelian




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Published on Nov 8, 2006

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.


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