The fundamental Group of the Torus is abelian

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

Category:

Howto & Style

Tags:

• topology
• math
• fundamental
• group
• torus
• abelian
• path
• homotopy
• bothmer
• fugru
• cg

[breaneainn]

If the cartesian symmetry of a torus is a mobius, then is is the fundamental group of a mobius a circle?

[bothmer]

@breaneainn: The fundamental group of the Toris is ZZ x ZZ. The fundamental group of the mobius strip is ZZ since it can be contracted to a circle.

[SpaceIceGirl]

this is really cool! your explanation made it an interesting little lesson. thanks.

[bothmer]

Glad you liked it!

[jedibill111]

MMMM...Donuts....

[dampf0Y0ente]

..the only video for the search-string "abelian" by now

hopefully this will change in the future =)

horaay educational youtube!!

[tothemesosphere]

exactly what I was looking for thanks! I've seen a non-direct proof that the fundamental group of a torus is ZxZ but I couldn't see how it was abelian...This is a really good visualisation.

[ihmc3jn09hk]

?? Is it true only when the idea of "vector" not involved?

[JVirago]

@dampf0Y0ente what does abelian mean?

[breaneainn]

@bothmer !! penny dropped, cheers! I get it, toris has two groups in third symmetry, mobius has one group in two symmetries. Far out, I'm not mensa material by any measure, but I had an art teacher 20 years ago that showed me the 3 curve toris dilemma, and it's bugged me ever since. Thanks again.

[Fersomling]

I do not know what you guys are talking about. Yay! Something new to apply my genius toward.

[sorrysonofa]

thanks man it's been a year since my comment I have ince corrected my ideas on the fundgrp of the torus :)

concerning commutativity, you could say that it is commu as the fundgrp of a topological grp, without even knowing its exact structure.

[jjsantos81]

@sorrysonofa the toru's fundamental group is ZxZ, cartesian product, since the torus is S^1xS^1 homotopy equivalent. Z is abelian with the usual sum, so the cartesian ZxZ is abelian with induced sum (a,b)+(c,d)=(a+c,b+d).

[Aziannic]

now I can make a donut by myself