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.
www-ifm.math.uni-hannover.de/~...

[@Sush]

I learned about the fundamental group during the second semester of my 3rd year of undergrad. Now I'm finishing my first semeter of 4th year (the last year) and this has helped a lot. At first, when I was taking this Geometry and Topology course I did not get this video. Now it's all good. Sometimes you just have to let things rest for a bit and everything will become clearer.

[@deepakmsmundayur3472]

Thank you so much! I had seen proof of this using lifting it to R² but I have been trying to visualise the fundamental group of torus in this way. Thanks for the great animation.

[@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).

[@nachorubiovera4604]

Actually the fundamental group of S^1 is the free group generated by an only one element (which is isomorphic to Z), so the fundamental group of the torus (which is isomorphic to S^1 x S^1) is the free group generated by two elements, which is not isomorphic to Z^2 because Z^2 is abelian and the free group of two elements maybe isn't abelian.

[@SpaceIceGirl]

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

[@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.

[@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.

[@breaneainn]

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

[@sergiorgio2000]

But this transformation isn't contunuous (as homotopy must be), it cut the connetion point and change the connect components (if get of a point) in the start

[@copernicus633]

The video would be more effective if it had audio which: Defines what a group is; Then shows what the group elements are for this topological situation; Then points out that it is an Abelian (commutative) group

[@JVirago]

@dampf0Y0ente what does abelian mean?

[@LinusPerssonsTube]

It means that a*b=b*a for all a,b in the group and where * is the group's operator.

[@francescos7361]

Thank you so much .

[@LoboLoko007]

someone please explain to me what is this video telling? looks cool

[@justin9571]

Morbius strip

[@sorrysonofa]

I thought the torus' fundamental group was the free product of Z with it self... and so is noncommutative. is it the direct product?

[@joseantoniofuentesmesa8865]

The fundamental group of a product X×Y with the product topology is isomorphic to the direct product of fundamental groups of X and Y. The torus is essentially S^1×S^1 and therefore its fundamental group is Z^2. An example of a topological space whose fundamental group is Z*Z (free product) is: two circunferences that touch just in a point

[@ihmc3jn09hk]

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

[@mamaladillo]

But, could someone show me a topologycal space without a abelian fundamental group?

[@cryptexify]

Yeah, the plane with two holes.

[@theoriginal639]

n-bouquet

[@kolomgorov]

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.

[@hulfe2514]

Are you okay youtube? because this isn't healthy behavior, why did you bring me here?

[@gert-jandugardein6836]

The torus with two holes.

[@Fersomling]

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

[@jokesKjokes]

This shit blows my mind

[@2t22tornadosiren]

That is not a donut. That is a torus.

[@jedibill111]

MMMM...Donuts....