The fundamental Group of the Torus is abelian

not exactly related to what the vid. is about, but wouldn't it be something similar to two trapezoids short-side-joined-together are required to make a torus, otherwise when it wraps the inner area is going to have loads of excess overlaps. i don't do higher-dimensional toplogy i just find it fascinating how people can think of processes (formulas) that describe actual-size wrapping of surfaces. but then it's likely textiles and measure knew all that, in the old ways (proportions to each other).

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

MMMM...Donuts....

That is not a donut. That is a torus.

Nice!

don't you mean homotopic rather than homeomorphic in your info.

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

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

You may be thinking of the figure eight space -- two circles joined together at a point. The fundamental group of a figure eight space is the free product of Z with itself. BTW, if you puncture the torus, i.e. delete a point, then the result is homotopy equivalent to a figure eight space, and so has the same fundamental group. You can see how the "missing point" would get in the way of the homotopy in the video.

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.