I am following the lecture 25 provided by Wildberger talking about fundamental groups. Here I have a problem that I can not understand. link:https://www.youtube.com/watch?v=E6f3I-RiWbk&t=490s at time 25:54
What I know so far is that the fundamental group of a torus would be like Z × Z. And he use the rectangle to explain it, but I think it is not right. He said that you could have a path like $a$$b$$a^{−1}b^{−1}$, and you could shrink it to the constant loop like what the picture drew. How could this shrinking process happen? I think the interior of the torus is empty, so you can not shrink it like a circle. If the path can shrink to the base point, That means a circle could shrink to a point like a disk, right? So I am confused and I hope someone could help me out.
I think you misunderstand what the square (with identified edges) represents. The square corresponds to the "surface" of the torus, and the edges corresponds to loops on the torus as marked with colors on the picture. So the nullhomotopy of $aba^{-1}b^{-1}$ that on the square is just shrinking to one of the corners also translates to a nullhomotopy on the torus.
Another way to see that $aba^{-1}b^{-1}$ must be trivial in $\pi_1(\mathbb{T})$ is that $\pi_1(\mathbb{T}) \cong \mathbb{Z} \times \mathbb{Z}$ is abelian, therefore any commutator vanishes.