#abelian

This video shows that the fundamental group of a torus (doughnut) is abelian. We can imagine the fundamental group as some sort of collection of loops. A loop is somehow a lasso with a fixed base point, i.e. the knot of the lasso. We consider two loops to be the same, if we can deform one into the other continuously. We can imagine this as pulling or shrinking the lasso into an other lasso. If there is no hole in the space, we can contract a lasso to a point. (We say that such a lasso is homotopic to the constant loop which stays at one point.) A loop can live on the surface of a doughnut or an other geometric object. So the fundamental group of a torus includes two different loops, which are not homotopic to the constant loop. One which goes around the torus and one which goes through the hole of the torus. We can represent any other loop by a combination of this two loops. For example we can represent a loop which goes twice through the hole of the torus by going two times the second loop described above. (Consider also that two loops are the same, if they can be deformed into each other continuously.) We see that the fundamental group somehow recognizes the holes of a space. The fundamental group of the torus is abelian, i.e. the sequence of the loops doesn’t matter. In particular it doesn’t matter, if you are going the loop (which we call m) through the hole of the torus first and then the loop around the torus (which we call n) or vice versa. The video shows this special property for exactly the loops described above on a square. This square forms a torus by gluing opposite edges together. If we want to show that the fundamental group is abelian, we have to show that we can continuously deform mn (first going n then m) into nm (first going m then n). This is shown in the video on the square.