I am studying algebraic topology and I am having a really hard time in trying to figure out this particular argument. While I somehow manage to understand the proofs that use retraction by deformation, or Van Kampen, I cannot understand those problems involving the identification of sides of a polygon. Could someone please explain how all of this work? I think I get what you do with a thorus (that famous square where you identify opposite edges), but when you have to deal with something more complicated like a double-thorus or even triple, I really can't see it. I mean, for instance the teacher told us that a double thorus can be seen by using an octagon (I think?) and identifying its edges in some way, but I can't see the logic behind it. And this is even worse https://en.wikipedia.org/wiki/Genus-three_surface
Thanks in advance for the help