I find it counter-intuitive to visualize finite fundamental groups.
For example, the fundamental group of the real projective plane is a group with two elements. That means there are two kinds of loops, one which you cannot shrink to a point, and the other, you can. But then, if you traverse this non-trivial loop twice, you end up with something that can be homotoped to the trivial loop.
Is there some way I can visualize this? Is there a way to visualize the projective plane? (I used to think the projective plane is like the pacman universe, but the pacman universe is more like a torus). Is there a visualizable surface where I can see this effect easily?