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?
The real projective plane can be viewed as a a square [0,1]x[0,1] with the edges identified as in the diagram above. Let b be our base point. The green loop going across the square from the 'b' in the lower left to the 'b' in the top right (which are identified) cannot be homotoped relative to its endpoints to the trivial loop. However if we compose it with the blue loop then the end of the green line no longer has to stick to 'b' during a homotopy relative to endpoints so we can 'detach' it as in the right figure and then this clearly homotopes to the trivial loop.
To clarify, the blue loop (although it doesn't look like it in my picture) is the same loop as the green one, if you can imagine drawing the green one, then rotating yourself to the opposite side of the square, then drawing the same loop.