# How does one visualize non-trivial finite fundamental groups?

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?

• Regarding the projective plane. There is a model $S^1/_\pm$ where you identify antipodal points on a sphere. This model can also be viewed as a half-sphere where the antipodal points of the border are identified, thus when you travel along a big circle and you traverse the border you jump on the opposite side of the sphere. Both models helped me a lot. – Patrick Abraham May 3 '18 at 14:41
• @PatrickAbraham What is the difference between this and the pacman universe, i.e, old videogames where the screen wraps around? – Agnishom Chattopadhyay May 3 '18 at 14:42
• mathoverflow.net/a/38220 this might be helpful – tschih May 3 '18 at 14:43
• If you use an arrow with a left- and a right-hand mark, you will see when you cross the border the left- and right-hand mark will swap places. This is also called non-orientable. – Patrick Abraham May 3 '18 at 14:45
• – Ethan Bolker May 3 '18 at 17:37