The real projective plane $\mathbb{RP}^2$ has fundamental group $C_2$. We can understand this via the universal covering mapping $S^2 \to \mathbb{RP}^2$ which identifies antipodal points: the contractible loops in $\mathbb{RP}^2$ lift to loops on $S^2$, while non-contractible loops lift to paths which connect a point with its antipodal point (and 'simultaneously' connects that antipodal point with the point, on the other side of the sphere). We can visualize this, and via this the group operation on $\pi_1(\mathbb{RP}^2)$.
Although this visualization is somewhat satisfying, it still intuitively bothers me that you can have a circle wrapped around something that you can't untie, but then doing the wrapping again does allow you to untie it. Certainly it seems that it cannot happen for subsets of $\mathbb R^3$ (or can it?), but since $\mathbb{RP}^2$ embeds into $\mathbb R^4$, it does happen in Euclidean space.
Since $\mathbb{RP}^2$ has low dimension, and thus seems relatively amenable to visualization, my question is:
How can I visualise the fundamental group of $\mathbb{RP}^2$ (or another space with a fundamental group with torsion) in such a way that I can geometrically understand how torsion in a fundamental group works?
Best Answer
View $\mathbb{RP}^2$ as the Möbius strip with a disk attached on its boundary circle. The core circle of the Möbius strip is the generator of the fundamental group. You cannot drag the core circle to its boundary - it's "stuck". To see this, try and perturb the core circle.
I've included a picture which shows how wrapping around twice gets you "unstuck".