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I've been thinking today about the fundamental group of the projective plane which is the cyclic group of two elements ($\pi_1(\mathbb{R}P^2)=\mathbb{Z}_2$). This means there is only one class of elements different from zero and that it has order 2.
What is the meaning of this in terms of loops? It seems that every non-zero loop is traversed in a different direction each time. Is this the right intuition? What is a good geometrical way to visualize this?
One way to visualize this is that $\mathbb{R}P^2$ is homeomorphic to what you get from the Mobius band $M$ and a disc $D$ by identifying the circles $\partial M$, $\partial D$ to a single circle $C$. This circle $C$ is homotopic to a point, through the disc $D$. The core circle of $M$ itself is not contractible to a point, but if instead you go twice around that circle then it is homotopic to $C$ and therefore is homotopic to a point.
