Visualzing a nontrivial element of $\pi(\mathbb{R}P^2)$

Question

Visualzing a nontrivial element of $\pi(\mathbb{R}P^2)$.

I'd like to visualize a nontrivial element of $\pi(\mathbb{R}P^2)$ whilst thinking of $\mathbb{R}P^2$ as $S^2$ with antipodal points being identified. What would one of these loops look like? Thanks.

Answer 1

Assuming $\pi$ means the fundamental group functor. Let's say we have as base point the northsouth pole of $\Bbb RP^2$. Then a nontrivial loop is simply a path that goes from the northsouth pole to the southnorth pole. The two antipodal points swap places.

Answer 2

$\mathbb{R}P^2$ is what you get when you attach a disk along the boundary of a Möbius strip, so what you get when you want to trivialize the boundary loop of a Möbius strip. One can physically verify that the loop corresponding to the boundary is twice the core circle of the Möbius strip.

This means that this core circle is what generates the fundamental group of projective space.