Prove that every continuous map $f:P^2\to S^1$, where $P^2$ is the projective plane, is nullhomotopic.
I think I need to use the fact that $\pi_1(P^2) = \mathbb{Z}/2\mathbb{Z}$ and covering space theory. The map $f$ induces a map $f_* : \pi_(P^2) = \mathbb{Z}/2\mathbb{Z} \to \pi_1(S^1) = \mathbb{Z}$, but I don't see why this map is trivial.