Without getting into Lefschetz theorem i can prove that any continuous map from $\mathbb{RP}^2 \to \mathbb{RP}^2$ has a fixed point using the general lifting criterion.
Does the same hold for real projective line? It doesn't seem to be true as with Lefschetz we can prove it for even dimensions. But i can't think of any map that has no fixed point? Is there a trivial one?
Best Answer
$\mathbb{RP}^1$ is homeomorphic to $S^1$, so we can map $S^1$ to itself by rotating counterclockwise by $\pi/2$ radians to get a map without fixed points.