Does every continuous function $f: \mathbb R P^2 \vee \mathbb R P^2 \to \mathbb R P^2 \vee \mathbb R P^2$ have a fixed point?
I don't really have a good feeling as to whether or not this is true. My first thought was to try and get a map $f':\mathbb R P^2 \to \mathbb RP^2$, which was easier to show has a fixed point, but I can't seem to restrict $f$ in an appropriate way.
If $X = \mathbb R P^2 \vee \mathbb R P^2$, then any map $f:X \to X$ induces a map $\varphi:\tilde X \to \tilde X$, where $\tilde X$ is the universal cover. I believe $\tilde X$ is a (infinite) wedge product of spheres. If I can show that $\varphi:\tilde X \to \tilde X$ has a fixed point, then it should follow that $f$ does, too, by doing some commutative diagram chasing. However, I can't prove that final fact (if it is true, at all).
I'm doing some algebraic topology studying on my own before I take the class next semester. This is coming from an old qual, so hints would be great. Thanks!