Restricting a map from $S^{2n-1}$ to $\mathbb{R}P^{2n-1}$ This might be really obvious, but I have constructed a continuous map $f:S^{2n-1} \rightarrow S^{2n-1}$ with no fixed points, and I want to use this to get a continuous map $g:\mathbb{R}P^{2n-1} \rightarrow \mathbb{R}P^{2n-1}$ with no fixed points (where $\mathbb{R}P^{2n-1}$ is the real projective plane). Is the idea that $\mathbb{R}P^{2n-1}$ can be defined as points on $S^{2n-1}$, with points identified with their antipodal points, so $\mathbb{R}P^{2n-1} \subseteq S^{2n-1}?$ Therefore, by simply restricting $f|_{\mathbb{R}P^{2n-1}}$ do we get the continuous map $g$?
Assuming this doesn't work, using the universal cover $p:S^{2n-1} \rightarrow \mathbb{R}P^{2n-1}, pf:\mathbb{R}P^{2n-1} \rightarrow \mathbb{R}P^{2n-1}$. But does $pf$ also not have fixed points if $f$ has no fixed points? I don't see why that necessarily would be true. If this doesn't work either, how should I get $g$? 

Hint: Let me put $m$ for $2n-1$ for brevity and generality. $\Bbb{R}P^m$ is a quotient space of $S^m$ obtained using the equivalence relation that identifies antipodal points. You need to show that your function $g : S^m \to S^m$ is compatible with the equivalence relation, so that it induces a function on the equivalence classes (pairs of antipodal points) that constitute the points of $\Bbb{R}P^m$. See the discussion about functions that "descend to the quotient" in the Wikipedia link.
