Problem: let M be a compact connected manifold, and let $f \colon M \to M$ be a local diffeomorphism with degree two. Prove that if $f$ has one fixed point, then there exists a diffeomorphism $g \colon M \to M$ that has no fixed points.
I'm not really sure how to begin the problem, any hints would be appreciated.
Since $f$ is a degree $2$ covering map, we see that $\chi(M) = 0$. By a nontrivial result, this means there's a vector field $X$ on $M$ with no zeroes. Let $g$ be the time-$\varepsilon$ flow of $X$.
However, it bothers me that the hypothesis we were given that $f$ has one fixed point seems irrelevant.