One of my graduate students asked me the following question, and I can't seem to answer it. Let $\Sigma_g$ denote a compact oriented genus $g$ surface. For which $g$ does there exist an orientation-preserving diffeomorphism $f\colon \Sigma_g \rightarrow \Sigma_g$ with the following two properties:

- $f$ has no fixed points.
- The action of $f$ on $H_1(\Sigma_g)$ fixes no nonzero elements.

Since $f$ has no fixed points, you can use the Lefschetz fixed point theorem to deduce that the trace of the action of $f$ on $H_1(\Sigma_g)$ must be $2$. From this, you can easily see that no such $f$ can occur for $g=0$ and $g=1$. However, I can't figure out what is going on here for $g \geq 2$.