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:


1. $f$ has no fixed points.
2. 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 $g \geq 2$.



Fixed-point free diffeomorphisms of surfaces fixing no homology classes
Goodwillie's construction (in genus two) generalises to all higher genus as follows.


Let $P\_n$ be the regular $n$-gon in the plane with vertices at roots of unity. When $n$ is even, we can glue opposite (and thus parallel) sides to obtain an oriented surface $F\_n$. Suppose that $n = 4g + 2$. In this case $F\_n$ has genus $g$; also the rotation by $2\pi / (4g + 2)$ induces a homeomorphism $f\_n$ of $F\_n$ with exactly one fixed point, at the origin.


Now we take copies of $F\_{4g + 2}$ and $F\_{4h + 2}$, remove small disks about the origin of each, and glue along the so created boundaries. The resulting connect sum $F$ has genus $g + h$. In a neighbourhood of the gluing we interpolate between the homeomorphisms $f\_{4g + 2}$ and $f\_{4h + 2}$ (this is called a "fractional Dehn twist" in some places). The resulting homeomorphism $f \colon F \to F$ has the desired properties.


