# Fixed-point free diffeomorphisms of surfaces fixing no homology classes

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 $$g \geq 2$$.

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.

• This is great, thanks!!! Jul 3 at 1:14

Here's an example with $$g=2$$. Let $$T$$ be the torus $$\mathbb C/L$$, where $$L$$ is the lattice spanned by $$1$$ and $$\zeta=e^{2\pi i/6}$$. Let $$f:T\to T$$ be induced by multiplication by $$\zeta$$. This is a diffeomorphism fixing one point $$0\in T$$ and fixing no non-zero elements of $$H_1(T)$$. Now remove a little disk centered at $$0$$ and stick together two copies of this punctured torus, and let the map act like $$f$$ on both copies.

But I don't immediately see how to learn anything about $$g>2$$ from this example.

• Your construction generalises to all genera greater than one, 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... Jun 30 at 8:02
• take copies of $F_{4g + 2}$ and $F_{4h + 2}$, remove small disks about the origin of each and glue. This gives a surface of genus $g + h$. Also, in a neighbourhood of the gluing we interpolate between the homeomorphsims $f_{4g + 2}$ and $f_{4h + 2}$ (this is called a "fractional Dehn twist" in some places). Jun 30 at 8:03
• Sam Nead: I hadn't thought of interpolating like that. You should make this an answer. Jun 30 at 10:48
• This is great, but Sam Nead's answer sounds even better! So I'm going to hold off on accepting an answer until he has a chance to write one. Jun 30 at 21:29