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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 $g \geq 2$.

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2 Answers 2

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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.

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    $\begingroup$ This is great, thanks!!! $\endgroup$
    – Robert
    Jul 3 at 1:14
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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.

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    $\begingroup$ 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... $\endgroup$
    – Sam Nead
    Jun 30 at 8:02
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    $\begingroup$ 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). $\endgroup$
    – Sam Nead
    Jun 30 at 8:03
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    $\begingroup$ Sam Nead: I hadn't thought of interpolating like that. You should make this an answer. $\endgroup$ Jun 30 at 10:48
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    $\begingroup$ 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. $\endgroup$
    – Robert
    Jun 30 at 21:29

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