Let $M$ be a compact 2-manifold of genus 2. Does there exist an orientation preserving homeomorphism $f:M\to M$, so that $f^n=id$ for some integer $n$, and $f$ doesn't have fixed points? 


Using Lefschetz formula and the cohomology ring structure, I can prove that the order of such $f$ must be a multiple of 4 or 6. I vaguely remember that I have seen such an $f$ before, but I'm not sure. Is there anyone familiar with this kind of examples?



Existence of orientation preserving, finite order self homeomorphism on a genus 2 surface without fixed point
The answer is yes, as it has been remarked in the previous answers. Here is a very explicit construction.


Let $T$ be the sphere $S^2 \subset \Bbb R^3$ with three disks removed. Take these three disks to be centered at the vertices of an equilateral triangle inscribed in the maximal circle $\{z = 0\} \cap S^2$ and with the same radius, so that they are cyclically permuted by the $\frac{2\pi}{3}$-rotation $R$ about the $z$-axis. So, the surface in question is $M = \partial (T \times [-1, 1])$. Let $f : M \to M$ be defined by $f(x, y, z, t) = (R(x,y,-z), -t)$. Hence, $f$ satisfies the conditions in the question, with $n = 6$.


