Does there exist homeomorphism without fixed points?

Question

Does there exist a homeomorphism of the unit disk with two holes $$\left\{(x,y):x^2+y^2 \le 1\right\} \setminus \left (\left \{(x,y):\left(x+ \frac 1 2 \right)^2+y^2 < \frac 1 {10} \right \} \bigcup \left \{(x,y):\left(x-\frac 1 2 \right)^2+y^2 < \frac 1 {10}\right\}\right)$$ on itself without any fixed point?

Answer 1

Yes. Your space is homeomorphic to the standard unit sphere $S^2\subseteq\mathbb R^3$ with three open disks removed. These disks can be chosen so that the rotation of $\mathbb R^3$ by $\frac{2\pi}3$ around some axis cyclically permutes them. This rotation, however, has two fixed points, where the axis of rotation intersects $S^2$. Compose it with a reflection across the plane perpendicular to your axis and you are done. (The disks can be chosen so that this reflection preserves them.)