TITLE: Fixed-point free map of the 2-sphere which has order 4
QUESTION [6 upvotes]: The antipodal involution of $\mathbb{S}^2$ clearly has no fixed points. However I cannot think up an example of homeomorphism of order 4 which has no fixed points. Could you help me?

REPLY [4 votes]: Consider the linear transformation of $\Bbb{R}^3$ corresponding to
$$
A=\left(\begin{array}{ccc}0&-1&0\\1&0&0\\0&0&-1\end{array}\right).
$$
The transformation is orthogonal, hence preserves lengths, hence maps $S^2$ to itself. It is clearly of order four. The eigenvalues of $A$ are $i,-i,-1$, so it has no fixed points.

This is in the set of homeomorphisms described by Antonio: a 90 degree rotation followed by a reflection.