Regular homeomorphisms of $\mathbb{R}^3$ and of $\mathbb{S}^3$

This paper is the paper announced in [Be2, References [2]]. We show that every compact abelian group of homeomorphisms of $\mathbb{R}^3$ is either zero-dimensional or equivalent to a subgroup of the orthogonal group O(3). We prove a similar result if we replace $\mathbb{R}^3$ by $\mathbb{S}^3$, and we study regular homeomorphisms that are conjugate to their inverses.