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$3$-sphere} \begin{itemize} \item $\so 4$ is isomorphic to the isometry group of $\S^3$. - \item Subgroups $\Gamma \leqslant \so 4$ define equivalence classes of orbits on $\S^3$ by the standard action of $\so 4$ on $\R^3$, i.e. $x \sim y$ iff $x = My$ for some $M \in \Gamma$. + \item Subgroups $\Gamma \leqslant \so 4$ define equivalence classes of orbits on $\S^3$ by the standard action of $\so 4$ on $\R^4$, i.e. $x \sim y$ iff $x = My$ for some $M \in \Gamma$. \item The obtain space is (\emph{sometimes}) a manifold. In particular, it is well defined and spherical for finite $\Gamma$. \item This can be easily generalised to the $n$-sphere. \end{itemize}