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> What is known about isometric actions on $\mathbb S^n$ such that the quotient space is homeomorphic to $\mathbb S^n$?
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**Comments.**


* I am mostly interested in (maybe trivial) properties of such actions for large $n$.
* I see that the orientation preserving part of Coxeter's group has this property.
* Originally I thought that any such action is generated by rotations around $\mathbb S^{n-2}$'s; now I see that there are other examples for $\mathbb S^3$; thanks to Lee Mosher.



Actions on Sⁿ with quotient Sⁿ
I think I have completely answered the question in the following form:


**Theorem**. *For a finite subgroup $\Gamma < O(n)$ the quotient space $S^{n-1}/\Gamma$ is homeomorphic to $S^{n-1}$ if and only if $\Gamma$ has the form
\begin{eqnarray\*}
 \Gamma = \Gamma\_{ps} \times P\_1 \times \ldots \times P\_k
\end{eqnarray\*}
for a pseudoreflection group $\Gamma\_{ps}$ and Poincaré groups $P\_i<SO(4)$, $i=1,\ldots,k$, such that the factors act in pairwise orthogonal spaces and such that $n>5$ if $k=1$.* 


cf. <http://arxiv.org/abs/1307.4875>.


Here, a pseudoreflection group is understood in the sense of Mikhailova and a Poincaré $P$ group comes from the binary icosahedral group in $SU(2)$, i.e. $S^3/P$ is Poincaré's homology sphere. 


