# When is a sphere quotient a sphere?

## Ben Blum-Smith

## October 24, 2017

A finite subgroup $G$ of the orthogonal group acts on the unit sphere
$\mathbb{S}^{n-1}$ in $\mathbb{R}^n$, and we can ask how the topology of the quotient
$\mathbb{S}^{n-1}/G$ is related to the structure of $G$. The first question in this
direction — for what $G$ is the quotient a sphere? — has recently been
answered by Christian Lange. The proof of his result is a delicate
case-based induction. A natural followup question is to find a uniform
proof. One possibility is to impose a $G$-invariant cell complex
structure on $\mathbb{S}^{n-1}$ and then to utilize tools from discrete geometry
such as shellings and discrete Morse theory. In this talk I will
present an outline of Lange's result and some notes toward this
followup program, focusing on subgroups of Coxeter groups.

My interest in the problem actually comes from the invariant theory of
finite groups, although this motivation will be discussed only in
passing in the talk.