# Given finite G⊂O(n)G\subset O(n), is there a “standard” cell structure on Sn−1S^{n-1} with GG acting cellularly?

Let $G\subset O(n)$ be a finite orthogonal group.

Is there a regular CW-complex structure on $S^{n-1}$ on which $G$ acts cellularly which is in any sense “natural”?

What I’m looking for is inspired by the following examples and ideally would generalize them:

(a) If $G$ is a reflection group, then the Coxeter complex, i.e. the regular CW structure given to $S^{n-1}$ by intersecting with all the reflecting hyperplanes.

(b) If $G$ is $\{\pm I\}$, then the standard cell-complex structure on $S^{n-1}$ with $2$ cells in each dimension from zero up.

(This question is a little soft because I’m not sure how much “naturality” it would be fair to ask for. Candidates include: (1) such that the action on top-dimensional cells is free and transitive [but this seems like it would be too much to hope for]; (2) such that the construction is functorial from the category of subgroups of $O(n)$ with inclusions as morphisms to the category of regular CW-complexes with cellular embeddings refinements as morphisms [again, maybe too much to hope for]; (3) a uniform general construction that specializes to both the special cases above. And I’m open to other ideas…)

For a first pass at an answer:

Fix a point of the sphere with trivial stabilizer in $G$ and take the Voronoi diagram of its orbit under $G$. The top-dimensional cells are the Voronoi cells, which are polyhedra, and the $d$-dimensional cells are the $d$-faces of these polyhedra.

This construction has property (1) – the action is free and transitive on top-dimensional cells. It does not have property (2). I believe it does have part of property (3) – generalizing the Coxeter complex – since reflecting hyperplanes should be the boundaries of Voronoi cells since the images of the point on either side of a given reflecting hyperplane are equidistant. However, it’s unsatisfactory re: the other part of property (3), since I don’t see how it constructs lower-dimensional cells (below $n-2$) for the example of $G=\mathbb{Z}/2$ acting by the antipodal map.

So, this is an imperfect but passable solution. (Tom Goodwillie shows in a comment that (1) and (2) are not simultaneously possible. More generally, (2) seems kind of ridiculous given how any reasonable cell decomposition for the antipodal map should depend on some arbitrary choice, since this map is central in $O(n)$ so cannot distinguish any points of the sphere. How could such a choice then be made functorial?) But I would be interested in others, so I’ll leave the question open for a while.