symmetric group of regular polyhedrons

Question

Let $\Delta^n$ be the regular $n$-simplex spanned by $(n+1)$ vertices, equipped with an Riemannian metric such that all the edges are of equal length. For example,

$\Delta^2$:

$\Delta^3$:

Let $c:=c(\Delta^n)$ be the center of $\Delta^n$. Consider the mapping space $$ M=\{f: \Delta^n\to \mathbb{R}^n\mid f\text{ is injective and isometric, and }f (c)=0\}. $$ I observe that $$ \text{simplicial isomorphism group of }\Delta^n\cong \text{symmetric group }S_{n+1} \text{ permuting on the vertices of }\Delta^n. $$ I define an equivalent relation $\sim$ on $M$ by setting $$ f\sim g $$ if and only if there exists a simplicial isomorphism $i$ of $\Delta^n$ such that $$ f=g\circ i. $$ I find that $M/\sim$ is called the polyhedral group of $\Delta^n$.

Question: Are there any references about the topological structure of this mapping space $M/\sim$?

I want to know the cohomology ring $ H^*(M/\sim;\mathbb{Z}_2) $?

Answer 1

It will make no difference if you define $\Delta^n$ in such a way that $c=0$. In that case it is not hard to see that any isometric embedding is linear, so $M=O(n)$. It follows that $M/\sim=O(n)/\Sigma_{n+1}=SO(n)/A_{n+1}$. We therefore have a Serre spectral sequence $$ H^i(A_{n+1};H^j(SO(n);\mathbb{Z}/2)) \Longrightarrow H^{i+j}(M/\sim;\mathbb{Z}/2). $$ In principle, the $E^2$ term involves the group cohomology of $H^*(SO(n))$ as a module over $A_{n+1}$. However, given any $g\in A_{n+1}$ we can choose a path in $SO(n)$ connecting $g$ to $1$, and it follows that the action of $g$ on $SO(n)$ is homotopic to the identity, and so acts as the identity on cohomology. Thus, the $E^2$ term can be rewritten as $$ H^i(A_{n+1};\mathbb{Z}/2) \otimes H^j(SO(n);\mathbb{Z}/2). $$ The mod $2$ cohomology groups of $SO(n)$ and $\Sigma_{n+1}$ are well known, although those of $\Sigma_{n+1}$ are complicated. It should not be hard to deduce the cohomology groups of $A_{n+1}$. However, I would guess that there are many differentials in the spectral sequence unless $n$ is very small.