# Can a spherical simplicial complex have more than one “central” inversion?

Let $$\Delta$$ be a finite connected simplicial complex. Call a simplicial map $$\phi:\Delta\to\Delta$$ an inversion if

1. $$\phi$$ is an involution, that is $$\phi\circ\phi=\mathrm{id}$$, and
2. $$\phi$$ is not fixing any face of $$\Delta$$ set-wise, except for the empty face and (potentially) $$\Delta$$ itself.

For example, if $$\Delta$$ is the boundary complex of a centrally symmetric polytope then such an inversion is induced by the action of the central inversion $$-I$$.

I have the following questions:

Qustions:

1. If $$\Delta$$ has more than one inversion, do they necessarily commute?
2. If $$\Delta$$ is spherical, can there be more than one inversion?
3. Are there more general conditions on $$\Delta$$ so that $$\Delta$$ has at most a single inversion?
• Let $\phi$ and $\psi$ be any noncommuting fixed point free involutions in the symmetric group $S_{2n}$. Regarding the $2n$ elements that are permuted by $S_{2n}$ as a 0-dimensional simplicial complex of $2n$ disjoint points, we get a negative answer to the first question. – Richard Stanley Mar 26 at 13:57
• @RichardStanley You are right of course! Allow me to add that I want to consider connected complexes. – M. Winter Mar 26 at 13:58
• One can turn my example into a connected one. Let $G$ be the graph obtained by taking a set $X$ of $2n$ disjoint points and adding two new vertices $u,v$ connected to all points in $X$. Let $\phi$ and $\psi$ act on $X$ as before while transposing $u$ and $v$. – Richard Stanley Mar 26 at 14:05
• @RichardStanley I now see how badly this can fail. Thank you! The more my focus is now on the other two questions. Is being simply connected enough to prevent this? What about trivial first homology? It might also be interesting to restrict to simplicial manifolds (a complex which is homeomorphic to a manifold). – M. Winter Mar 26 at 14:08