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

- $\phi$ is an involution, that is $\phi\circ\phi=\mathrm{id}$, and
- $\phi$ 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:

- If $\Delta$ has more than one inversion, do they necessarily commute?
- If $\Delta$ is spherical, can there be more than one inversion?
- Are there more general conditions on $\Delta$ so that $\Delta$ has at most a single inversion?

connectedcomplexes. $\endgroup$ – M. Winter Mar 26 at 13:58