Rigidity of Symmetric Simplicial Complexes and the Lower Bound Theorem

We show that, if $\mathrm{\Gamma }$ is a point group of ${\mathbb{R}}^{k+1}$ of order two for some $k\ge 2$ and $\mathcal{S}$ is a $k$-pseudomanifold which has a free automorphism of order two, then either $\mathcal{S}$ has a $\mathrm{\Gamma }$-symmetric infinitesimally rigid realisation in ${\mathbb{R}}^{k+1}$ or $k=2$ and $\mathrm{\Gamma }$ is a half-turn rotation group.This verifies a conjecture made by Klee, Nevo, Novik and Zhang for the case when $\mathrm{\Gamma }$ is a point-inversion group. Our result implies that Stanley’s lower bound theorem for centrally symmetric polytopes extends to pseudomanifolds with a free simplicial involution, thus verifying (the inequality part) of another conjecture of Klee, Nevo, Novik and Zheng. Both results actually apply to a much larger class of simplicial complexes, namely the circuits of the simplicial matroid. The proof of our rigidity result adapts earlier ideas of Fogelsanger to the setting of symmetric simplicial complexes.