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Rigidity of Symmetric Simplicial Complexes and the Lower Bound Theorem

James Cruickshank, Bill Jackson, Shinichi Tanigawa

TL;DR

This work extends rigidity theory to $\mathbb{Z}_2$-symmetric simplicial complexes, proving that every $\mathbb{Z}_2$-symmetric $k$-circuit admits a $\Gamma$-symmetric infinitesimally rigid realisation in $\mathbb{R}^{k+1}$ except in the exceptional case $k=2$ with a half-turn symmetry, where flexibility persists. The authors adapt Fogelsanger’s decomposition to the symmetric setting and develop new tools for frameworks with partial symmetry, enabling a symmetric proof of a lower bound theorem analogous to Stanley’s in the symmetric context. These results apply to the circuits of the simplicial matroid, yielding a symmetric extension of the Lower Bound Theorem to pseudomanifolds and providing structural insights into equality cases via symmetry. The methods also connect rigidity with the combinatorics of symmetric $k$-complexes, offering a framework to explore more general symmetry groups and higher-dimensional faces.

Abstract

We show that, if $Γ$ is a point group of $\mathbb{R}^{k+1}$ of order two for some $k\geq 2$ and $\mathcal S$ is a $k$-pseudomanifold which has a free automorphism of order two, then either $\mathcal S$ has a $Γ$-symmetric infinitesimally rigid realisation in $\mathbb{R}^{k+1}$ or $k=2$ and $Γ$ is a half-turn rotation group.This verifies a conjecture made by Klee, Nevo, Novik and Zhang for the case when $Γ$ 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.

Rigidity of Symmetric Simplicial Complexes and the Lower Bound Theorem

TL;DR

This work extends rigidity theory to -symmetric simplicial complexes, proving that every -symmetric -circuit admits a -symmetric infinitesimally rigid realisation in except in the exceptional case with a half-turn symmetry, where flexibility persists. The authors adapt Fogelsanger’s decomposition to the symmetric setting and develop new tools for frameworks with partial symmetry, enabling a symmetric proof of a lower bound theorem analogous to Stanley’s in the symmetric context. These results apply to the circuits of the simplicial matroid, yielding a symmetric extension of the Lower Bound Theorem to pseudomanifolds and providing structural insights into equality cases via symmetry. The methods also connect rigidity with the combinatorics of symmetric -complexes, offering a framework to explore more general symmetry groups and higher-dimensional faces.

Abstract

We show that, if is a point group of of order two for some and is a -pseudomanifold which has a free automorphism of order two, then either has a -symmetric infinitesimally rigid realisation in or and is a half-turn rotation group.This verifies a conjecture made by Klee, Nevo, Novik and Zhang for the case when 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.
Paper Structure (19 sections, 25 theorems, 28 equations, 2 figures)

This paper contains 19 sections, 25 theorems, 28 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on Rigidity and Symmetry
  3. Infinitesimal rigidity
  4. Rigidity under symmetry
  5. Gluing properties
  6. Vertex splitting
  7. Background on Simplicial Complexes
  8. $\Gamma$-rigidity of Simplicial Circuits with a Non-adjacent Vertex Pairing
  9. ${\mathbb Z}_2$-Symmetric Simplicial Complexes
  10. $\mathbb{Z}_2$-irreducible cycles and their structural properties
  11. $\mathbb{Z}_2$-symmetric Fogelsanger decomposition
  12. Main rigidity theorem
  13. The Lower Bound Theorem
  14. Closing Remarks
  15. Characterising $\Gamma_{1,3}$-rigid simplicial $2$-circuits
  16. ...and 4 more sections

Key Result

Lemma 2.1

Suppose that $0 \leq t \leq d-1$ and let $\Gamma=\Gamma_{t,d}$. If a $\mathbb{Z}_2$-symmetric graph $(G,\ast)$ is $\Gamma$-rigid and $|V(G)|\geq 2d$, then

Figures (2)

  • Figure 1: The Bricard octahedron.
  • Figure 2: The 1-skeleton of a $\mathbb Z_2$-irreducible $2$-cycle ${\cal S}$, which is not a simplicial 2-circuit. We have ${\cal S}={\cal T}\sqcup {\cal T}^*$ where ${\cal T}=\{\{v_1,v_2,v_3\},\{v_1,v_2,v_4\},\{v_1,v_3,v_4\},\{v_1^*,v_2,v_3\},\{v_1^*,v_2,v_4\},\{v_1^*,v_3,v_4\}\}$ is the simplicial 2-circuit given by the boundary complex of the hexahedron.

Theorems & Definitions (62)

  • Example 1
  • Lemma 2.1
  • proof
  • Example 2
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • Lemma 2.4
  • proof
  • Claim 2.5
  • ...and 52 more