Rigidity of symmetric simplicial complexes and the lower bound theorem

Cruickshank, James ; Jackson, Bill ; Tanigawa, Shin-ichi

Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e4

Résumé

We show that if $\Gamma $ 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 $\Gamma $-symmetric infinitesimally rigid realisation in ${\mathbb R}^{k+1}$ or $k=2$ and $\Gamma $ is a half-turn rotation group. This verifies a conjecture made by Klee, Nevo, Novik and Zheng for the case when $\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 automorphism of order 2, 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.

Détail

DOI : 10.1017/fms.2024.150

Bibliographie

[1] Adin, R. M., ‘On face numbers of rational simplicial polytopes with symmetry’, Adv. Math. 115(2) (1995), 269–285. Google Scholar | DOI

[2] Asimow, L. and Roth, B., ‘The rigidity of graphs’, Trans. Amer. Math. Soc. 245 (1978), 279–289. Google Scholar | DOI

[3] Barnette, D., ‘A proof of the lower bound conjecture for convex polytopes’, Pacific J. Math. 46(2) (1973), 349–354. Google Scholar | DOI

[4] Bricard, R., ‘Mémoire sur la théorie de l’octaèdre articulé’, J. Math. Pures Appl. 3 (1897), 113–148. Google Scholar

[5] Connelly, R., ‘A counterexample to the rigidity conjecture for polyhedra’, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 333–338. Google Scholar | DOI

[6] Cruickshank, J., Jackson, B. and Tanigawa, S., ‘Global rigidity of triangulated manifolds’, Adv. Math. 458 (2024), 109953. Google Scholar | DOI

[7] Fogelsanger, A. L., ‘The generic rigidity of minimal cycles’, PhD thesis, Cornell University, 1988. Google Scholar

[8] Gromov, M., Partial Differential Relations (Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]) vol. 9 (Springer-Verlag, Berlin, 1986). Google Scholar

[9] Jorge, H., ‘Combinatorics of polytopes with a group of linear symmetries of prime power order’, Discrete Comput. Geom. 30 (2003), 529–542. Google Scholar | DOI

[10] Kalai, G., ‘Rigidity and the lower bound theorem. I’, Invent. Math. 88(1): (1987), 25–151. Google Scholar | DOI

[11] Klee, S., Nevo, E., Novik, I. and Zheng, H., ‘A lower bound theorem for centrally symmetric simplicial polytopes’, Discrete Comput. Geom. 61(3) (2019), 541–561. Google Scholar | DOI

[12] Novik, I. and Zheng, H., ‘Highly neighborly centrally symmetric spheres’, Adv. Math. 370 (2020), 107238. Google Scholar | DOI

[13] Sanyal, R., Werner, A. and Ziegler, G. M., ‘On Kalai’s conjectures concerning centrally symmetric polytopes’, Discrete Comput. Geom. 41(2) (2009), 183–198. Google Scholar | DOI

[14] Schlenker, J.-M., ‘A rigidity criterion for non-convex polyhedra’, Discrete Comput. Geom. 33(2) (2005), 207–221. Google Scholar | DOI

[15] Schulze, B. and Tanigawa, S., ‘Infinitesimal rigidity of symmetric bar-joint frameworks’, SIAM J. Discrete Math. 29(3) (2015), 1259–1286. Google Scholar | DOI

[16] Schulze, B. and Whiteley, W., ‘Rigidity of symmetric frameworks’, in Toth, J. G. C. D., O’Rourke, J., eds., Handbook of Discrete and Computational Geometry, third edn. (Chapman & Hall CRC, 2018). Google Scholar

[17] Servatius, B., ‘Polyhedra in 3-space’, in Handbook of Geometric Constraint Systems Principles (CRC Press, 2018), chapter 13. Google Scholar

[18] Stanley, R. P., ‘The number of faces of a simplicial convex polytope’, Adv. Math. 35(3) (1980), 236–238, . Google Scholar | DOI

[19] Stanley, R. P., ‘On the number of faces of centrally-symmetric simplicial polytopes’, Graphs Combin. 3(1) (1987), 55–66. Google Scholar | DOI

[20] Tay, T., ‘Lower-bound theorems for pseudomanifolds’, Discrete Comput. Geom. 13(2) (1995), 203–216. Google Scholar | DOI

[21] Whiteley, W., ‘Cones, infinity and one-story buildings’, Structural Topology 8 (1983). Google Scholar

[22] Whiteley, W., ‘Infinitesimally rigid polyhedra. I. Statics of frameworks’, Trans. Amer. Math. Soc. 285(2) (1984), 431–465. Google Scholar | DOI

[23] Whiteley, W., ‘Vertex splitting in isostatic frameworks’, Structural Topology 16 (1990), 23–30. Google Scholar