×

A homological interpretation of skeletal ridigity. (English) Zbl 0968.52019

In their previous articles [Eur. J. Comb. 16, No. 4, 381-403 (1995; Zbl 0834.52009) and ibid., No. 5, 503-523 (1995; Zbl 0834.52010)], the authors together with N. White investigated skeletal rigidity of simplicial complexes introducing certain vector spaces, namely the spaces of stresses, loads and motions of degree \(r\). In the article under review, they give a homological interpretation for those vector spaces and apply this to express the \(g\)-vector of the simplicial complex in terms of their dimensions.
Starting from a given embedding of a simplicial complex \(\Delta\) in projective space, the authors define skeletal chain complexes \({\mathcal R}_r\) extracted from the \((r-1)\)-skeleton of \(\Delta\) such that the stresses, unresolved loads and non-trivial motions of degree \(r\) occur as homology groups of certain degrees. If a \(d\)-dimensional complex \(\Delta\) is embedded into \(d+1\)-dimensional space then the \(r\)th entry of the \(g\)-vector of \(\Delta\) is nothing but the Euler characteristic of \({\mathcal R}_r\).
In fact, essentially the same chain complexes for the case that \(\Delta\) arises from a fan were previously introduced by M. Ishida in the context of toric varieties [see Tôhoku Math. J. 32, 111-146 (1980; Zbl 0454.14021], and they were analyzed by T. Oda for simplicial \(d\)-spheres realized in \(d\)- and \(d+1\)-dimensional space [see J. Pure Appl. Algebra 71, No. 2/3, 265-286 (1991; Zbl 0780.52011)]. Motivated by the attempt to find an elementary algebraic proof for the positivity of the \(g\)-vector of a polytopal simplicial fan, Oda proved vanishing of homology groups of sufficiently low degree and duality results.
In the present article, among other things Oda’s results are extended to the case of a \(d\)-dimensional Cohen-Macaulay simplicial complex realized in \(d\)- or \(d+1\)-dimensional space in a sufficiently general position. As a corollary, the authors obtain for a homology \(d\)-sphere that \(g_r\) equals the dimension of the space of \(r\)-stresses minus the dimension of the space of unresolved \(r\)-loads. Moreover, duality implies that the space of \(r\)-stresses is isomorphic to the space of non-trivial \((d+1-r)\)-motions.
The last section of the article contains the conjecture that any homology \(d\)-sphere even admits an \(r\)-rigid embedding into \(d+1\)-dimensional space (for \(r\leq [d+1/2]\)), i.e. such that all unresolved loads are zero. This is known to be true for \(r=2\), a fact generalizing the theorem of Cauchy on rigidity of the \(1\)-skeleton of a \(3\)-dimensional convex polytope. If the conjecture would be true in general that would immediately imply positivity of the \(g\)-vector of any homology sphere.

MSC:

52C25 Rigidity and flexibility of structures (aspects of discrete geometry)
55U15 Chain complexes in algebraic topology
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Alexandrov, A. D., Konvex Polyeder, (1958), Akademie Verlag Berlin
[2] Billera, L., Homology of smooth splines: generic triangulations and a conjecture of strang, Trans. Amer. Math. Soc., 310, 325-340, (1988) · Zbl 0718.41017
[3] Billera, L.; Lee, C., A proof of the sufficiency of Mcmullen’s conditions for f-vectors of simplicial convex polytopes, J. Combin. Theory A, 31, 237-255, (1981) · Zbl 0479.52006
[4] Cauchy, A., Deuxiéme memoire sur LES polygons et LES polyédres, J. École Polytech., 9, 87-98, (1813)
[5] Crapo, H., Concurrence geometries, Adv. Math., 54, 278-301, (1984) · Zbl 0593.51008
[6] Crapo, H.; Ryan, J., Spatial realizations of linear scenes, Structural Topology, 13, 33-68, (1986) · Zbl 0612.51004
[7] Crapo, H.; Whiteley, W., Stresses on frameworks and motions of panel structures: A projective geometric introduction, Structural Topology, 6, 42-82, (1982)
[8] Crapo, H.; Whiteley, W., Plane stresses and projected polyhedra I: the basic pattern, Structural Topology, 20, 55-78, (1993) · Zbl 0793.52006
[9] Dehn, M., Über die starrheit konvexer polyeder, Math. Ann., 77, 466-473, (1916) · JFM 46.1115.01
[10] Doubilet, P.; Rota, G.-C.; Stein, J., On the foundations of combinatorial theorey IX: combinatorial methods in invariant theory, Stud. Appl. Math., 57, 185-216, (1974) · Zbl 0426.05009
[11] P. Filliman, Face Numbers of pl-Spheres, preprint, MSI, Cornell University, Ithaca, NY.
[12] A. Fogelsanger, The Generic Rigidity of Minimal Cycles, Ph.D. thesis, Department of Mathematics, Cornell University, Ithaca, NY.
[13] Gluck, H., Almost all simply connected surfaces are rigid, Lecture Notes in Math., (1975), Springer-Verlag Berlin/New York, p. 225-239
[14] Ishida, M., Torus embeddings and dualizing complexes, Tôhoku Math. J., 30, 11-146, (1980) · Zbl 0454.14021
[15] Kalai, G., Rigidity and the lower bound theorem I, Invent. Math., 88, 125-151, (1987) · Zbl 0624.52004
[16] Kallay, M., Indecomposable polyoptes, Israel J. Math., 41, 235-243, (1981)
[17] Lee, C., Some recent results on convex polytopes, Contemp. Math., 114, 3-19, (1990)
[18] Lee, C., P.L.-spheres, convex polytopes, and stress, Discrete Comput. Geom., 15, 389-421, (1996) · Zbl 0856.52009
[19] McMullen, P., The number of faces of simplicial polytopes, Israel J. Math., 9, 559-570, (1971) · Zbl 0209.53701
[20] McMullen, P., On simple polytopes, Invent. Math., 113, 419-444, (1993) · Zbl 0803.52007
[21] McMullen, P., Weights on polytopes, Discrete Comput. Geom., 15, 363-388, (1996) · Zbl 0849.52011
[22] McMullen, P.; Walkup, D. W., A generalized lower-bound conjecture for simplicial polytopes, Mathematika, 18, 264-273, (1971) · Zbl 0233.52003
[23] Munkres, Elements of Algebraic Topology, (1984), Addison-Wesley Reading · Zbl 0673.55001
[24] Oda, T., Simple convex polytopes and the strong Lefschetz theorem, J. Pure Appl. Algebra, 71, 265-286, (1991) · Zbl 0780.52011
[25] Rotman, J. J., An Introduction to Homological Algebra, (1979), Academic Press New York/San Francisco London · Zbl 0441.18018
[26] Stanley, R., The number of faces of a simplicial convex polytope, Adv. Math., 35, 236-238, (1980) · Zbl 0427.52006
[27] Stanley, R., Combinatorics and Commutative Algebra, Progress in Mathematics, 41, (1983), Birkhäuser Boston, p. 21-69
[28] Tay, T.-S., Skeletal rigidity of PL-spheres, Contemp. Math., 197, 387-400, (1996)
[29] Tay, T.-S.; White, N.; Whiteley, W., Skeletal rigidity of simplicial complexes I, European J. Combin., 16, 381-403, (1995) · Zbl 0834.52009
[30] Tay, T.-S.; White, N.; Whiteley, W., Skeletal rigidity of simplicial complexes II, European J. Combin., 16, 503-523, (1995) · Zbl 0834.52010
[31] Tay, T.-S.; Whiteley, W., Generating isostatic frameworks, Structural Topology, 11, 21-69, (1985) · Zbl 0574.51025
[32] White, N.; Whiteley, W., The algebraic geometry of stresses in frameworks, SIAM J. Algebraic Discrete Methods, 4, 481-511, (1983) · Zbl 0542.51022
[33] Whiteley, W., Motions, stresses and projected polyhedra, Structural Topology, 7, 13-38, (1982) · Zbl 0536.51014
[34] Whiteley, W., Cones infinity and 1-story buildings, Structural Topology, 7, 53-70, (1983) · Zbl 0545.51017
[35] Whiteley, W., Infinitesimally rigid polyhedra I: statics of frameworks, Trans. Amer. Math. Soc., 285, 431-465, (1984) · Zbl 0518.52010
[36] Whiteley, W., Rigidity and polarity I: statics of sheet structures, Geom. Dedicata, 329-362, (1987) · Zbl 0618.51006
[37] Whiteley, W., Rigidity and polarity II: weavings lines and plane tensegrity frameworks, Geom. Dedicata, 30, 255-279, (1989) · Zbl 0675.51008
[38] Whiteley, W., Vertex splitting in isostatic frameworks, Structural Topology, 16, 23-30, (1991) · Zbl 0724.52014
[39] W. Whiteley, Parallel Redrawings of Configurations in 3-Space, preprint, Department of Mathematics and Statistics, York University, North York, Ontario.
[40] Whiteley, W., Some matroids from discrete applied geometry, Contemp. Math., 197, 171-312, (1996) · Zbl 0860.05018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.