Skeletal rigidity of simplicial complexes. II
[This review comprises both, part I [the authors, ibid., No. 4, 381-403 (1995)] and part II of this paper.] If \(\Delta\) is a simplicial \((d-1)\)-complex, in particular the boundary complex of a simplicial \(d\)-polytope, let \(\Delta^{(r)}\) denote its family of \(r\)-simplices, write \(f_r : = |\Delta^{(r)} |\), and \[ g_r = g_r (\Delta, d) : = \sum^{r - 1}_{j = - 1} (- 1)^{r + j + 1} {d - j \choose d - r - 1} f_j. \] The \(g\)-theorem for simplicial \(d\)-polytopes characterizes all their possible sequences \((f_0, \ldots, f_{d - 1})\) of face numbers; it gets its name from being posed in terms of the \(g\)-numbers. The original proof of its necessity by \textit{R. P. Stanley} [Adv. Math. 35, 236-238 (1980; Zbl 0427.52006)] used deep techniques from algebraic geometry. A proof within convexity (working with the dual simple polytopes) was found by the reviewer [Invent. Math. 113, No. 2, 419-444 (1993; Zbl 0803.52007)]; the present work contributes to a possible approach using concepts of stress and rigidity. A crucial part of the \(g\)-theorem is the inequality \(g_r \geq 0\) for \(r \leq (d + 1)/2\); the case \(r = 2\) is the lower bound theorem. \textit{G. Kalai} [Invent. Math. 88, 125-151 (1987; Zbl 0624.52004)] showed that \(g_2 = f_1 - df_0 + {d + 1 \choose 2} \geq 0\) follows from the infinitesimal rigidity of the edge-skeleton of a simplicial \(d\)-polytope when \(d \geq 3\). \textit{C. W. Lee} [NATO ASI Ser. Ser. C, Math. Phys. Sci. 440, 249-271 (1994; Zbl 0809.52014)] has generalized the 2-stresses used in Kalai's argument to \(r\)-stresses. Now the present authors have introduced definitions of \(r\)-rigidity matrices, three in Part I of the paper, and two more in Part II. These have isomorphic cokernels (the spaces of \(r\)-stresses), but otherwise describe different features of rigidity and related by more purely combinatorial concepts such as homology and the face-ring of \(\Delta\). Part I is largely scene-setting, and is devoted to introducing the basic concepts such as stress, (trivial and non-trivial) motions and static and kinematic rigidity, and further notions such as loads. Part II begins by showing that cones on complexes inherit \(r\)-rigidity. There then follows a discussion of the relationships between the basic concepts just listed. The central result in Part II is that the inequality \(g_r \geq 0\) for polytopal \(\Delta\) is implied by the (as yet only conjectured) \(r\)- rigidity of \(\Delta\); if this holds for all \(r \leq (d + 1)/2\), then the \(g\)-theorem follows. Included in this part are many conjectures about the connexions (as well as details of what is already known about them); these provide pointers to future directions of research.
scientific article; zbMATH DE number 798915
scientific article