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**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.

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.

Reviewer: Annette A’Campo-Neuen (Konstanz)

### MSC:

52C25 | Rigidity and flexibility of structures (aspects of discrete geometry) |

55U15 | Chain complexes in algebraic topology |

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\textit{T.-S. Tay} and \textit{W. Whiteley}, Adv. Appl. Math. 25, No. 1, 102--151 (2000; Zbl 0968.52019)

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