Skeletal rigidity of simplicial complexes, I
Skeletal rigidity of simplicial complexes, I
复制标题
单纯复合体的骨骼刚度，I
DOI：
10.1016/0195-6698(95)90019-5
复制
发表时间：
1995
期刊：
影响因子：
--
通讯作者：
W. Whiteley
中科院分区：
文献类型：
--
作者：
T. Tay;N. White;W. Whiteley
这是一个两部分的论文的第一部分，第二部分将出现在本杂志的后期问题。无穷小刚性的概念涉及在d维欧氏空间中实现的图（或1维单纯复形，我们将其视为棒-关节框架）。我们将这种符号推广到高维单纯复形的r-刚性，再次实现在d维空间。粗略地说，r-刚性意味着没有非平凡的r-运动，而r-运动相当于给每个E（r-2）维单形分配一个速度向量，使得（r-1）-单形的所有（r-1）维体积瞬时保持不变。在本文的这一部分中，我们给出了三个不同但等价的r-运动和r-应力相关概念的基本公式，在第二部分中给出了另外两个公式。在一种特殊情况下，我们也给出了这些概念的同调解释。这种同调解释可以推广到一般情况，这将在以后的论文中完成。本文的动机是希望了解g-定理的组合学，g-定理是单纯多面体的所有可能的f-向量的表征。g-定理的关键部分是不等式gr 0，也称为广义下界定理，其中r ≤ [（d + 1）2]，其中gr是单纯d-多胞形的边界复形Δ的g-向量中的第r个元素。我们证明了这个不等式和完全g-定理是由所有多面体Δ的r-刚性（r ≤ [（d + 1）2]）所隐含的。这个多面体r-刚性是被证明的，但没有被证明。这与g-定理的连接的细节载于第二部分的文件。
This is the first part of a two-part paper, with the second part to appear in a later issue of this journal. The concept of infinitesimal rigidity concerns a graph (or a 1-dimensional simplicial complex, which we regard as a bar-and-joint framework) realized in d-dimensional euclidean space. We generalize this notation to r-rigidity of higher-dimensional simplicial complexes, again realized in d-dimensional space. Roughly speaking, r-rigidity means lack of non-trivial r-motion, and an r-motion amounts to assigning a velocity vector to each E (r − 2)-dimensional simplex in such a way that all (r − 1)-dimensional volumes of (r − 1)-simplices are instantaneously preserved. We give three different, but equivalent, elementary formulations of r-motions and the related idea of r-stresses in this part of the paper, and two additional ones in Part II. We also give a homological interpretation of these concepts, in a special case. This homological interpretation can be extended to the general case, which will be done in a later paper. The motivation for this paper is the desire to understand the combinatorics of the g-theorem, which is the characterization of all the possible f-vectors of simplicial polytopes. The crucial part of the g-theorem is the inequality gr⩾ 0, also known as the generalized lower bound theorem, for r ≤ [(d + 1) 2] , where gris the rth entry in the g-vector of the boundary complex Δ of a simplicial d-polytope. We show that this inequality and, indeed, the full g-theorem, is implied by the r-rigidity of all polytopal Δ for r ≤ [(d + 1) 2] . This polytopal r-rigidity is conjectured, but not proven. The details of this connection with the g-theorem are contained in the second part of the paper.