Skeletal rigidity of simplicial complexes, I

Abstract

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 g r ⩾ 0, also known as the generalized lower bound theorem, for r ≤ [(d + 1) 2] , where g r is 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.