On traces of \(d\)-stresses in the skeletons of lower dimensions of piecewise-linear \(d\)-manifolds (Q5946639)

This interesting paper deals with \(d\)-stresses on piecewise-linear realizations of \(d\)-manifolds (or homology manifolds) in Euclidean space. These can be interpreted in terms of rectilinear realizations of the combinatorial dual graph (so-called reciprocals). As one of the main results, a polynomial mapping \(p_k\) of degree \(d-k-1\) from the \(d\)-stresses to the \(k\)-stresses is constructed. Furthermore, a generalization of Maxwell's theorem on stresses incluced by projections of polytopes is obtained which is based on these polynomial mappings. A number of conjectures are discussed. One of them states that the mappings \(p_k\) are injective for any realization in general position. In the paper various connections to the theory of rigidity and the theory of convex polytopes are explained.