On Traces ofd-stresses in the Skeletons of Lower Dimensions of Piecewise-lineard-manifolds

We show how a d -stress on a piecewise-linear realization of an oriented (non-simplicial, in general)d -manifold in Rdnaturally induces stresses of lower dimensions on this manifold, and discuss implications of this construction to the analysis of self-stresses in spatial frameworks. The mappings we construct are not linear, but polynomial. In the 1860–70s J. C. Maxwell described an interesting relationship between self-stresses in planar frameworks and vertical projections of polyhedral 2-surfaces. We offer a spatial analog of Maxwell’s correspondence based on our polynomial mappings. By applying our main result we derive a class of three-dimensional spider webs similar to the two-dimensional spider webs described by Maxwell. We also conjecture an important property of our mappings that is based on the lower bound theorem (g2(d + 1) = dim Stress2 ≥ 0) for d -pseudomanifolds generically realized in Rd + 1.