On traces of $d$-stresses in the skeletons of lower dimensions of homology $d$-manifolds

TL;DR

This paper explores how $d$-stresses in homology $d$-manifolds induce lower-dimensional stresses, establishing polynomial mappings and analogs of Maxwell's framework, leading to new insights into spatial frameworks and spider web structures.

Contribution

It introduces a polynomial mapping framework for $d$-stresses in homology manifolds and provides a partial Maxwell correspondence for spatial frameworks, with applications to spider web structures.

Findings

Derived polynomial mappings from $d$-stresses to lower dimensions.

Established a partial analog of Maxwell's correspondence in 3D frameworks.

Constructed a class of 3D spider webs similar to 2D Maxwell spider webs.

Abstract

We show how a $d$-stress on a piecewise-linear realization of an oriented (non-simplicial, in general) $d$-manifold in \rd naturally induces stresses of lower dimensions on this manifold, and discuss implications of this construction to the analysis of self-stresses in spatial frameworks. The constructed mappings are not linear, but polynomial. In 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 partial analog of Maxwell correspondence for self-stresses in spatial frameworks and vertical projections of 3-dimensional surfaces based on our construction of polynomial mappings. Applying this theorem we derive a class of three-dimensional spider webs similar to the family of two-dimensional spider webs described by Maxwell. In addition, we conjecture an important property of our mappings which is supported by a heuristic count based on the lower bound theorem ($g_2(d+1)=dim\:Stress_2 \ge 0$) for $d$-pseudomanifolds generically realized in ${\R}^{d+1}$ (Fogelsanger).