Rigidity of Balanced Minimal Cycle Complexes

TL;DR

This paper extends the rigidity and lower bound theorems for balanced minimal cycle complexes to higher dimensions, demonstrating their rigidity properties and exploring infinitesimal rigidity in balanced simplicial complexes.

Contribution

It generalizes the balanced lower bound theorem and rigidity results to balanced minimal cycle complexes for all dimensions d ≥ 3.

Findings

Rigidity of subgraphs induced by three colors in balanced minimal cycle complexes for d ≥ 3.

Balanced homology (d-1)-manifolds can be realized as infinitesimally rigid frameworks in ℝ^d with vertex placement on coordinate axes.

Extension of rigidity results to non-generic realizations of balanced and a-balanced simplicial complexes.

Abstract

A $(d-1)$-dimensional simplicial complex $\Delta$ is balanced if its graph $G(\Delta)$ is $d$-colorable. Klee and Novik obtained the balanced lower bound theorem for balanced normal $(d-1)$-pseudomanifolds $\Delta$ with $d\geq3$ by showing that the subgraph of $G(\Delta)$ induced by the vertices colored in $T$ is rigid in $\mathbb{R}^3$ for any $3$ colors $T$. We show that the same rigidity result, and thus the balanced lower bound theorem, holds for balanced minimal $(d-1)$-cycle complexes with $d \geq 3$. Motivated by the Stanley's work on a colored system of parameters for the Stanley-Reisner ring of balanced simplicial complexes, we further investigate the infinitesimal rigidity of non-generic realization of balanced, and more broadly $\bm{a}$-balanced, simplicial complexes. Among other results, we show that for $d \geq 4$, a balanced homology $(d-1)$-manifold can be realized as an infinitesimally rigid framework in $\mathbb{R}^d$ such that each vertex of color $i$ lies on the $i$th coordinate axis.