2023/10/08 by Ryoshun Oba, Oba, Ryoshun · 1 citation
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2310.05005
openalex publication_date 2023/10/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A (d-1)-dimensional simplicial complex Δ is balanced if its graph G(Δ) is d-colorable. Klee and Novik obtained the balanced lower bound theorem for balanced normal (d-1)-pseudomanifolds Δ with d≥3 by showing that the subgraph of G(Δ) induced by the vertices colored in T is rigid in ℝ3 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 ≥ 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 \bma-balanced, simplicial complexes. Among other results, we show that for d ≥ 4, a balanced homology (d-1)-manifold can be realized as an infinitesimally rigid framework in ℝd such that each vertex of color i lies on the ith coordinate axis.