2017/06/01 by Alan Lew
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Algebra over a field #Cohomology #Combinatorics #Connection (principal bundle) #Dimension (graph theory) #Eigenvalues and eigenvectors #Flag (linear algebra) #Geometry #Graph theory and applications #Mathematics #Matroid #Physics #Position (finance) #Pure mathematics #Quantum mechanics #Topological and Geometric Data Analysis #Type (biology) #math.CO
paper · pdf · doi:10.1093/imrn/rny115
published as International Mathematics Research Notices, rny115 (2018)
arxiv created 2017/06/01 · openalex publication_date 2018/05/12 · arxiv updated 2019/10/16 · openalex created_date 2020/11/23 · openalex updated_date 2026/05/21
Abstract Let X be a simplicial complex on n vertices without missing faces of dimension larger than d. Let Lk denote the k-Laplacian acting on real k-cochains of X and let μ k(X) denote its minimal eigenvalue. We study the connection between the spectral gaps μ k(X) for k≥ d and μ d-1(X). In particular, we establish the following vanishing result: if μ d-1(X)>(1-\binomk+1d-1)n, then Hj (X;ℝ )=0 for all d-1≤ j ≤ k. As an application we prove a fractional extension of a Hall-type theorem of Holmsen, Martínez-Sandoval, and Montejano for general position sets in matroids.