Eigenvalue growth of the discrete Hodge Laplacian across dimensions

Abstract

We prove several bounds on the largest and smallest eigenvalues of the combinatorial Hodge Laplacian $Δ^H_k$ of a finite simplicial complex $Σ.$ As a consequence, we obtain new vanishing criteria for cohomology groups $H^k(Σ,\mathbb{R)}$ and confirm a conjecture of O on the dimensional monotonicity of the largest eigenvalue.