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 ${\Delta }_k^H$ of a finite simplicial complex $\Sigma .$ As a consequence, we obtain new vanishing criteria for cohomology groups $H^k(\Sigma ,\mathbb{R})$ and confirm a conjecture of O on the dimensional monotonicity of the largest eigenvalue.

generalized Corollary 4.5 and Corollary 6.2, added Lemma 4.6 to relate our results to those of Lew, corrected some typos