Eigenvalue growth of the discrete Hodge Laplacian across dimensions

Philipp Bartmann; Matthias Keller

Eigenvalue growth of the discrete Hodge Laplacian across dimensions

Combinatorics 2026-08-11 v1 Geometric Topology

Authors: Philipp Bartmann , Matthias Keller

Abstract

We prove several bounds on the largest and smallest eigenvalues of the combinatorial Hodge Laplacian $\Delta^H_k$ 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.

Keywords

combinatorial Hodge Laplacian eigenvalue bounds simplicial complex cohomology groups dimensional monotonicity

Cite

@article{arxiv.2608.11170,
  title  = {Eigenvalue growth of the discrete Hodge Laplacian across dimensions},
  author = {Philipp Bartmann and Matthias Keller},
  journal= {arXiv preprint arXiv:2608.11170},
  year   = {2026}
}