A Cheeger-Type Inequality on Simplicial Complexes

In this paper, we consider a variation on Cheeger numbers related to the coboundary expanders recently defined by Dotterer and Kahle. A Cheeger-type inequality is proved, which is similar to a result on graphs due to Fan Chung. This inequality is then used to study the relationship between coboundary expanders on simplicial complexes and their corresponding eigenvalues, complementing and extending results found by Gundert and Wagner. In particular, we find these coboundary expanders do not satisfy natural Buser or Cheeger inequalities.

💡 Research Summary

This paper investigates higher‑dimensional analogues of the classical Cheeger inequality, focusing on simplicial complexes and the two natural Laplacians that arise from the chain complex (boundary operator) and the co‑chain complex (coboundary operator). The authors first recall the graph case: the Cheeger constant h and the algebraic connectivity λ satisfy 2h ≥ λ ≥ h²/(2 max deg). They then ask whether a similar relationship holds for the “coboundary Cheeger numbers” recently introduced by Dotterer and Kahle.

The paper’s main contributions are twofold.

1. Negative result for the co‑chain complex.
   For an m‑dimensional simplicial complex X (m > 1) let λ_{m‑1} denote the smallest non‑zero eigenvalue of the (m‑1)‑dimensional up‑Laplacian L^{up}{m‑1}=∂{m} δ_{m‑1} (the coboundary Laplacian) and let h_{m‑1} be the (m‑1)‑dimensional coboundary Cheeger number defined over ℤ₂. The authors prove that there are no universal constants p₁, p₂, C such that either
   C·(h_{m‑1})^{p₁} ≥ λ_{m‑1} or λ_{m‑1} ≥ C·(h_{m‑1})^{p₂}·max_{s∈S_{m‑1}} d_s
   holds for all complexes. In other words, the Cheeger inequality fails in both directions for the coboundary Laplacian. This extends earlier probabilistic constructions of Gundert and Wagner (which required torsion in homology) by providing explicit torsion‑free counterexamples. The paper presents two families of complexes: one with h_{m‑1}=0 but λ_{m‑1}>c>0, and another with h_{m‑1}>0 while λ_{m‑1}=0. Thus, neither side of a Cheeger‑type bound can be salvaged in general for the co‑chain setting.

2. Positive result for the chain complex.
   Turning to the boundary Laplacian L_{m}=∂{m+1} δ{m}+δ_{m‑1} ∂{m}, the authors obtain a genuine Cheeger‑type inequality under mild topological assumptions. If X is an orientable pseudomanifold, or more generally a non‑branching constructible simplicial m‑ball (i.e., each (m‑1)‑simplex is incident to at most two m‑simplices and the complex can be built by gluing constructible pieces), then the m‑dimensional boundary Cheeger number h{m} and the spectral gap γ_{m} (the smallest non‑zero eigenvalue of L_{m}) satisfy
   h_{m} ≥ γ_{m} ≥ h_{m}² /