Higher Dimensional Discrete Cheeger Inequalities

TL;DR

This paper extends the Cheeger inequality to higher-dimensional simplicial complexes, providing two different proofs that establish spectral-expansion relationships beyond the well-understood graph case.

Contribution

It proves the Cheeger inequality for arbitrary simplicial complexes, generalizing previous results limited to complexes with complete skeletons, using two distinct proof techniques.

Findings

Established the Cheeger inequality for all finite simplicial complexes.

Provided two different proofs with unique methods and potential for further strengthening.

Extended spectral-expansion relationships from graphs to higher dimensions.

Abstract

For graphs there exists a strong connection between spectral and combinatorial expansion properties. This is expressed, e.g., by the discrete Cheeger inequality, the lower bound of which states that $\lambda(G) \leq h(G)$, where $\lambda(G)$ is the second smallest eigenvalue of the Laplacian of a graph $G$ and $h(G)$ is the Cheeger constant measuring the edge expansion of $G$. We are interested in generalizations of expansion properties to finite simplicial complexes of higher dimension (or uniform hypergraphs). Whereas higher dimensional Laplacians were introduced already in 1945 by Eckmann, the generalization of edge expansion to simplicial complexes is not straightforward. Recently, a topologically motivated notion analogous to edge expansion that is based on $\mathbb{Z}_2$-cohomology was introduced by Gromov and independently by Linial, Meshulam and Wallach. It is known that for this generalization there is no higher dimensional analogue of the lower bound of the Cheeger inequality. A different, combinatorially motivated generalization of the Cheeger constant, denoted by $h(X)$, was studied by Parzanchevski, Rosenthal and Tessler. They showed that indeed $\lambda(X) \leq h(X)$, where $\lambda(X)$ is the smallest non-trivial eigenvalue of the ($(k-1)$-dimensional upper) Laplacian, for the case of $k$-dimensional simplicial complexes $X$ with complete $(k-1)$-skeleton. Whether this inequality also holds for $k$-dimensional complexes with non-complete $(k-1)$-skeleton has been an open question. We give two proofs of the inequality for arbitrary complexes. The proofs differ strongly in the methods and structures employed, and each allows for a different kind of additional strengthening of the original result.