Isoperimetric Inequalities in Simplicial Complexes

TL;DR

This paper introduces a notion of combinatorial expansion for high-dimensional simplicial complexes, establishing spectral connections similar to graph theory, including Cheeger inequalities and expander mixing lemmas, with applications to random complexes.

Contribution

It extends spectral graph theory concepts to high-dimensional simplicial complexes, providing new inequalities and links to geometric overlap and random complex models.

Findings

Established a Cheeger-type inequality for simplicial complexes

Proved a high-dimensional Expander Mixing Lemma

Connected spectral properties to geometric overlap in complexes

Abstract

In graph theory there are intimate connections between the expansion properties of a graph and the spectrum of its Laplacian. In this paper we define a notion of combinatorial expansion for simplicial complexes of general dimension, and prove that similar connections exist between the combinatorial expansion of a complex, and the spectrum of the high dimensional Laplacian defined by Eckmann. In particular, we present a Cheeger-type inequality, and a high-dimensional Expander Mixing Lemma. As a corollary, using the work of Pach, we obtain a connection between spectral properties of complexes and Gromov's notion of geometric overlap. Using the work of Gunder and Wagner, we give an estimate for the combinatorial expansion and geometric overlap of random Linial-Meshulam complexes.