Higher dimensional discrete Cheeger inequalities

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 direct 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-com \-plete $(k-1)$ -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.