Spectral gap bounds for the simplicial Laplacian and an application to random complexes

Abstract: In this article, we derive two spectral gap bounds for the reduced Laplacian of a general simplicial complex. Our two bounds are proven by comparing a simplicial complex in two different ways with a larger complex and with the corresponding clique complex respectively. Both of these bounds generalize the result of Aharoni et al. (2005) \cite{ABM} which is valid only for clique complexes. As an application, we investigate the thresholds for vanishing of cohomology of the neighborhood complex of the Erd\"{o}s-R\'enyi random graph. We improve the upper bound derived in Kahle (2007) \cite{kahle} by a logarithmic factor using our spectral gap bounds and we also improve the lower bound via finer probabilistic estimates than those in Kahle (2007) \cite{kahle}.