  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}.
