The Spectral Gaps of Generalized Flag Complexes and a Geometric Hall-type Theorem. (1st June 2018) - Lew, Alan
Abstract: Let $X$ be a simplicial complex on $n$ vertices without missing faces of dimension larger than $d$ . Let $L_{k}$ denote the $k$ -Laplacian acting on real $k$ -cochains of $X$ and let $μ _{k}(X)$ denote its minimal eigenvalue. We study the connection between the spectral gaps $μ _{k}(X)$ for $k≥ d$ and $μ _{d-1}(X)$ . In particular, we establish the following vanishing result: if $μ _{d-1}(X)>\big(1-\binom{k+1}{d}^{-1}\big)n$, then $\tilde{H}^{j}\left (X;{\mathbb{R}}\right )=0$ for all $d-1≤ j ≤ k$ . As an application we prove a fractional extension of a Hall-type theorem of Holmsen, Martínez-Sandoval, and Montejano for general position sets in matroids.