The Spectral Gaps of Generalized Flag Complexes and a Geometric Hall-type Theorem
Alan Lew · International Mathematics Research Notices · 2018
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 $\mu _{k}(X)$ denote its minimal eigenvalue. We study the connection between the spectral gaps $\mu _{k}(X)$ for $k\geq d$ and $\mu _{d-1}(X)$. In particular, we establish the following vanishing result: if $\mu _{d-1}(X)>\big(1-\binom{k+1}{d}^{-1}\big)n$, then $\tilde{H}^{j}\left (X;{\mathbb{R}}\right )=0$ for all $d-1\leq j \leq k$. 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.