Spectral gaps of simplicial complexes without large missing faces

Let $X$ be a simplicial complex on $n$ vertices without missing faces of dimension larger than $d$. Let $L_j$ denote the $j$-Laplacian acting on real $j$-cochains of $X$ and let ${\mu }_j(X)$ denote its minimal eigenvalue. We study the connection between the spectral gaps ${\mu }_k(X)$ for $k\ge d$ and ${\mu }_{d-1}(X)$. In particular, we establish the following vanishing result: If ${\mu }_{d-1}(X)>(1-{\left(\genfrac{}{}{0px}{}{k+1}{d}\right)}^{-1})n$, then ${\tilde{H}}^j(X;\mathbb{R})=0$ for all $d-1\le j\le k$. As an application we prove a fractional extension of a Hall-type theorem of Holmsen, Mart'inez-Sandoval and Montejano for general position sets in matroids.