  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 $k\geq d$ and
$\mu_{d-1}(X)$. In particular, we establish the following vanishing result: If
$\mu_{d-1}(X)>(1-\binom{k+1}{d}^{-1})n$, then $\tilde{H}^{j}(X;\mathbb{R})=0$
for all $d-1\leq j \leq k$. 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.
