# Spectral gaps of simplicial complexes without large missing faces

## TL;DR

This paper investigates the spectral properties of simplicial complexes without large missing faces, establishing conditions under which certain cohomology groups vanish and applying these results to a fractional Hall-type theorem in matroids.

## Contribution

It introduces new spectral gap bounds for simplicial complexes without large missing faces and connects these bounds to topological and combinatorial properties.

## Key findings

- Spectral gap conditions imply vanishing of cohomology groups.
- Established a fractional Hall-type theorem for general position sets in matroids.
- Provided bounds relating eigenvalues of Laplacians to topological features.

## Abstract

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.