The Spectral Gaps of Generalized Flag Complexes and a Geometric Hall-type Theorem. (1st June 2018)

• Record Type: Journal Article
• Authors: Lew, Alan
• Is Part Of: International mathematics research notices. Volume 2020:Number 11(2020)
• Page Start: 3364
• Page End: 3395
• Publication Date: 2018-06-01
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rny115
• Languages: English
• ISSNs: 1073-7928

Abstract

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.