The Spectral Gaps of Generalized Flag Complexes and a Geometric Hall-type Theorem. (1st June 2018) - Lew, Alan
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 μ _k(X) denote its minimal eigenvalue. We study the connection between the spectral gaps μ _k(X) for k≥ d and μ _d-1(X) . In particular, we establish the following vanishing result: if μ _d-1(X)>(1-k+1d^-1)n, then H^j(X;R)=0 for all d-1≤ j ≤ 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.