Bounds for entries of γ-vectors of flag homology spheres

We present some enumerative and structural results for ag homology spheres. For a ag homology sphere δ, we show that its γ-vector γ^δ =(1, γ₁, γ₂, ...) satisfues: γ_j =0 for all j γ₁, γ₂ ≤ (γ¹₂, γ_γ1 ϵ 0, 1, and γ_γ1-1 ϵ 0, 1, 2, γ₁, supporting a conjecture of Nevo and Petersen. Further we characterize the possible structures for δ in extremal cases. As an application, the techniques used produce infunitely many f-vectors of ag balanced simplicial complexes that are not γ-vectors of ag homology spheres (of any dimension); these are the furst examples of this kind. In addition, we prove a ag analog of Perles' 1970 theorem on k-skeleta of polytopes with "few" vertices, specifucally, the number of combinatorial types of κ-skeleta of ag homology spheres with γ₁ ≤ b of any given dimension, is bounded independently of the dimension.