Bounds for Entries of $\gamma$-Vectors of Flag Homology Spheres

Abstract

We present some enumerative and structural results for flag homology spheres. For a flag homology sphere $\Delta$, we show that its $\gamma$-vector $\gamma^\Delta{\,=\,}(1,\gamma_1,\gamma_2,\ldots)$ satisfies: $\gamma_j{\,=\,}0$ for all $j>\gamma_1$, $\gamma_2\leq\binom{\gamma_1}{2}$, $\gamma_{\gamma_1}\in\{0,1\}$, and $\gamma_{\gamma_1-1}\in\{0,1,2,\gamma_1\}$, supporting a conjecture of Nevo and Petersen. Further we characterize the possible structures for $\Delta$ in extremal cases. As an application, the techniques used produce infinitely many $f$-vectors of flag balanced simplicial complexes that are not $\gamma$-vectors of flag homology spheres (of any dimension); these are the first examples of this kind. In addition, we prove a flag analog of Perles' 1970 theorem on $k$-skeleta of polytopes with “few” vertices, specifically, the number of combinatorial types of $k$-skeleta of flag homology spheres with $\gamma_1\leq b$ of any given dimension, is bounded independently of the dimension.