Induced equators in flag spheres

Abstract

We propose a combinatorial approach to the following strengthening of Gal's conjecture: $\gamma (\Delta )\ge \gamma (E)$ coefficientwise, where $\Delta$ is a flag homology sphere and $E\subseteq \Delta$ an induced homology sphere of codimension $1$. We provide partial evidence in favor of this approach, and prove a nontrivial nonlinear inequality that follows from the above conjecture, for boundary complexes of flag $d$-polytopes: $h_1(\Delta )h_i(\Delta )\ge (d-i+1)h_{i-1}(\Delta )+(i+1)h_{i+1}(\Delta )$ for all $0\le i\le d$.

12 pages