Lower bounds on the $g$-numbers of spheres without large missing faces

TL;DR

This paper establishes new lower bounds on the $g$-numbers of simplicial spheres without large missing faces, linking combinatorial properties to geometric and topological invariants.

Contribution

It extends bounds on $g$-numbers for these spheres, relates them to graph independence numbers, and proves inequalities among $g$-numbers, especially for flag and 4-spheres.

Findings

Flag $(d-1)$-spheres satisfy $g_2 \\geq (1/2 - \\delta(d))f_0$ with $\\delta(d) o 0$ as $d o \\infty$.

Initial segments of the $g$-vector form a level sequence for certain spheres.

For 4-spheres without missing faces of dimension > 2, $g_2 \\geq \\frac{2}{5}f_0 - \\frac{6}{5}$.

Abstract

We establish several new lower bounds on the $g$-numbers of simplicial spheres without large missing faces. For this class of spheres, we derive bounds on the $g$-numbers in terms of the independence numbers of their graphs, extending a result of Chudnovsky and Nevo. As a consequence, we show that flag $(d-1)$-spheres -- and more generally, flag normal $(d-1)$-pseudomanifolds -- satisfy $g_2\geq (1/2-\delta(d))f_0$, where $\delta(d)$ is a function of $d$ with $\delta(d)\to 0$ as $d\to \infty$. We further prove that, for simplicial $(d-1)$-spheres without large missing faces, an initial segment of the $g$-vector forms a level sequence, yielding additional inequalities among the $g$-numbers. Finally, we show that simplicial $4$-spheres without missing faces of dimension greater than two satisfy $g_2\geq \frac{2}{5}f_0 - \frac{6}{5}$.