On the generalized lower bound conjecture for polytopes and spheres

TL;DR

This paper proves the remaining part of the generalized lower bound conjecture for simplicial polytopes and extends the property to certain simplicial spheres with the weak Lefschetz property.

Contribution

It provides a proof for the second part of the conjecture and generalizes the property to simplicial spheres with the weak Lefschetz property.

Findings

Proof of the remaining part of the conjecture for polytopes.

Extension of the property to simplicial spheres with the weak Lefschetz property.

Advancement in understanding the combinatorial structure of polytopes and spheres.

Abstract

In 1971, McMullen and Walkup posed the following conjecture, which is called the generalized lower bound conjecture: If $P$ is a simplicial $d$-polytope then its $h$-vector $(h_0,h_1,...,h_d)$ satisfies $h_0 \leq h_1 \leq ... \leq h_{\lfloor \frac d 2 \rfloor}$. Moreover, if $h_{r-1}=h_r$ for some $r \leq \frac d 2$ then $P$ can be triangulated without introducing simplices of dimension $\leq d-r$. The first part of the conjecture was solved by Stanley in 1980 using the hard Lefschetz theorem for projective toric varieties. In this paper, we give a proof of the remaining part of the conjecture. In addition, we generalize this property to a certain class of simplicial spheres, namely those admitting the weak Lefschetz property.