On the generalized lower bound conjecture for 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 $h_0\le h_1\le ...\le h_{\lfloor \frac{d}{2}\rfloor }$. Moreover, if $h_{r-1}=h_r$ for some $r\le \frac{d}{2}$ then $P$ can be triangulated without introducing simplices of dimension $\le d-r$. 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.

14 pages, improved presentation