Nonpolytopal nonsimplicial lattice spheres with nonnegative toric g-vector

TL;DR

This paper constructs numerous nonpolytopal, nonsimplicial lattice spheres with nonnegative toric g-vectors, providing evidence for Stanley's conjecture by using Bier spheres over multiplex face posets.

Contribution

It introduces a new class of nonpolytopal, nonsimplicial Gorenstein* meet semi-lattices with nonnegative toric g-vectors, expanding the understanding of lattice spheres.

Findings

Constructed many examples of nonpolytopal nonsimplicial lattice spheres.

Confirmed nonnegativity of the toric g-vector in these examples.

Supported Stanley's conjecture through these constructions.

Abstract

We construct many nonpolytopal nonsimplicial Gorenstein* meet semi-lattices with nonnegative toric g-vector, supporting a conjecture of Stanley. These are formed as Bier spheres over the face posets of multiplexes, polytopes constructed by Bisztriczky as generalizations of simplices.