Simplicial vs. cubical spheres, polyhedral products and the Nevo-Petersen conjecture

TL;DR

This paper classifies certain polytopes related to Murai and Bier spheres, proves flag properties, and confirms the Nevo-Petersen conjecture for specific classes of flag homology spheres, advancing understanding of polyhedral and combinatorial structures.

Contribution

It provides a classification of polytopes associated with Murai and Bier spheres and verifies the Nevo-Petersen conjecture for flag Murai spheres.

Findings

Flag Murai spheres are exactly nerve complexes of flag nestohedra.

Flag Murai spheres satisfy the Nevo-Petersen conjecture on γ-vectors.

A Bier sphere is minimally non-Golod iff it is a nerve complex of a truncation polytope.

Abstract

We prove that a Murai sphere is flag if and only if it is a nerve complex of a flag nestohedron and classify all the polytopes arising in this way. Our classification implies that flag Murai spheres satisfy the Nevo-Petersen conjecture on $\gamma$-vectors of flag homology spheres. We continue by showing that a Bier sphere is minimally non-Golod if and only if it is a nerve complex of a truncation polytope different from a simplex and classify all the polytopes arising in this way. Finally, the notion of a cubical Bier sphere is introduced based on the polyhedral product construction, and we study combinatorial and geometrical properties of these cubical complexes.