Symmetric (co)homology polytopes

TL;DR

This paper generalizes symmetric edge polytopes from graphs to arbitrary simplicial complexes, exploring their geometric and topological properties and providing triangulations using Gr"obner basis techniques.

Contribution

It introduces a new class of centrally symmetric polytopes from simplicial complexes and analyzes their properties, extending previous graph-based results.

Findings

Analysis of the integer decomposition property

Characterization of facets and reflexivity

Triangulation of polytopes using Gr"obner basis techniques

Abstract

Symmetric edge polytopes are a recent and well-studied family of centrally symmetric polytopes arising from graphs. In this paper, we introduce a generalization of this family to arbitrary simplicial complexes. We show how topological properties of a simplicial complex can be translated into geometric properties of such polytopes, and vice versa. We study the integer decomposition property, facets and reflexivity of these polytopes. Using Gr\"obner basis techniques, we obtain a (not necessarily unimodular) triangulation of these polytopes. Due to the tools we use, most of our results hold in the more general setting of arbitrary centrally symmetric polytopes.