Symmetric (co)homology polytopes

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 Gröbner 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.

💡 Research Summary

The paper introduces a new class of centrally symmetric lattice polytopes, called symmetric (co)homology polytopes, which generalize the well‑studied symmetric edge polytopes that arise from graphs. The construction starts with an arbitrary simplicial complex Δ on a vertex set V. By assigning to each vertex a pair of opposite unit vectors ±e_i in ℝ^|V| and taking the convex hull of all vectors of the form ±e_i ± e_j for every edge {i,j} belonging to Δ (and more generally for every simplex of Δ), the authors obtain a centrally symmetric polytope P(Δ). This definition reduces to the classical symmetric edge polytope when Δ is the clique complex of a graph, but it works for any finite simplicial complex, thus providing a bridge between combinatorial topology and the geometry of lattice polytopes.

The first major contribution is a topological‑geometric dictionary. The authors prove that several homological invariants of Δ control fundamental geometric properties of P(Δ). In particular, if Δ is Cohen–Macaulay (or satisfies weaker homological conditions such as the vanishing of certain reduced homology groups), then P(Δ) enjoys the integer decomposition property (IDP): every lattice point in the k‑th dilate k·P(Δ) can be expressed as a sum of k lattice points from P(Δ). The proof proceeds by interpreting lattice points of P(Δ) as formal integer combinations of the vertices of Δ and then using the chain complex of Δ to split these combinations. Conversely, the authors show that failure of IDP forces the existence of non‑trivial homology in Δ, establishing a two‑way correspondence.

Next, the paper gives a complete facet description. By examining the minimal non‑faces of Δ, the authors write down explicit supporting hyperplanes for each facet of P(Δ). Each facet is defined by a 0‑1 inequality of the form Σ_{i∈S} x_i – Σ_{j∈T} x_j ≤ 1, where S∪T corresponds to a minimal non‑face. This description allows them to identify when P(Δ) is reflexive: the polar polytope P(Δ)° is again a lattice polytope. They prove that reflexivity holds for a broad family of complexes, including all pure flag complexes and, in particular, the clique complexes of chordal graphs. The reflexivity result recovers known facts about symmetric edge polytopes of bipartite graphs and extends them to higher‑dimensional complexes.

A substantial technical portion is devoted to Gröbner basis methods and triangulations. The authors encode the vertex coordinates of P(Δ) into a toric ideal I_Δ ⊂ k