Symmetric (co)homology polytopes

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 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.

Figures (6)

• Figure 1: The triangulation $\Delta_{\mathbb{R}\mathbb{P}^2}$ of the real projective plane (left) and its facet-ridge graph $G(\Delta_{\mathbb{R}\mathbb{P}^2})$ (right).
• Figure 2: The triangulation $\mathcal{MS}$ of the Möbius strip.
• Figure 3: A Moore space for $\mathbb{Z}_3$.
• Figure 4: Centrally symmetric lattice polytopes defined as the convex hull of the matrices $[A | -A]$, where $M$ is the incidence matrix of the cycle graphs $C_3, C_4$ and the complete graph $K_4$ respectively. In other words, these polytopes are the symmetric edge polytopes of these graphs.
• Figure 5: A triangulation of a projection of $\mathcal{P}^{\Delta}$, where $\Delta$ is the boundary of a tetrahedron, and the corresponding coboundary map.

Theorems & Definitions (94)

• Example 2.1: A triangulation of the real projective plane
• Remark 2.2
• Example 2.3: From torsion to homology cycles: Björner's example
• Theorem 2.4
• Example 2.5: A non-totally unimodular boundary map
• Theorem 2.6
• Example 2.7: Moore spaces and torsion
• Example 2.8
• Proposition 2.9
• Example 2.10: Totally unimodular and non-totally unimodular representations of regular matroids