[Paper Review] Projectivities in Simplicial Complexes and Colorings of Simple Polytopes

Michael Joswig|ArXiv.org|Feb 23, 2001

Advanced Combinatorial Mathematics10 references4 citations

TL;DR

This paper introduces the group of projectivities Π(Δ) for strongly connected simplicial complexes, a combinatorial invariant that captures structural symmetries independent of topology. It applies this framework to characterize simple polytopes with chromatic number γ(P) = d—specifically, those where every 2-face has an even number of vertices—proving s(P) = n − d, which resolves a key case in toric manifold theory and links to edge-coloring duality in d-regular bipartite graphs.

ABSTRACT

For each strongly connected finite-dimensional (pure) simplicial complex we construct a finite group, the group of projectivities of the complex, which is a combinatorial but not a topological invariant. This group is studied for combinatorial manifolds and, in particular, for polytopal simplicial spheres. The results are applied to a coloring problem for simplicial (or, dually, simple) polytopes which arises in the area of toric algebraic varieties.

Motivation & Objective

• To define and study the group of projectivities Π(Δ) as a combinatorial invariant of finite-dimensional, strongly connected simplicial complexes.
• To characterize the chromatic number γ(P) of simple d-polytopes P via combinatorial conditions on their 2-faces.
• To resolve a case of Problem 4.4.1 from Buchstaber and Panov by showing s(P) = n − d when γ(P) = d.
• To establish a connection between proper facet colorings of simple polytopes and proper edge colorings of their vertex-edge graphs.

Proposed method

• Constructs the group of projectivities Π(Δ) from local combinatorial data and the fundamental group of the complex.
• Applies the projectivity group to combinatorial manifolds and polytopal spheres, particularly boundaries of convex polytopes.
• Uses duality between simple polytopes and simplicial spheres to translate results on facet coloring to polytopal structure.
• Relies on the equivalence between even 2-faces and bipartite vertex-edge graphs in simple polytopes.
• Applies results from graph theory, including Vizing’s and König’s theorems on edge coloring in d-regular bipartite graphs.
• Uses constructibility and shellability of polytopal complexes to relate cycle spaces to 2-faces and support the chromatic characterization.

Experimental results

Research questions

• RQ1 When does a simple d-polytope P satisfy γ(P) = d, and what combinatorial condition characterizes such polytopes?
• RQ2 How is the maximal free rank s(P) of a torus action on the Davis-Januszkiewicz manifold Z_P related to the chromatic number γ(P)?
• RQ3 Can the group of projectivities Π(Δ) be computed from local combinatorial data and the fundamental group of a simplicial complex?
• RQ4 What is the relationship between proper facet colorings of a simple polytope and proper edge colorings of its 1-skeleton?

Key findings

• A simple d-polytope satisfies γ(P) = d if and only if every 2-face has an even number of vertices.
• For such polytopes, the maximal free torus rank s(P) equals n − d, where n is the number of facets.
• The group of projectivities Π(Δ) is a finite combinatorial invariant of strongly connected simplicial complexes, independent of topological structure.
• A proper d-coloring of the facets of a simple d-polytope induces a proper d-edge coloring of its 1-skeleton, consistent with König’s theorem on bipartite d-regular graphs.
• The cycle space of the vertex-edge graph of a constructible polytopal complex is generated by the 2-faces, supporting the chromatic and cycle-based characterizations.
• The result confirms that s(P) = n − d is the maximal possible value, and this case is combinatorially characterized by even 2-faces.