Dehn Sommerville Manifolds

TL;DR

Dehn-Sommerville manifolds are a class of simplicial complexes that generalize discrete manifolds, sharing key properties such as symmetries, invariance under refinements, and relationships with Euler characteristic, with implications for their combinatorial and topological structure.

Contribution

This paper introduces and analyzes Dehn-Sommerville manifolds, highlighting their properties, symmetries, invariance under operations, and their relation to classical manifold invariants.

Findings

Dehn-Sommerville manifolds satisfy symmetry properties reducing f-vector complexity.

They are invariant under edge refinement, Barycentric refinement, and Cartesian products.

Odd-dimensional Dehn-Sommerville manifolds are flat and form a monoid under join.

Abstract

Dehn-Sommerville manifolds are a class of finite abstract simplicial complexes that generalize discrete manifolds. Despite a simpler definition in comparison to manifolds, they still share most properties of manifolds. They especially satisfy all Dehn-Sommerville symmetries telling that half of the f-vector entries are redundant. They also share other properties with q-manifolds: for every Dehn-Sommerville q-manifold G and any function g: V(G) to A={0, ..., k} with positive k, the set of x such that g(x) contains A is a Dehn-Sommerville (q-k)-manifold if not empty. We also see that for Dehn-Sommerville q-manifolds, all higher characteristics w_m(G) agree with Euler characteristic that the chromatic number is bounded above by 2q+2 and that odd-dimensional Dehn-Sommerville manifolds are flat and form a monoid under the join operation. In general, Dehn-Sommerville manifolds are invariant under edge refinement, Barycentric refinement and Cartesian products.