Revisiting Generalizations of the Dehn--Sommerville Relations

TL;DR

This paper extends and unifies various Dehn--Sommerville relations across different classes of complexes, providing new generalized formulas with uniform proofs based on $ ilde h$-polynomial evaluations.

Contribution

It introduces generalized Dehn--Sommerville relations for reciprocal and balanced complexes, unifying previous results with simple, uniform proofs.

Findings

Derived Dehn--Sommerville relations for reciprocal complexes.

Extended relations to general balanced simplicial complexes.

Provided uniform proofs based on $ ilde h$-polynomial evaluations.

Abstract

We revisit several known versions of the Dehn--Sommerville relations in the context of: homology manifolds, semi-Eulerian complexes, general simplicial complexes, balanced semi-Eulerian complexes and general completely balanced complexes. In addition, we present Dehn--Sommerville relations for reciprocal complexes and general balanced simplicial complexes; which slightly generalize some of the previous results. Our proofs are uniform, and are based on two simple evaluations of the $\tilde h$-polynomial: one that recovers the $\tilde f$-polynomial, and one that counts faces according to certain multiplicities.