Revisiting Generalizations of the Dehn-Sommerville Relations

In this survey-like article, 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 ̃h-poly-
nomial: one that recovers the ̃f -polynomial, and one that counts faces according to certain multiplicities.