Revisiting Generalizations of the Dehn--Sommerville Relations

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.

20 pages, two figures. Comments are very welcome. arXiv admin note: text overlap with arXiv:2103.04769

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