A classification of the face numbers of Buchsbaum simplicial posets

TL;DR

This paper characterizes the face number vectors of Buchsbaum simplicial posets by proving that the known inequalities are both necessary and sufficient for given Betti numbers.

Contribution

It provides a complete characterization of the face vectors of Buchsbaum simplicial posets with specified Betti numbers, extending previous inequalities to sufficiency.

Findings

Necessary inequalities are also sufficient for face vectors.

Complete characterization of face numbers for Buchsbaum simplicial posets.

Extension of known results to prescribed Betti numbers.

Abstract

The family of Buchsbaum simplicial posets generalizes the family of simplicial cell manifolds. The $h'$-vector of a simplicial complex or simplicial poset encodes the combinatorial and topological data of its face numbers and the reduced Betti numbers of its geometric realization. Novik and Swartz showed that the $h'$-vector of a Buchsbaum simplicial poset satisfies certain simple inequalities; in this paper we show that these necessary conditions are in fact sufficient to characterize the $h'$-vectors of Buchsbaum simplicial posets with prescribed Betti numbers.