Face numbers of pseudomanifolds with isolated singularities

TL;DR

This paper studies the face numbers of pseudomanifolds with isolated singularities, deriving relations and bounds, and exploring Hilbert functions of face rings, advancing understanding of their combinatorial and algebraic properties.

Contribution

It introduces new Dehn-Sommerville relations and lower bounds for pseudomanifolds with isolated singularities, and characterizes Hilbert functions of face rings in these cases.

Findings

Derived Dehn-Sommerville relations for pseudomanifolds with isolated singularities.

Established lower bound theorems for these complexes.

Provided formulas for Hilbert functions of face rings in specific cases.

Abstract

We investigate the face numbers of simplicial complexes with Buchsbaum vertex links, especially pseudomanifolds with isolated singularities. This includes deriving Dehn-Sommerville relations for pseudomanifolds with isolated singularities and establishing lower bound theorems when the singularities are also homologically isolated. We give formulas for the Hilbert function of a generic Artinian reduction of the face ring when the singularities are homologically isolated and for any pure two-dimensional complex. Some examples of spaces where the $f$-vector can be completely characterized are described. Let $\Delta$ and $\Delta'$ be two simplicial complexes that are homeomorphic and have the same $f$-vector. In \cite{MNS} the question is raised of whether or not the Hilbert functions of generic Artinian reductions of their face rings are identical. We prove that this is the case if the spaces have isolated singularities and are PL-homeomorphic.