Gorenstein rings through face rings of manifolds

TL;DR

This paper investigates the algebraic structure of face rings of homology manifolds, establishing Gorenstein properties and linking them to the sphere g-conjecture and its generalizations, with implications for combinatorial enumeration.

Contribution

It introduces new algebraic results on face rings of manifolds, proving Gorenstein properties and connecting them to the manifold g-conjecture and its consequences.

Findings

Computed the socle of face rings of homology manifolds

Verified a quotient of the face ring is Gorenstein

Established a case of Kalai's manifold g-conjecture for certain manifolds

Abstract

The face ring of a homology manifold (without boundary) modulo a generic system of parameters is studied. Its socle is computed and it is verified that a particular quotient of this ring is Gorenstein. This fact is used to prove that the sphere $g$-conjecture implies all enumerative consequences of its far reaching generalization (due to Kalai) to manifolds. A special case of Kalai's manifold $g$-conjecture is established for homology manifolds that have a codimension-two face whose link contains many vertices.