Lefschetz Properties and Basic Constructions on Simplicial Spheres

TL;DR

This paper proves that the strong-Lefschetz property, related to the $g$-conjecture for homology spheres, is preserved under key constructions like join, connected sum, and stellar subdivisions, advancing the understanding of simplicial spheres.

Contribution

It demonstrates that the strong-Lefschetz property is maintained through important combinatorial constructions on homology spheres, aiding in the proof of the $g$-conjecture.

Findings

Strong-Lefschetz property preserved under join

Strong-Lefschetz property preserved under connected sum

Strong-Lefschetz property preserved under stellar subdivisions

Abstract

The well known $g$-conjecture for homology spheres follows from the stronger conjecture that the face ring over the reals of a homology sphere, modulo a linear system of parameters, admits the strong-Lefschetz property. We prove that the strong-Lefschetz property is preserved under the following constructions on homology spheres: join, connected sum, and stellar subdivisions. The last construction is a step towards proving the $g$-conjecture for piecewise-linear spheres.