Squeezed Complexes

TL;DR

This paper introduces squeezed complexes derived from shifted order ideals, exploring their combinatorial and algebraic properties, including vertex decomposability, Lefschetz properties, and a new invariant called the singularity index.

Contribution

It constructs a new family of simplicial complexes called squeezed complexes, analyzes their properties, and introduces the singularity index as a measure of manifold-likeness.

Findings

Squeezed complexes are vertex decomposable.

They satisfy conditions for the weak and strong Lefschetz properties.

The singularity index decreases to zero as the parameter t increases.

Abstract

Given a shifted order ideal $U$, we associate to it a family of simplicial complexes $(\Delta_t(U))_{t\geq 0}$ that we call squeezed complexes. In a special case, our construction gives squeezed balls that were defined and used by Kalai to show that there are many more simplicial spheres than boundaries of simplicial polytopes. We study combinatorial and algebraic properties of squeezed complexes. In particular, we show that they are vertex decomposable and characterize when they have the weak or the strong Lefschetz property. Moreover, we define a new combinatorial invariant of pure simplicial complexes, called the singularity index, that can be interpreted as a measure of how far a given simplicial complex is from being a manifold. In the case of squeezed complexes $(\Delta_t(U))_{t\geq 0}$, the singularity index turns out to be strictly decreasing until it reaches (and stays) zero if $t$ grows.