Expansion of a simplicial complex

TL;DR

This paper introduces the expansion of a simplicial complex, generalizing graph expansion, and studies how key algebraic and topological properties are preserved or related through this process.

Contribution

It defines the expansion of a simplicial complex and analyzes its effects on properties like vertex decomposability, shellability, Cohen-Macaulayness, and homological invariants.

Findings

Expansion preserves vertex decomposable and shellable properties.

In some cases, Cohen-Macaulayness is preserved under expansion.

Homological invariants of Stanley-Reisner rings relate between a complex and its expansion.

Abstract

For a simplicial complex $\Delta$, we introduce a simplicial complex attached to $\Delta$, called the expansion of $\Delta$, which is a natural generalization of the notion of expansion in graph theory. We are interested in knowing how the properties of a simplicial complex and its Stanley-Reisner ring relate to those of its expansions. It is shown that taking expansion preserves vertex decomposable and shellable properties and in some cases Cohen-Macaulayness. Also it is proved that some homological invariants of Stanley-Reisner ring of a simplicial complex relate to those invariants in the Stanley-Reisner ring of its expansions.