Extending simplicial complexes: Topological and combinatorial properties

TL;DR

This paper introduces three gluing techniques for hypergraphs that allow constructing larger hypergraphs with controlled topological and combinatorial properties, enabling the creation of broad classes of simplicial complexes with desired features.

Contribution

It presents novel gluing methods to extend simplicial complexes while preserving properties like Cohen-Macaulayness and shellability, offering new tools for topological and combinatorial construction.

Findings

Developed three gluing techniques for hypergraphs

Controlled topological properties in extended complexes

Constructed broad classes of complexes with desired properties

Abstract

Given an arbitrary hypergraph $\mathcal{H}$, we may glue to $\mathcal{H}$ a family of hypergraphs to get a new hypergraph $\mathcal{H}'$ having $\mathcal{H}$ as an induced subhypergraph. In this paper, we introduce three gluing techniques for which the topological and combinatorial properties (such as Cohen-Macaulayness, shellability, vertex-decomposability etc.) of the resulting hypergraph $\mathcal{H}'$ is under control in terms of the glued components. This enables us to construct broad classes of simplicial complexes containing a given simplicial complex as induced subcomplex satisfying nice topological and combinatorial properties. Our results will be accompanied with some interesting open problems.