Collapsibility of simplicial complexes of hypergraphs

TL;DR

This paper investigates the collapsibility properties of simplicial complexes derived from hypergraphs, establishing bounds based on hypergraph rank and intersection properties, which advances understanding in combinatorial topology.

Contribution

It introduces bounds on collapsibility for complexes formed from hypergraphs with specific intersection and covering properties, extending prior topological combinatorics results.

Findings

Complexes of hypergraphs with bounded covering number are 1-collapsible.

Complexes of pairwise intersecting hypergraphs are 2-collapsible.

Provides new bounds linking hypergraph properties to topological collapsibility.

Abstract

Let $\mathcal{H}$ be a hypergraph of rank $r$. We show that the simplicial complex whose simplices are the hypergraphs $\mathcal{F}\subset\mathcal{H}$ with covering number at most $p$ is $\left(\binom{r+p}{r}-1\right)$-collapsible, and the simplicial complex whose simplices are the pairwise intersecting hypergraphs $\mathcal{F}\subset\mathcal{H}$ is $\frac{1}{2}\binom{2r}{r}$-collapsible.