On the gap between representability and collapsibility

TL;DR

This paper explores the differences between three geometric and topological notions of simplicial complexes—representability, collapsibility, and Leray property—by constructing examples that highlight their gaps and implications.

Contribution

It constructs explicit complexes demonstrating the strict inclusions among d-representable, d-collapsible, and d-Leray complexes for all positive integers d.

Findings

Constructed a 2d-Leray complex not (3d-1)-collapsible.

Constructed a d-collapsible complex not (2d-2)-representable.

Proved the nerve of sets of size at most d is d-collapsible.

Abstract

A simplicial complex K is called d-representable if it is the nerve of a collection of convex sets in R^d; K is d-collapsible if it can be reduced to an empty complex by repeatedly removing a face of dimension at most d-1 that is contained in a unique maximal face; and K is d-Leray if every induced subcomplex of K has vanishing homology of dimension d and larger. It is known that d-representable implies d-collapsible implies d-Leray, and no two of these notions coincide for d greater or equal to 2. The famous Helly theorem and other important results in discrete geometry can be regarded as results about d-representable complexes, and in many of these results "d-representable" in the assumption can be replaced by "d-collapsible" or even "d-Leray". We investigate "dimension gaps" among these notions, and we construct, for all positive integers d, a 2d-Leray complex that is not (3d-1)-collapsible and a d-collapsible complex that is not (2d-2)-representable. In the proofs we obtain two results of independent interest: (i) The nerve of every finite family of sets, each of size at most d, is d-collapsible. (ii) If the nerve of a simplicial complex K is d-representable, then K embeds in R^d.