Dimension Gaps between Representability and Collapsibility

Abstract

A simplicial complex $\mathsf{K}$is called d -representable if it is the nerve of a collection of convex sets in ℝd ; $\mathsf{K}$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 $\mathsf{K}$is d -Leray if every induced subcomplex of $\mathsf{K}$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≥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 construct, for all d≥1, 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 $\mathsf{K}$is d-representable, then $\mathsf{K}$ embeds in ℝd .