On the gap between representability and collapsibility

A simplicial complex K is called d-representable if it is the nerve of a collection of convex sets in R; 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 ≥ 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 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 K is d-representable, then K embeds in R.