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 }= 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.' Zbytek anotace chybí kvůli limitu počtu znaků.