Leray Numbers of Tolerance Complexes

Abstract

Let K be a simplicial complex on vertex set V. K is called d-Leray if the homology groups of any induced subcomplex of K are trivial in dimensions d and higher. K is called d -collapsible if it can be reduced to the void complex by sequentially removing a simplex of size at most d that is contained in a unique maximal face. Motivated by results of Eckhoff and of Montejano and Oliveros on "tolerant" versions of Helly's theorem, we define the t -tolerance complex of K, T-t (K), as the simplicial complex on vertex set V whose simplices are formed as the union of a simplex in K and a set of size at most t. We prove that for any d and t there exists a positive integer h(t, d) such that, for every d-collapsible complex K, the t-tolerance complex T-t(K) is h(t, d)- Leray. As an application, we present some new tolerant versions of the colorful Helly theorem.