Leray Numbers of Tolerance Complexes

Abstract

AbstractLet 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, $${\mathcal {T}}_{t}(K)$$ 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 $${\mathcal {T}}_t(K)$$ T t ( K ) is h(t, d)-Leray. As an application, we present some new tolerant versions of the colorful Helly theorem.