Leray numbers of tolerance complexes

TL;DR

This paper introduces a new class of complexes called tolerance complexes and proves that they are Leray under certain conditions, extending Helly-type theorems with tolerance parameters.

Contribution

It establishes a bound on Leray numbers of tolerance complexes derived from d-collapsible complexes, linking topological properties with tolerance parameters.

Findings

Existence of a function h(t,d) bounding the Leray number of tolerance complexes.

Tolerance complexes of d-collapsible complexes are h(t,d)-Leray.

New tolerant versions of the colorful Helly theorem are derived.

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. We define the $t$-tolerance complex of $K$, $\mathcal{T}_t(K)$, 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 $\mathcal{T}_t(K)$ is $h(t,d)$-Leray. The definition of the complex $\mathcal{T}_t(K)$ is motivated by results of Montejano and Oliveros on "tolerant" versions of Helly's theorem. As an application, we present some new tolerant versions of the colorful Helly theorem.