Leray Numbers of Tolerance Complexes

Abstract

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 , $${\mathcal {T}}_{t}(K)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , 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 $${\mathcal {T}}_t(K)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is h ( t , d )-Leray. As an application, we present some new tolerant versions of the colorful Helly theorem.

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