耐受复合物的勒雷数

摘要：

设$K$是顶点集$V$上的单纯复形.如果$K$的任何诱导子复形的同调群在维数$d$或更高的维数上是平凡的，则$K$称为$d$ -Leray。$K$称为$d$ -可折叠的，如果它可以通过顺序地移除包含在唯一极大面中的大小至多为$d$的单形而被简化为空复形。定义$K$的$t$ -容差复形$\mathcal{T}_t(K)$为顶点集$V$上的单纯复形，其单纯形由$K$中的一个单纯形与大小至多为$t$的一个集合的并集构成.证明了对任何$d$和$t$，存在正整数$h(t,d)$，使得对任何$d$ -可塌复形$K$，$t$ -容差复形$\mathcal{T}_t(K)$是$h(t,d)$ -Leray.复$\mathcal{T}_t(K)$的定义是由蒙特哈诺和奥利维罗斯关于“容忍”版本的海利定理的结果所激发的。作为应用，我们提出了一些新的宽容版本的彩色Helly定理。

英文摘要：

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.

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