论文标题
通过光谱序列的相对Leray数字
Relative Leray numbers via spectral sequences
论文作者
论文摘要
令$ \ mathbb {f} $为固定字段，让$ x $成为顶点集合$ v $的简单复合体。 leray number $ l（x; \ mathbb {f}）$是最小$ d $，因此对于所有$ i \ geq d $和$ s \ subset v $，诱导的复杂$ x [s] $满足$ \ tilde {h} _i { Leray数字在制定和证明拓扑型Helly型定理中起作用。对于两个复合物$ x，y $上的同一个顶点集$ v $，定义相对leray number $ l_y（x; \ mathbb {f}）$作为最小$ d $，以便$ \ tilde {h} _i { y $。在本文中，我们将拓扑色彩的Helly定理扩展到相对设置。我们的主要工具是用于通过几何晶格索引的配合物交点的光谱序列。
Let $\mathbb{F}$ be a fixed field and let $X$ be a simplicial complex on the vertex set $V$. The Leray number $L(X;\mathbb{F})$ is the minimal $d$ such that for all $i \geq d$ and $S \subset V$, the induced complex $X[S]$ satisfies $\tilde{H}_i(X[S];\mathbb{F})=0$. Leray numbers play a role in formulating and proving topological Helly type theorems. For two complexes $X,Y$ on the same vertex set $V$, define the relative Leray number $L_Y(X;\mathbb{F})$ as the minimal $d$ such that $\tilde{H}_i(X[V \setminus σ];\mathbb{F})=0$ for all $i \geq d$ and $σ\in Y$. In this paper we extend the topological colorful Helly theorem to the relative setting. Our main tool is a spectral sequence for the intersection of complexes indexed by a geometric lattice.
