Relative Leray numbers via spectral sequences

Abstract

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 $i\ge d$ 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 ${\tilde{H}}_i(X[V\setminus \sigma ];\mathbb{F})=0$ for all $i\ge d$ and $\sigma \in Y$. 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.

7 pages

References in corpus (1)

• Leray numbers of projections and a topological Helly type theorem