RELATIVE LERAY NUMBERS VIA SPECTRAL SEQUENCES

Abstract

Let F be a fixed field and let X be a simplicial complex on the vertex set V. The Leray number L ( X ; F ) is the minimal d such that for all i ⩾ d and S ⊂ V , the induced complex X [ S ] satisfies H ∼ i ( X [ S ] ; 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 ; F ) as the minimal d such that H ∼ i ( X [ V ∖ τ ] ; F ) = 0 for all i ⩾ d and τ ∈ 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.