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] satisfiesH i (X [S]; F) = 0. Leray numbers play a role in formulating and proving topological Hellytype 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 thatH 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. §1. Introduction. Let F be a fixed field and let X be a simplicial complex on the vertex set V . All homology and cohomology appearing in the sequel will be with F coefficients. The induced subcomplex of X on a subset S ⊂ V is X [S] = {σ ∈ X : σ ⊂ S}.