RELATIVE LERAY NUMBERS VIA SPECTRAL SEQUENCES. Issue 3 (26th June 2021)

• Record Type: Journal Article
• Authors: Kalai, Gil; Meshulam, Roy
• Is Part Of: Mathematika. Volume 67:Issue 3(2021)
• Page Start: 730
• Page End: 737
• Publication Date: 2021-06-26
• Subjects: 52A35 -- 55U10; Mathematics -- Periodicals; 510.5
• DOI: 10.1112/mtk.12103
• Languages: English
• ISSNs: 0025-5793

Abstract

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.