On the homological dimension of lattices
type: Item
instance of: scholarly article
title: On the homological dimension of lattices
zbMATH Open document ID: 1159.06006
author: Roy Meshulam
published in: Order
publication date: 2008-09-09T00:00:00Z
review text: It is shown that if \(L\) is a finite lattice, \(\widehat L = L\setminus\{\hat0,\hat1\}\), and \(\tilde H_{k-2}(\Delta(\widehat L)) \neq 0\) (where \(\Delta(\widehat L)\) is the order complex of \(\widehat L\)), then \(|L| \geq 2^k\), with equality if and only if \(L\) is the Boolean lattice having \(2^k\) elements. As a corollary it is shown that the sum of the reduced Betti numbers of \(\Delta(\widehat L)\), where \(|L|=n\), is at most \(n^{\log_2 n}\), strengthening the result of Edelman and Kahn that the value of the Möbius invariant of \(L\) is bounded by \(n^{\log_2 n}\).
reviewed by: James F. Lawrence
Mathematics Subject Classification ID: 06B05
Mathematics Subject Classification ID: 05D99
Mathematics Subject Classification ID: 05E99
zbMATH DE Number: 5323318
zbMATH Keywords: lattice
zbMATH Keywords: order complex
zbMATH Keywords: homology
MaRDI profile type: Publication
full work available at URL: s11083-008-9086-4
OpenAlex ID: W2092604713
cites work: Q4871782
cites work: Convex hulls of \(f\)- and \(\beta\)-vectors
cites work: Homotopy properties of the poset of nontrivial p-subgroups of a group
cites work: Posets with maximal Möbius function
DOI: 10.1007/S11083-008-9086-4
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description: scientific article; zbMATH DE number 5323318