On the homological dimension of lattices
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 \(n^{\log_2 n}\).
scientific article; zbMATH DE number 5323318