Iterated homology of simplical complexes
type: Item
instance of: scholarly article
title: Iterated homology of simplical complexes
zbMATH Open document ID: 0981.55010
DOI: 10.1023/A:1011216008191
author: Art M. Duval
author: Lauren L. Rose
published in: Journal of Algebraic Combinatorics
publication date: 2002-03-21T00:00:00Z
review text: The authors develop an iterated homology theory which is a variant of one due to Kalai, for simplicial complexes. For a simplicial complex of dimension \(d-1\) and each \(r=0,1,\dots,d\), they define \(r\)th iterated homology groups which coincide with the ordinary homology if \(r=0\). The notion of a shifted complex is central to the development of iterated homology and any simplicial complex can be transformed to a shifted simplicial complex by using a method called algebraic shifting, which preserves Betti numbers and other algebraic properties of the original complex. It may be mentioned that iterated homology is not a topological invariant. For a shellable simplicial complex, the restriction numbers of the shelling are defined and the authors relate these numbers to iterated Betti numbers. Finally, the authors give a characterization of depth of a simplicial complex in terms of its iterated Betti numbers.
reviewed by: Goutam Mukherjee
Mathematics Subject Classification ID: 55U10
Mathematics Subject Classification ID: 13F55
Mathematics Subject Classification ID: 55N99
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 1563246
zbMATH Keywords: shellability
zbMATH Keywords: depth
zbMATH Keywords: shifted complex
zbMATH Keywords: simplicial complex
zbMATH Keywords: algebraic shifting
zbMATH Keywords: Betti numbers
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1563246