Iterated homology of simplical complexes (Q1840659)

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.