Abstract
We develop an iterated homology theory for simplicial complexes. This theory is a variation on one due to Kalai. For Δ a simplicial complex of dimension i>d − 1, and each i>r e 0, …,i>d, we define i>rth iterated homology groups of Δ. When i>r e 0, this corresponds to ordinary homology. If Δ is a cone over Δ′, then when i>r e 1, we get the homology of Δ′. If a simplicial complex is (nonpure) shellable, then its iterated Betti numbers give the restriction numbers, i>hi>k,j, of the shelling. Iterated Betti numbers are preserved by algebraic shifting, and may be interpreted combinatorially in terms of the algebraically shifted complex in several ways. In addition, the depth of a simplicial complex can be characterized in terms of its iterated Betti numbers.
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