Geodesic
Iterated homology of simplical complexes
Journal of Algebraic Combinatorics, Tome 12 (2000) no. 3, pp. 279-294.

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Summary: We develop an iterated homology theory for simplicial complexes. Thistheory is a variation on one due to Kalai. For $Delta$ a simplicial complex of dimension $d - 1$, and each $r = 0, \dots , d$, we define rth iterated homology groups of $Delta$. When $r = 0$, this corresponds to ordinary homology. If $Delta$ is a cone over $Deltaprime$, then when $r = 1$, we get the homology of $Deltaprime$. If a simplicial complex is (nonpure) shellable, then its iterated Betti numbers give the restriction numbers, $h _{ k,j }$, 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.
Keywords: shellability, algebraic shifting, depth, Betti numbers, simplicial complex 
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     title = {Iterated homology of simplical complexes},
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Duval, Art M.; Rose, Lauren L. Iterated homology of simplical complexes. Journal of Algebraic Combinatorics, Tome 12 (2000) no. 3, pp. 279-294. http://geodesic.mathdoc.fr/item/JAC_2000__12_3_a1/