| Abstract: | We develop an iterated homology theory for simplicial complexes. Thistheory is a variation on one due to Kalai. For  a simplicial complex of dimension d – 1, and each r = 0, ...,d, we define rth iterated homology groups of . When r = 0, this corresponds to ordinary homology. If is a cone over  , then when r = 1, we get the homology of . 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. |
