### Abstract

We develop an iterated homology theory for simplicial complexes. This
theory is a variation on one due to Kalai. For \Delta a simplicial
complex of dimension d-1, and each r=0,...,d, we define *
r**th* iterated homology groups of \Delta. When r=0, this
corresponds to ordinary homology. If \Delta is a cone over \Delta',
then when r=1, we get the homology of \Delta'. 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.