Abstract
We find strong necessary conditions on the f-vectors, Betti
sequences, and relative Betti sequence of a pair of simplicial
complexes. We also present an example showing that these conditions
are not sufficient. If only the difference between two Betti
sequences is specified, and not the individual Betti sequences, then
the characterization is complete, and the characterization of all
pairs of simplicial complexes matches the characterization of pairs of
near-cones. Our necessary conditions rely upon a combinatorial
decomposition of pairs of simplicial complexes that reflects the
homology and relative homology of the complexes.