Iterated Homology and Decompositions of Simplicial Complexes



Kalai has conjectured that a simplicial complex can be partitioned into Boolean algebras at least as roughly as a shifting-preserving collapse sequence of its algebraically shif\-ted complex. In particular, then, a simplicial complex could (conjecturally) be partitioned into Boolean intervals whose sizes are indexed by its iterated Betti numbers, a generalization of ordinary homology Betti numbers. This would imply a long-standing conjecture made (separately) by Garsia and Stanley concerning partitions of Cohen-Macaulay complexes into Boolean intervals. We prove a relaxation of Kalai's conjecture, showing that a simplicial complex can be partitioned into recursively defined spanning trees of Boolean intervals indexed by its iterated Betti numbers.