Decomposing k-fold acyclic complexes

In 1993 Stanley showed that if a simplicial complex is acyclic over some field, then its face poset can be decomposed into disjoint rank-\$1\$ boolean intervals whose minimal faces together form a subcomplex. Stanley further conjectured that complexes which possess a higher notion of acyclicity could be decomposed in a similar way using boolean intervals of higher rank. The focus of this talk is an explicit counterexample to this conjecture. Time permitting, we will prove both a weaker version and a special case of the original conjecture.