On Stanley's Conjecture on k-fold Acyclic Complexes
Joseph Doolittle, Bennet Goeckner · arXiv (Cornell University) · 2018
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 with a higher notion of acyclicity could be decomposed in a similar way using boolean intervals of higher rank. We prove a weaker version of this conjecture using algebraic shifting and iterated homology. We also provide an explicit counterexample to the original conjecture. This example is constructed from a relative example using similar techniques as those in [Duval, Goeckner, Klivans, and Martin 2016] and [Juhnke-Kubitzke and Venturello 2017].