Séminaire Lotharingien de Combinatoire, 82B.46 (2019), 9 pp.
Joseph Doolittle and Bennet Goeckner
Resolving Stanley's conjecture on k-fold acyclic complexes
Abstract.
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 provide an explicit counterexample to this conjecture. We also prove both a weaker version and a special case of the original conjecture.
Received: November 15, 2018.
Accepted: February 17, 2019.
Final version: April 1, 2019.
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