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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