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Resolving Stanley's conjecture on $k$-fold acyclic complexes

  • Author(s): Doolittle, Joseph;
  • Goeckner, Bennet
  • et al.

Published Web Location Commons 'BY' version 4.0 license

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 a version of the conjecture for boolean trees and show that the original conjecture holds when this notion of acyclicity is as high as possible.

Mathematics Subject Classifications: 05E45, 55U10

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