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arXiv · 1811.08518

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

Joseph Doolittle·Bennet Goeckner
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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 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.

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Joseph Doolittle, Bennet Goeckner. 2018-11-20. Resolving Stanley's conjecture on $k$-fold acyclic complexes. https://arxiv.org/abs/1811.08518

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