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

Joseph Doolittle,Bennet Goeckner

doi:10.5070/c61055379

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.Mathematics Subject Classifications: 05E45, 55U10

Highlights

The interplay between combinatorial and topological properties of simplicial complexes has been a subject of great interest for researchers for many decades
One beautiful result due to Stanley connects the homology of the geometric realization of a complex to a well-behaved decomposition of its face poset
The face poset of ∆ can be written as the disjoint union of rank 1 boolean intervals such that the minimal faces of these intervals together form a subcomplex of ∆

Summary

Introduction

The interplay between combinatorial and topological properties of simplicial complexes has been a subject of great interest for researchers for many decades (see, e.g., [1, 13, 16]). The face poset of ∆ can be written as the disjoint union of rank 1 boolean intervals such that the minimal faces of these intervals together form a subcomplex of ∆ This theorem was generalized by Stanley [15, Proposition 2.1] and Duval [4, Theorem 1.1]. Using results from Kalai’s algebraic shifting [12], Stanley [15, Proposition 2.3] further showed that the f -polynomial of a k-fold acyclic complex can be written as f (∆, t) = (1 + t)kf (Γ, t). This result gives a witness for the behavior observed in (1.1).

Preliminaries

Construction

Boolean Trees

Open Questions