MathDB

The shifted-complex polyhedral product wedge decomposition conjecture

About 19 years old · traced to

Let KK be a shifted simplicial complex, let (CXi,Xi,xi)(CX_i,X_i,x_i) be based pairs indexed by the vertices of KK, and write Z(K;(CX‾,X‾))Z(K;(\underline{CX},\underline{X})) for the associated polyhedral product. For a subset II of the vertices, let KIK_I denote the full subcomplex on II, let X^I\widehat{X}^I denote the smash product of the spaces XiX_i for i∈Ii\in I, and let ∣KI∣∗X^I|K_I|*\widehat{X}^I denote their join. Shifted-complex wedge decomposition conjecture. The spaces

Z(K;(CX‾,X‾))and⋁I∉K∣KI∣∗X^IZ(K;(\underline{CX},\underline{X}))\quad\text{and}\quad\bigvee_{I\notin K}|K_I| *\widehat{X}^I

are of the same homotopy type. This would extend the preceding suspension decomposition to an unsuspended homotopy equivalence for shifted complexes; related unsuspended wedge-of-spheres results are known in special cases, but the stated general assertion is presented as a conjecture here.

References

Primary source

A. Bahri, M. Bendersky, F. R. Cohen and S. Gitler, “The polyhedral product functor: a method of computation for moment-angle complexes, arrangements and related spaces”, arXiv:0711.4689 (2008).

homotopy-theorymoment-angle-complexespolyhedral-products
Posted

Progress summary

Refreshed
Claimed solved

A 2011 paper claims to prove the conjecture for the standard class of pointed connected spaces, so no open case is recorded there.

The conjecture, posed by Bahri, Bendersky, Cohen, and Gitler, asks for an unsuspended wedge decomposition of the polyhedral product for shifted complexes. The cited 2011 work states that this assertion is true, in the setting of pointed connected CW-complexes.

2011 affirmative proof

Iriye and Kishimoto state and prove that, for shifted KK and pointed connected CW-complexes XiX_i, Z(K;(CX‾,X‾))≃⋁I∉K∣KI∣∗X^IZ(K; (\underline{CX},\underline{X})) \simeq \bigvee_{I\notin K}|K_I|*\widehat{X}^{I}. Later work describes the conjecture as affirmatively resolved and records the result for polyhedral products and moment-angle complexes.

Current status (as of September 2026): The conjecture is claimed solved for pointed connected CW-complexes, while the cited theorem does not establish the statement for arbitrary based spaces beyond those hypotheses.

Sources

Solutions 0

No solutions have been posted yet.