Delooping of the loop-space decomposition for shifted and flag–chordal complexes

Establish that for any simplicial complex K that is either shifted or a flag complex with chordal 1-skeleton, and for any maps f_i: X_i → A_i with each X_i contractible and each A_i simply connected, the loop-space decomposition Ω f^K_co ≃ ∏_{b ∈ B_{[m]}, I_b ∉ K} Ω Map_*(Σ |K_{I_b}|, Σ Ω A_1^{∧ l_1(b)} ∧ ··· ∧ Ω A_m^{∧ l_m(b)}) lifts to a space-level product decomposition; that is, prove a homotopy equivalence f^K_co ≃ ∏_{b ∈ B_{[m]}, I_b ∉ K} Map_*(Σ |K_{I_b}|, Σ Ω A_1^{∧ l_1(b)} ∧ ··· ∧ Ω A_m^{∧ l_m(b)}).

Background

Theorem 4.2 gives a general loop-space decomposition for polyhedral coproducts when the domain spaces X_i are contractible, expressing Ω f^K_co as a product of looped mapping spaces indexed by missing faces of K via Hall basis data.

For polyhedral products, analogous suspension splittings are known to desuspend (i.e., admit space-level decompositions) for certain classes of complexes, including shifted complexes and flag complexes with chordal 1-skeleton. Motivated by these precedents, the authors conjecture that the dual loop-space decomposition for polyhedral coproducts deloops for the same classes of K.