Adams–Hilton Models and Higher Whitehead Brackets for Polyhedral Products

We construct Adams–Hilton models for the polyhedral products of spheres $(\underline {S})^{\mathcal K}$ and Davis–Januszkiewicz spaces $(\mathbb C\mathrm P^\infty )^{\mathcal K}$. We show that in these cases the Adams–Hilton model can be chosen so that it coincides with the cobar construction of the homology coalgebra. We apply the resulting models to the study of iterated higher Whitehead products in $(\mathbb C\mathrm P^\infty )^{\mathcal K}$. Namely, we explicitly construct a chain in the cobar construction that represents the homology class of the Hurewicz image of a Whitehead product.