Iterated higher Whitehead products and Adams–Hilton models for polyhedral products

In this talk I discuss higher Whitehead products, invariants in unstable homotopy theory, which are considered in the context of the studying Davis—Januszkiewicz spaces and moment-angle complexes.
It is known that rational homotopy groups of loop space form the homotopy Lie algebra in which the Jacobi identity holds. There is a structure of $L_{\infty}$ algebra, the generalization of Lie algebra for which we have n-ary brackets that satisfy the generalized Jacobi identities. In general, we do not know, what relations hold for (canonical) higher Whitehead products.
In this talk I introduce an algebraic construction that gives us chains in the cobar–construction of the homology of Davis—Januszkiewicz space representing Hurewicz images of higher iterated Whitehead products. For this purpose we exhibit Adams–Hilton models for Davis—Januszkiewicz spaces and polyhedral product of spheres (for arbitrary simplicial complexes). Using these chains one can obtain relations on (canonical) higher Whitehead products.
There is a minimal simplicial complex $\mathcal{K} = \partial \Delta(\partial \Delta(1,2,3), 4,5)$, for which iterated Whitehead products $[[\mu_1, \mu_2, \mu_3], \mu_4, \mu_5]$ is defined. In this talk I represent the full description of Pontryagin algebra and homotopy Lie algebra of Davis—Januszkiewicz space for this $\mathcal{K}$, using Whitehead products. We will see that relations on Whitehead products have the form of L-infinity identities.