We study the question of realisability of iterated higher Whitehead products with a given form of nested brackets by simplicial complexes, using the notion of the moment-angle complex Z_K. Namely, we say that a simplicial complex K realises an iterated higher Whitehead product w if w is a nontrivial element of the homotopy group $\pi_*(Z_K)$.

The combinatorial approach to the question of realisability uses the operation of substitution of simplicial complexes: for any iterated higher Whitehead product w we describe a simplicial complex $\partial\Delta_w$ that realises w. Furthermore, for a particular form of brackets inside w, we prove that $\partial\Delta_w$ is the smallest complex that realises w. We also give a combinatorial criterion for the nontriviality of the product w. In the proof of nontriviality we use the Hurewicz image of w in the cellular chains of Z_K and the description of the cohomology product of Z_K.

The second approach is algebraic: we use the coalgebraic versions of the Koszul and Taylor complex for the face coalgebra of Z_K to describe the canonical cycles corresponding to iterated higher Whitehead products w. This gives another criterion for realisability of w.

The talk is based on a joint work with Semyon Abramyan, arXiv:1901.07918.