Higher Whitehead products in moment-angle complexes and substitution of simplicial complexes

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 \(\pi_*(Z_K)\). 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 \(K\) to describe the canonical cycles corresponding to iterated higher Whitehead products \(w\). This gives another criterion for realisability of \(w\).