Higher Whitehead products in moment-angle complexes and substitution of simplicial complexes (Q2285391)

With a simplicial complex \(K\) on the set \([m]=\{1,\ldots, m\}\), a moment-angle complex \(\mathcal Z_K\) can be associated as a cellular subcomplex of the product space \((D^2)^m\). An interesting but extremely difficult problem is to understand the homotopy type of \(\mathcal Z_K\) or its homotopy groups \(\pi_*(\mathcal Z_K)\). Many topological and geometrical properties of moment-angle complexes are often encoded in nontrivial combinatorial features of simplicial complexes \(K\). In this paper, the authors study the question of realisability of iterated higher Whitehead products in \(\pi_*(\mathcal Z_K)\) in two ways: combinatorially by introducing certain substitutions of simplicial complexes and algebraically by looking at the coalgebraic versions of the Koszul and Taylor complexes of the face coalgebra of \(K\), which is a dual of the Stanley-Reisner algebra or the face ring of \(K\). Both approaches produce criteria (combinatorial and algebraic) for realisability of an iterated higher Whitehead product as a nontrivial element of \(\pi_*(\mathcal Z_K)\).