Iterated Higher Whitehead products in topology of moment-angle complexes

In this paper we study the topological structure of moment-angle complexes \(\mathcal{Z_K}\). We consider two classes of simplicial complexes. The first class \(B_{\Delta}\) consists of simplicial complexes \(\mathcal{K}\) for which \(\mathcal{Z_K}\) is homotopy equivalent to a wedge spheres. The second class \(W_{\Delta}\) consists of \(\mathcal{K}\in B_{\Delta}\) such that all spheres in the wedge are realized by iterated higher Whitehead products. Buchstaber and Panov asked if it is true that \(B_{\Delta} = W_{\Delta}\). In this paper we show that this is not the case. Namely, we give an example of a simplicial complex whose corresponding moment-angle complex is homotopy equivalent to a wedge of spheres, but there is a sphere which cannot be realized by any linear combination of iterated higher Whitehead products. On the other hand we show that class \(W_{\Delta}\) is large enough. Namely, we show that the class \(W_{\Delta}\) is closed with respect to two explicitly defined operations on simplicial complexes. Then using these operations we prove that there exists a simplicial complex that realizes any given iterated higher Whitehead product. Also we describe the smallest simplicial complex that realizes an iterated product with only two pairs of nested brackets.