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Working paper

Iterated Higher Whitehead products in topology of moment-angle complexes

math. arxive. Cornell University, 2017. No. 1708.01694.
In this paper we study the topological structure of moment-angle complexes ZK. We consider two classes of simplicial complexes. The first class BΔ consists of simplicial complexes K for which ZK is homotopy equivalent to a wedge spheres. The second class WΔ consists of K∈BΔ such that all spheres in the wedge are realized by iterated higher Whitehead products. Buchstaber and Panov asked if it is true that BΔ=WΔ. 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 WΔ is large enough. Namely, we show that the class WΔ 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.