ИСТИНА

The polyhedral product is defined as a colimit of a diagram in TOP over the face category of a simplicial complex K. There is a parallel construction of topological groups, called the graph product, defined as a colimit in TGRP and depending only on the 1-skeleton of K. The two constructions can be related by the classifying space functor B and the loop functor \Omega. When K is a flag complex, the classifying space of a graph product is the corresponding polyhedral product, and the loops on a polyhedral product is the corresponding graph product. This is not the case when K is not flag; the presence of higher Whitehead brackets obstructs the functors B and \Omega to preserve colimits. They do preserve homotopy colimits though, in respective categories TGRP and TOP, which leads to a homotopy decomposition of loops on a polyhedral product X^K. A closer look on higher Whitehead brackets leads to some interesting observation on the structure of the loops space on polyhedral products X^K and moment-angle complexes Z_K=(D^2,S^1)^K. In particular, we prove that when Z_K is a wedge of spheres, each sphere in the wedge is a lift of an iterated higher Whitehead bracket of the canonical 2-spheres in (CP^\infty)^K. This should be generalisable to arbitrary polyhedral products of the form (X,A)^K. (The latter part is joint with Jelena Grbic.)