Iterated Whitehead products in the homotopy groups of polyhedral products

Abstract

We study structure within the homotopy groups of the Davis-Januszkiewicz space DJ(K) associated with a simplicial complex K. The inclusion of each vertex in K induces a map from the two-sphere into DJ(K). These maps generate a quasi-Lie subalgebra QL(K) via the Whitehead product and a Pi-subalgebra S(K) via the Whitehead product and composition. We describe the quasi-Lie subalgebra QL(K), and show that the Pi-subalgebra S(K) coincides with the whole of the homotopy groups of DJ(K) if and only if K is a flag complex. Extensions to more general polyhedral products are also considered.