Online Talk

This is a joint work with Taras Panov and Stephen Theriault. We study iterated Whitehead brackets of canonical two-dimensional elements t1,...,tm in the homotopy groups of Davis-Januszkiewicz spaces. By a result of Panov and Ray, rationally there are only the relations of partial commutativity: [ti,ti]=0 for all i and [ti,tj]=0 for i,j adjacent. The integral situation is more complicated: from the Barcus-Barratt identity, we deduce that [ti,[ti,tj]] is a nontrivial 2-torsion element which is equal to the composition of [ti,tj] with the suspended Hopf element.

More generally, we show that any iterated Whitehead bracket of ti's can be obtained from "GPTW elements" by taking Whitehead brackets and compositions with iterated Hopf elements. If the 1-skeleton of the simplicial complex is chordal, we describe the additive structure of the quasi-Lie algebra generated by t1,...,tm. It has nontrivial 2- and 3-torsion, but no p-torsion for p>3.