Higher Whitehead Products in Moment—Angle Complexes and Substitution of Simplicial Complexes

Abstract

We study the question of realisability of iterated higher Whitehead products with a given form of nested brackets by simplicial complexes, using the notion of the moment–angle complex \({{\cal Z}_{\cal K}}\). Namely, we say that a simplicial complex \({\cal K}\) realises an iterated higher Whitehead product w if w is a nontrivial element of \({\pi _*}\left( {{{\cal Z}_{\cal K}}} \right)\). The combinatorial approach to the question of realisability uses the operation of substitution of simplicial complexes: for any iterated higher Whitehead product w we describe a simplicial complex ∂Δ_w that realises w. Furthermore, for a particular form of brackets inside w, we prove that ∂Δ_w is the smallest complex that realises w. We also give a combinatorial criterion for the nontriviality of the product w. In the proof of nontriviality we use the Hurewicz image of w in the cellular chains of \({{\cal Z}_{\cal K}}\) and the description of the cohomology product of \({{\cal Z}_{\cal K}}\). The second approach is algebraic: we use the coalgebraic versions of the Koszul and Taylor complexes for the face coalgebra of \({\cal K}\) to describe the canonical cycles corresponding to iterated higher Whitehead products w. This gives another criterion for realisability of w.