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 $Z_K$. Namely, we say that a simplicial complex $K$ realises an iterated higher Whitehead product $w$ if $w$ is a nontrivial element of $\pi_*(Z_K)$. 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 $\partial\Delta_w$ that realises $w$. Furthermore, for a particular form of brackets inside $w$, we prove that $\partial\Delta_w$ 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 $Z_K$ and the description of the cohomology product of $Z_K$. The second approach is algebraic: we use the coalgebraic versions of the Koszul and Taylor complex for the face coalgebra of $K$ to describe the canonical cycles corresponding to iterated higher Whitehead products $w$. This gives another criterion for realisability of $w$.