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Article

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

Proceedings of the Steklov Institute of Mathematics. 2019. Vol. 1. No. 305. P. 1-21.
Abramyan S., Panov T. E.

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