Higher Whitehead Products in Moment—Angle Complexes and Substitution of Simplicial Complexes
Semyon A. Abramyan, Taras Evgenievich Panov · Proceedings of the Steklov Institute of Mathematics · 2019
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 ∂Δw that realises w. Furthermore, for a particular form of brackets inside w, we prove that ∂Δw 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.