2019/01/31 by Semyon Abramyan, Taras Panov · 1 citation
Mathematics · #math.AT #math.CO
paper · pdf · doi:10.1134/s0081543819030015
published as Proceedings of the Steklov Institute of Mathematics 305 (2019), 1-21 · 22 pages
arxiv created 2019/10/05 · arxiv updated 2019/10/08
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 ZK. Namely, we say that a simplicial complex K realises an iterated higher Whitehead product w if w is a nontrivial element of π_*(ZK). 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 ZK and the description of the cohomology product of ZK. 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.