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

• Published in: Proceedings of the Steklov Institute of Mathematics 2019-05, Vol.305 (1), p.1-21
• Main Authors: Abramyan, Semyon A., Panov, Taras E.
• Format: Article
• Language: English
• Subjects: Brackets; Combinatorial analysis; Criteria; Homology; Mathematics; Mathematics and Statistics; Questions; Substitutes
• ISSN: 0081-5438; 1531-8605
• DOI: 10.1134/S0081543819030015

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 π * ( 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 ∂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 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 complexes 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 .