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Higher Whitehead Products in Moment–Angle Complexes and Substitution of Simplicial Complexes
Semyon A. Abramyana,
Taras E. Panovbcd a Laboratory of Algebraic Geometry and Its Applications, National Research University Higher School of Economics, ul. Usacheva 6, Moscow, 119048 Russia
b Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991 Russia
c Institute for Theoretical and Experimental Physics of National Research Centre “Kurchatov Institute,” Bol'shaya Cheremushkinskaya ul. 25, Moscow, 117218 Russia
d Institute for Information Transmission Problems (Kharkevich Institute), Russian Academy of Sciences, Bol'shoi Karetnyi per. 19, str. 1, Moscow, 127051 Russia
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
$\mathcal Z_\mathcal K$. Namely, we say that a simplicial complex
$\mathcal K$ realises an iterated higher Whitehead product
$w$ if
$w$ is a nontrivial element of
$\pi _*(\mathcal Z_\mathcal K)$. 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
$\mathcal Z_\mathcal K$ and the description of the cohomology product of
$\mathcal Z_\mathcal K$. 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
$w$.
UDC:
515.143+
515.146 Received: December 25, 2018Revised: March 4, 2019Accepted: March 6, 2019
DOI:
10.4213/tm3995
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