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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2019, Volume 305, Pages 7–28
DOI: https://doi.org/10.4213/tm3995
(Mi tm3995)
 

This article is cited in 7 scientific papers (total in 7 papers)

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
Full-text PDF (346 kB) Citations (7)
References:
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 $w$.
Funding agency Grant number
Russian Foundation for Basic Research 18-51-50005
17-01-00671
Ministry of Education and Science of the Russian Federation 5-100
HSE Basic Research Program
Simons Foundation
The first author was partially supported by the HSE Basic Research Program, the Russian Academic Excellence Project ‘5-100’, the Russian Foundation for Basic Research (project no. 18-51-50005), and the Simons Foundation. The second author was partially supported by the Russian Foundation for Basic Research (project nos. 17-01-00671 and 18-51-50005) and the Simons Foundation.
Received: December 25, 2018
Revised: March 4, 2019
Accepted: March 6, 2019
English version:
Proceedings of the Steklov Institute of Mathematics, 2019, Volume 305, Pages 1–21
DOI: https://doi.org/10.1134/S0081543819030015
Bibliographic databases:
Document Type: Article
UDC: 515.143+515.146
Language: Russian
Citation: Semyon A. Abramyan, Taras E. Panov, “Higher Whitehead Products in Moment–Angle Complexes and Substitution of Simplicial Complexes”, Algebraic topology, combinatorics, and mathematical physics, Collected papers. Dedicated to Victor Matveevich Buchstaber, Corresponding Member of the Russian Academy of Sciences, on the occasion of his 75th birthday, Trudy Mat. Inst. Steklova, 305, Steklov Math. Inst. RAS, Moscow, 2019, 7–28; Proc. Steklov Inst. Math., 305 (2019), 1–21
Citation in format AMSBIB
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\by Semyon~A.~Abramyan, Taras~E.~Panov
\paper Higher Whitehead Products in Moment--Angle Complexes and Substitution of Simplicial Complexes
\inbook Algebraic topology, combinatorics, and mathematical physics
\bookinfo Collected papers. Dedicated to Victor Matveevich Buchstaber, Corresponding Member of the Russian Academy of Sciences, on the occasion of his 75th birthday
\serial Trudy Mat. Inst. Steklova
\yr 2019
\vol 305
\pages 7--28
\publ Steklov Math. Inst. RAS
\publaddr Moscow
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\vol 305
\pages 1--21
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  • This publication is cited in the following 7 articles:
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    Труды Математического института имени В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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