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 Tr. Mat. Inst. Steklova, 2019, Volume 305, Pages 7–28 (Mi tm3995)

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 $w$.

 Funding Agency Grant Number Russian Foundation for Basic Research 18-51-5000517-01-00671 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.

DOI: https://doi.org/10.4213/tm3995

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English version:
Proceedings of the Steklov Institute of Mathematics, 2019, 305, 1–21

Bibliographic databases:

UDC: 515.143+515.146
Revised: March 4, 2019
Accepted: March 6, 2019

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, Tr. Mat. Inst. Steklova, 305, Steklov Math. Inst. RAS, Moscow, 2019, 7–28; Proc. Steklov Inst. Math., 305 (2019), 1–21

Citation in format AMSBIB
\Bibitem{AbrPan19}
\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 Tr. Mat. Inst. Steklova
\yr 2019
\vol 305
\pages 7--28
\publ Steklov Math. Inst. RAS
\mathnet{http://mi.mathnet.ru/tm3995}
\crossref{https://doi.org/10.4213/tm3995}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2019
\vol 305
\pages 1--21
\crossref{https://doi.org/10.1134/S0081543819030015}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85073514581}