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Tr. Mat. Inst. Steklova, 2019, Volume 305, Pages 174–196 (Mi tm4016)  

On Higher Massey Products and Rational Formality for Moment–Angle Manifolds over Multiwedges

Ivan Yu. Limonchenko

National Research University Higher School of Economics, ul. Myasnitskaya 20, Moscow, 101000 Russia

Abstract: We prove that certain conditions on multigraded Betti numbers of a simplicial complex $K$ imply the existence of a higher Massey product in the cohomology of a moment–angle complex $\mathcal Z_K$, and this product contains a unique element (a strictly defined product). Using the simplicial multiwedge construction, we find a family $\mathcal F$ of polyhedral products being smooth closed manifolds such that for any $l,r\geq 2$ there exists an $l$-connected manifold $M\in \mathcal F$ with a nontrivial strictly defined $r$-fold Massey product in $H^*(M)$. As an application to homological algebra, we determine a wide class of triangulated spheres $K$ such that a nontrivial higher Massey product of any order may exist in the Koszul homology of their Stanley–Reisner rings. As an application to rational homotopy theory, we establish a combinatorial criterion for a simple graph $\Gamma $ to provide a (rationally) formal generalized moment–angle manifold $\mathcal Z_P^J=(\underline {D}^{2j_i},\underline {S}^{2j_i-1})^{\partial P^*}$, $J=(j_1,…,j_m)$, over a graph-associahedron $P=P_{\Gamma }$, and compute all the diffeomorphism types of formal moment–angle manifolds over graph-associahedra.

Funding Agency Grant Number
HSE Basic Research Program
Ministry of Education and Science of the Russian Federation 5-100
This work was supported by the HSE Basic Research Program and the Russian Academic Excellence Project “5-100.”


DOI: https://doi.org/10.4213/tm4016

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English version:
Proceedings of the Steklov Institute of Mathematics, 2019, 305, 161–181

Bibliographic databases:

UDC: 515.143.5
MSC: Primary 13F55, 55S30, Secondary 52B11
Received: October 30, 2018
Revised: December 25, 2018
Accepted: March 14, 2019

Citation: Ivan Yu. Limonchenko, “On Higher Massey Products and Rational Formality for Moment–Angle Manifolds over Multiwedges”, 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, 174–196; Proc. Steklov Inst. Math., 305 (2019), 161–181

Citation in format AMSBIB
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\by Ivan~Yu.~Limonchenko
\paper On Higher Massey Products and Rational Formality for Moment--Angle Manifolds over Multiwedges
\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 174--196
\publ Steklov Math. Inst. RAS
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm4016}
\crossref{https://doi.org/10.4213/tm4016}
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\jour Proc. Steklov Inst. Math.
\yr 2019
\vol 305
\pages 161--181
\crossref{https://doi.org/10.1134/S008154381903009X}
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