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JOURNALS // Trudy Matematicheskogo Instituta imeni V.A. Steklova // Archive

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

This article is cited in 8 papers

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 $\mathcal Z_P^J=(\underline {D}^{2j_i},\underline {S}^{2j_i-1})^{\partial P^*}$, $J=(j_1,\dots ,j_m)$, over a graph-associahedron $P=P_{\Gamma }$, and compute all the diffeomorphism types of formal moment–angle manifolds over graph-associahedra.

Keywords: polyhedral product, moment–angle manifold, simplicial multiwedge, Stanley–Reisner ring, Massey product, graph-associahedron.

UDC: 515.143.5

MSC: Primary 13F55, 55S30, Secondary 52B11

Received: October 30, 2018
Revised: December 25, 2018
Accepted: March 14, 2019

DOI: 10.4213/tm4016


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

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