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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, 2018Revised: December 25, 2018Accepted: March 14, 2019
DOI:
10.4213/tm4016