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Izv. RAN. Ser. Mat., 2019, Volume 83, Issue 6, Pages 3–62 (Mi izv8927)  

Massey products, toric topology and combinatorics of polytopes

V. M. Buchstabera, I. Yu. Limonchenkob

a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
b National Research University "Higher School of Economics", Moscow

Abstract: In this paper we introduce a direct family of simple polytopes $P^{0} {\subset}  P^{1} {\subset}{\kern1pt}{\cdots}$ such that for any $2 {\leq} k {\leq} n$ there are non-trivial strictly defined Massey products of order $k$ in the cohomology rings of their moment-angle manifolds $\mathcal Z_{P^n}$. We prove that the direct sequence of manifolds $*\subset S^{3}\hookrightarrow…\hookrightarrow\mathcal Z_{P^n}\hookrightarrow\mathcal Z_{P^{n+1}} {\hookrightarrow} {\cdots}$ has the following properties: every manifold $\mathcal Z_{P^n}$ is a retract of $\mathcal Z_{P^{n+1}}$, and one has inverse sequences in cohomology (over $n$ and $k$, where $k\to\infty$ as $n\to\infty$) of the Massey products constructed. As an application we get that there are non-trivial differentials $d_k$, for arbitrarily large $k$ as $n\to\infty$, in the Eilenberg–Moore spectral sequence connecting the rings $H^*(\Omega X)$ and $H^*(X)$ with coefficients in a field, where $X=\mathcal Z_{P^n}$.

Keywords: polyhedral product, moment-angle manifold, Massey product, Lusternik–Schnirelmann category, polytope family, flag polytope, generating series, nestohedron, graph-associahedron.

Funding Agency Grant Number
Russian Foundation for Basic Research 17-01-00671
18-51-50005
Ministry of Education and Science of the Russian Federation 5-100
The research of the first author was supported by the Russian Foundation for Basic Research (grants nos. 17-01-00671 and 18-51-50005). The research of the second author was carried out within the University Basic Research Programme of the Higher School of Economics and was funded by the Russian Academic Excellence Project ‘5-100’.


DOI: https://doi.org/10.4213/im8927

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English version:
Izvestiya: Mathematics, 2019, 83:6, 1081–1136

Bibliographic databases:

UDC: 515.143
MSC: Primary 13F55, 14M25, 55S30; Secondary 52B11
Received: 24.04.2019

Citation: V. M. Buchstaber, I. Yu. Limonchenko, “Massey products, toric topology and combinatorics of polytopes”, Izv. RAN. Ser. Mat., 83:6 (2019), 3–62; Izv. Math., 83:6 (2019), 1081–1136

Citation in format AMSBIB
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