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

Ivan Yu. Limonchenko

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebraic topology, combinatorics, and mathematical physics, Tome 305 (2019), pp. 174-196

Résumé

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.

Détail

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