Topology of polyhedral products over simplicial multiwedges

2017/11/01 by Limonchenko, Ivan
#13F55 #55S30 (Primary) 52B11 (Secondary) #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.1711.00461

Abstract

We prove that certain conditions on multigraded Betti numbers of a simplicial complex K imply existence of a higher Massey product in cohomology of a moment-angle-complex \mathcal Z_K, which contains a unique element (a strictly defined product). Using the simplicial multiwedge construction, we find a family F of polyhedral products being smooth closed manifolds such that for any l,r≥ 2 there exists an l-connected manifold M∈\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 Koszul homology of their Stanley--Reisner rings. As an application to rational homotopy theory, we establish a combinatorial criterion for a simple graph Γ to provide a (rationally) formal generalized moment-angle manifold \mathcal Z_P^J=(D^2j_i,S^2j_i-1)^{∂ P^*}, J=(j₁,…,j_m) over a graph-associahedron P=P_Γ and compute all the diffeomorphism types of formal moment-angle manifolds over graph-associahedra.

Related

• Non-trivial higher Massey products in moment-angle complexes
• Massey products in cohomology of moment-angle manifolds corresponding to\n Pogorelov polytopes
• Direct families of polytopes with nontrivial Massey products
• Embeddings of moment-angle manifolds and sequences of Massey products
• LS-category of moment-angle manifolds and higher order Massey products