Moment-angle manifolds, 2-truncated cubes and Massey operations

Abstract

We construct a family of manifolds, one for each $n\ge 2$, having a nontrivial Massey $n$-product in their cohomology. These manifolds turn out to be smooth closed 2-connected manifolds with a compact torus action called moment-angle manifolds ${\mathcal{Z}}_P$, whose orbit spaces are simple $n$-dimensional polytopes $P$ obtained from a $n$-cube by a sequence of truncations of faces of codimension 2 only (2-truncated cubes). Moreover, the polytopes $P$ are flag nestohedra but not graph-associahedra. We compute some bigraded Betti numbers ${\beta }^{-i,2(i+1)}(Q)$ for an associahedron $Q$ in terms of its graph structure and relate it to the structure of the loop homology (Pontryagin algebra) $H_{\ast }(\Omega {\mathcal{Z}}_Q)$. We also study triple Massey products in $H^{\ast }({\mathcal{Z}}_Q)$ for a graph-associahedron $Q$.

19 pages, 6 figures. To appear in Ch. Ann. of Math., Ser. B. Minor changes, misprints corrected. Main results of Section 4 were published in Rus. Math. Surveys

References in corpus (2)

• Matroid polytopes, nested sets and Bergman fans
• The polyhedral product functor: a method of computation for moment-angle complexes, arrangements and related spaces

Cited by in corpus (4)

• Massey products, toric topology and combinatorics of polytopes
• Topology of polyhedral products over simplicial multiwedges
• Real toric manifolds associated with chordal nestohedra
• Zoo of monotone Lagrangians in $\mathbb{C}{P}^n$