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

摘要

We construct a family of manifolds, one for each $n\geq 2$, having anontrivial Massey $n$-product in their cohomology. These manifolds turn out tobe smooth closed 2-connected manifolds with a compact torus action calledmoment-angle manifolds $\mathcal Z_P$, whose orbit spaces are simple$n$-dimensional polytopes $P$ obtained from a $n$-cube by a sequence oftruncations of faces of codimension 2 only (2-truncated cubes). Moreover, thepolytopes $P$ are flag nestohedra but not graph-associahedra. We compute somebigraded Betti numbers $\beta^{-i,2(i+1)}(Q)$ for an associahedron $Q$ in termsof its graph structure and relate it to the structure of the loop homology(Pontryagin algebra) $H_{*}(\Omega\mathcal Z_Q)$. We also study triple Masseyproducts in $H^{*}(\mathcal Z_Q)$ for a graph-associahedron $Q$.