Locally standard torus actions and sheaves over Buchsbaum posets

Abstract

We consider a sheaf of exterior algebras on a simplicial poset $S$ and introduce a notion of homological characteristic function. Two natural objects are associated with these data: a graded sheaf $\mathcal{I}$ and a graded cosheaf $\widehat{\Pi}$. When $S$ is a homology manifold, we prove the isomorphism $H^{n-1-p}(S;\mathcal{I})\cong H_{p}(S;\widehat{\Pi})$ which can be considered as an extension of the Poincare duality. In general, there is a spectral sequence $E^2_{p,q}\cong H^{n-1-p}(S;\mathcal{U}_{n-1+q}\otimes \mathcal{I})\Rightarrow H_{p+q}(S;\widehat{\Pi})$, where $\mathcal{U}_*$ is the local homology stack on $S$. This spectral sequence, in turn, extends Zeeman--McCrory spectral sequence. This sheaf-theoretical result is applied to toric topology. We consider a manifold $X$ with a locally standard action of a compact torus and acyclic proper faces of the orbit space. A principal torus bundle $Y$ is associated with $X$, so that $X\cong Y/\sim$. The orbit type filtration on $X$ is covered by the topological filtration on $Y$. We prove that homological spectral sequences associated with these two filtrations are isomorphic in many nontrivial positions.