Locally standard torus actions and sheaves over Buchsbaum posets

TL;DR

This paper develops a sheaf-theoretic framework extending Poincaré duality and spectral sequences for simplicial posets, and applies it to analyze torus actions on manifolds with acyclic orbit spaces.

Contribution

It introduces a homological characteristic function and related sheaves, extending duality and spectral sequences, with applications to toric topology and locally standard torus actions.

Findings

Established an isomorphism extending Poincaré duality for homology manifolds.

Derived a spectral sequence extending Zeeman–McCrory spectral sequence.

Proved isomorphism of homological spectral sequences for orbit type filtrations.

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.