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 $\hat{\Pi }$. When $S$ is a homology manifold, we prove the isomorphism $H^{n-1-p}(S;\mathcal{I})\cong H_p(S;\hat{\Pi })$ which can be considered as an extension of the Poincare duality. In general, there is a spectral sequence $E_{p,q}^2\cong H^{n-1-p}(S;{\mathcal{U}}_{n-1+q}\otimes \mathcal{I})\Rightarrow H_{p+q}(S;\hat{\Pi })$, where ${\mathcal{U}}_{\ast }$ 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.

23 pages. Several typos were corrected and the numbering of theorems changed

References in corpus (3)

• The fundamental group and Betti numbers of toric origami manifolds
• Cohomology of toric origami manifolds with acyclic proper faces
• Homology of torus spaces with acyclic proper faces of the orbit space

Cited by in corpus (2)

• Locally standard torus actions and h'-vectors of simplicial posets
• Topological model for h"-vectors of simplicial manifolds