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Let $X$ be 2n-dimensional compact manifold with a locally standard action ofa compact torus. The orbit space $X/T$ is a manifold with corners. Suppose thatall proper faces of $X/T$ are acyclic. In the paper we study the homologicalspectral sequence $E^*_{*,*}\Rightarrow H_*(X)$ corresponding to the filtrationof $X$ by orbit types. When the free part of the action is not twisted, wedescribe the whole spectral sequence in terms of homology and combinatorialstructure of $X/T$. In this case we describe the kernel and the cokernel of thenatural map $k[X/T]/(l.s.o.p.) \to H_*(X)$, where $k[X/T]$ is a face ring of$X/T$ and $(l.s.o.p.)$ is the ideal generated by a linear system of parameters(this ideal appears as the image of $H^{>0}(BT)$ in equivariant cohomology.There exists a natural double grading on $H_*(X)$, which satisfies bigradedPoincare duality. This general theory is applied to compute homology groups oforigami toric manifolds with acyclic proper faces of the orbit space. A numberof natural generalizations is considered. These include Buchsbaum simplicialcomplexes and posets. h'- and h''-numbers of simplicial posets appear as theranks of certain terms in the spectral sequence $E^*_{*,*}$. In particular,using topological argument we show that Buchsbaum posets have nonnegativeh''-vectors. The proofs of this paper rely on the theory of cellular sheaves.We associate to a torus space certain sheaves and cosheaves on the underlyingsimplicial poset, and observe an interesting duality between these objects.This duality seems to be a version of Poincare-Verdier duality between cellularsheaves and cosheaves.

Homology of torus spaces with acyclic proper faces of the orbit space