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

Abstract

Let $X$ be 2n-dimensional compact manifold with a locally standard action of a compact torus. The orbit space $X/T$ is a manifold with corners. Suppose that all proper faces of $X/T$ are acyclic. In the paper we study the homological spectral sequence $E^*_{*,*}\Rightarrow H_*(X)$ corresponding to the filtration of $X$ by orbit types. When the free part of the action is not twisted, we describe the whole spectral sequence in terms of homology and combinatorial structure of $X/T$. In this case we describe the kernel and the cokernel of the natural 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 bigraded Poincare duality. This general theory is applied to compute homology groups of origami toric manifolds with acyclic proper faces of the orbit space. A number of natural generalizations is considered. These include Buchsbaum simplicial complexes and posets. h'- and h''-numbers of simplicial posets appear as the ranks of certain terms in the spectral sequence $E^*_{*,*}$. In particular, using topological argument we show that Buchsbaum posets have nonnegative h''-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 underlying simplicial poset, and observe an interesting duality between these objects. This duality seems to be a version of Poincare-Verdier duality between cellular sheaves and cosheaves.