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Locally standard torus actions and \(h'\)-numbers of simplicial posets. (English) Zbl 1361.57030

An action of a torus \(T^n\) on a manifold \(M^{2n}\) is called locally standard if it is locally modelled on the standard representation of \(T^n\) on \(\mathbb{C}^n\). The orbit space of a locally standard torus action is a manifold with corners \(Q\). In the paper under review the following situation is considered:
The union of the principal orbits in \(M\) is a trivial torus bundle over the interior of \(Q\); and
all proper faces of \(Q\) are acyclic

For this situation a homological spectral sequence associated to the orbit filtration is considered. The ranks of the groups in the spectral sequence and the Betti numbers of \(M\) are computed. It turns out that these numbers depend only on the topology and combinatorics of \(Q\), but not on the isotropy groups of the action.
Several generalizations of the above results are also disscussed. As a consequence a new proof of the non-negativity of the \(h''\)-numbers of a Buchsbaum simplicial poset is deduced.

MSC:

57N65 Algebraic topology of manifolds
55R20 Spectral sequences and homology of fiber spaces in algebraic topology
05E45 Combinatorial aspects of simplicial complexes
06A07 Combinatorics of partially ordered sets
18G40 Spectral sequences, hypercohomology
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