On the Cohomology of Torus Manifolds

A torus manifold is an even-dimensional manifold acted on by a half-dimensional torus with non-empty fixed point set and some additional orientation data. It may be considered as a far-going generalisation of toric manifolds from algebraic geometry. The orbit space of a torus manifold has a reach combinatorial structure, e.g., it is a manifold with corners provided that the action is locally standard. Here we investigate relationships between the cohomological properties of torus manifolds and the combinatorics of their orbit quotients. We show that the cohomology ring of a torus manifold is generated by two-dimensional classes if and only if the quotient is a homology polytope. In this case one retrieves the familiar picture from toric geometry: the equivariant cohomology is the face ring of the nerve simplicial complex and the ordinary cohomology is obtained by factoring out certain linear forms. In a more general situation, we show that the odd-degree cohomology of a torus manifold vanishes if and only if the orbit space is face-acyclic. Although the cohomology is no longer generated in degree two under these circumstances, it is still possible to identify the equivariant cohomology with the face ring of an appropriate simplicial poset.