Cohomology of Open Torus Manifolds

The notion of an open torus manifold is introduced. A compact open torus manifold is a torus manifold introduced earlier. It is shown that the equivariant cohomology ring of an open torus manifold $M$ is the face ring of a simplicial poset when every face of the orbit space $Q$ is acyclic. This result extends an earlier result by Masuda and Panov, and the proof here is more direct. Reisner's theorem is then applied to our setting, and a necessary and sufficient condition is given for the equivariant cohomology ring of $M$ to be Cohen–Macaulay in terms of the orbit space $Q$.