Geodesic
Cohomology of Open Torus Manifolds
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometric topology, discrete geometry, and set theory, Tome 252 (2006), pp. 158-166

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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 $Q$. 
@article{TM_2006_252_a13,
     author = {M. Masuda},
     title = {Cohomology of {Open} {Torus} {Manifolds}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {158--166},
     publisher = {mathdoc},
     volume = {252},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_2006_252_a13/}
}
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M. Masuda. Cohomology of Open Torus Manifolds. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometric topology, discrete geometry, and set theory, Tome 252 (2006), pp. 158-166. http://geodesic.mathdoc.fr/item/TM_2006_252_a13/