A class of torus manifolds with nonconvex orbit space

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We study a class of smooth torus manifolds whose orbit space has the struc- ture of a simple polytope with holes. We prove that these manifolds have stable almost complex structure and give combinatorial formula for some of their Hirzebruch genera. They have (invariant) almost complex structure if they admit positive omniorientation. In dimension four, we calculate the homology groups, construct symplectic structure on a large class of these manifolds, and give a family which is symplectic but not complex.