A class of torus manifolds with nonconvex orbit space

Abstract

We study a class of smooth torus manifolds whose orbit space has the combinatorial structure of a simple polytope with holes. We construct moment angle manifolds for such polytopes with holes and use them to prove that the associated torus manifolds admit stable almost complex structure. We give a combinatorial formula for the Hirzebruch ${\chi }_y$ genus of these torus manifolds. We show that they have (invariant) almost complex structure if they admit positive omniorientation. We give examples of almost complex manifolds that do not admit a complex structure. When the dimension is four, we calculate the homology groups and describe a method for computing the cohomology ring.

Minor changes in previous version, 14 pages, 3 figures, to appear in Proceedings of the American Mathematical Society

References in corpus (1)

• On the cohomology of torus manifolds

Cited by in corpus (6)

• Some Aspects of Equivariant LS-category
• Locally standard torus actions and sheaves over Buchsbaum posets
• Cohomology of toric origami manifolds with acyclic proper faces
• On integral cohomology of certain orbifolds
• A generalization of moment-angle manifolds with non-contractible orbit spaces
• Orientability and equivariant oriented cobordism of 2-torus manifolds