Cohomology of toric origami manifolds with acyclic proper faces

摘要

A toric origami manifold is a generalization of a symplectic toric manifold(or a toric symplectic manifold). The origami symplectic form is allowed todegenerate in a good controllable way in contrast to the usual symplectic form.It is widely known that symplectic toric manifolds are encoded by Delzantpolytopes, and the cohomology and equivariant cohomology rings of a symplectictoric manifold can be described in terms of the corresponding polytope.Recently, Holm and Pires described the cohomology of a toric origami manifold$M$ in terms of the orbit space $M/T$ when $M$ is orientable and the orbitspace $M/T$ is contractible. But in general the orbit space of a toric origamimanifold need not be contractible. In this paper we study the topology oforientable toric origami manifolds for the wider class of examples: we requirethat every proper face of the orbit space is acyclic, while the orbit spaceitself may be arbitrary. Furthermore, we give a general description of theequivariant cohomology ring of torus manifolds with locally standard torusactions in the case when proper faces of the orbit space are acyclic and thefree part of the action is a trivial torus bundle.