Cohomology of toric origami manifolds with acyclic proper faces

Abstract

Toric origami manifolds are generalizations of symplectic toric manifolds, where the origami symplectic form, in contrast to the usual sym- plectic form, is allowed to degenerate in a good controllable way. It is widely known that symplectic toric manifolds are encoded by Delzant polytopes. The cohomology and equivariant cohomology rings of a symplectic toric manifold can be described in terms of the corresponding polytope. One can obtain a similar description for the cohomology of a toric origami manifold M in terms of the orbit space M=T when M is orientable and the orbit space M=T is con- tractible; this was done by Holm and Pires in (9). Generally, the orbit space of a toric origami manifold need not be contractible. In this paper we study the topology of orientable toric origami manifolds for the wider class of examples: we require that any proper face of the orbit space is acyclic, while the orbit space itself may be arbitrary. Furthermore, we give a general description of the equivariant cohomology ring of torus manifolds with locally standard torus action in the case when proper faces of the orbit space are acyclic and the free part of the action is a trivial torus bundle.