The topology of toric origami manifolds

TL;DR

This paper studies the topology of toric origami manifolds with acyclic templates and co-orientable folds, showing their cohomology is concentrated in even degrees and they satisfy GKM conditions, linking them to torus manifolds.

Contribution

It proves that certain toric origami manifolds have even-degree cohomology and satisfy GKM theory, connecting them to known classes of torus manifolds.

Findings

Cohomology is concentrated in even degrees.

Equivariant cohomology satisfies GKM description.

Provides examples of Masuda and Panov's torus manifolds.

Abstract

A folded symplectic form on a manifold is a closed 2-form with the mildest possible degeneracy along a hypersurface. A special class of folded symplectic manifolds are the origami symplectic manifolds, studied by Cannas da Silva, Guillemin and Pires, who classified toric origami manifolds by combinatorial origami templates. In this paper, we examine the topology of toric origami manifolds that have acyclic origami template and co-orientable folding hypersurface. We prove that the cohomology is concentrated in even degrees, and that the equivariant cohomology satisfies the GKM description. Finally we show that toric origami manifolds with co-orientable folding hypersurface provide a class of examples of Masuda and Panov's torus manifolds.