  The notion of a toric origami manifold, which weakens the notion of a
symplectic toric manifold, was introduced by Cannas da Silva-Guillemin-Pires
\cite{ca-gu-pi11} and they show that toric origami manifolds bijectively
correspond to origami templates via moment maps, where an origami template is a
collection of Delzant polytopes with some folding data. Like a fan is
associated to a Delzant polytope, a multi-fan introduced in \cite{ha-ma03} and
\cite{masu99} can be associated to an oriented origami template. In this paper,
we discuss their relationship and show that any simply connected compact smooth
4-manifold with a smooth action of $T^2$ can be a toric origami manifold. We
also characterize products of even dimensional spheres which can be toric
origami manifolds.

Symplectic Geometry;Algebraic Topology