  Toric origami manifolds are characterized by origami templates, which are
combinatorial models built by gluing polytopes together along facets. In this
paper, we examine the topology of orientable toric oigami manifolds with
coorientable folding hypersurface. We determine the fundamental group. In our
previous paper [HP], we studied the ordinary and equivariant cohomology rings
of simply connected toric origami manifolds. We conclude this paper by
computing some Betti numbers in the non-simply connected case.
