Geodesic
Toric origami structures on quasitoric manifolds
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometry, topology, and applications, Tome 288 (2015), pp. 16-37

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We construct quasitoric manifolds of dimension $6$ and higher which are not equivariantly homeomorphic to any toric origami manifold. All necessary topological definitions and combinatorial constructions are given, and the statement is reformulated in discrete geometrical terms. The problem reduces to the existence of planar triangulations with certain coloring and metric properties. 
@article{TM_2015_288_a1,
     author = {A. A. Aizenberg and M. Masuda and Seonjeong Park and Haozhi Zeng},
     title = {Toric origami structures on quasitoric manifolds},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {16--37},
     publisher = {mathdoc},
     volume = {288},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2015_288_a1/}
}
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A. A. Aizenberg; M. Masuda; Seonjeong Park; Haozhi Zeng. Toric origami structures on quasitoric manifolds. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometry, topology, and applications, Tome 288 (2015), pp. 16-37. http://geodesic.mathdoc.fr/item/TM_2015_288_a1/