Toric origami structures on quasitoric manifolds
  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 existence of planar triangulations with certain coloring and metric properties. 
Anton Ayzenberg, Mikiya Masuda, Seonjeong Park, Haozhi Zeng
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