On manifolds defined by 4-colourings of simple 3-polytopes

Let P be the class of combinatorial 3-dimensional simple polytopes P, different from a tetrahedron, without 3- and 4-belts of facets. By the results of Pogorelov and Andreev, a polytope P admits a realisation in Lobachevsky space L³ with right dihedral angles if and only if P ∈ P. We consider two families of smooth manifolds defined by regular 4-colourings of Pogorelov polytopes P: six-dimensional quasitoric manifolds over P and three-dimensional small covers of P; the latter are also known as three-dimensional hyperbolic manifolds of Loebell type. We prove that two manifolds from either of the families are diffeomorphic if and only if the corresponding 4-colourings are equivalent.