Canonical geometrization of orientable $3$-manifolds defined by vector-colourings of $3$-polytopes

In short geometrization conjecture of W.,Thurston (finally proved by G.~Perelman) says that any oriented $3$-manifold can be canonically partitioned into pieces, which have a geometric structure of one of the eight types. In the seminal paper (1991) M.,W.,Davis and T.,Januszkiewicz introduced a wide class of $n$-dimensional manifolds – small covers over simple $n$-polytopes. We give a complete answer to the following problem: to build an explicit canonical decomposition for any orientable $3$-manifold defined by a vector-colouring of a simple $3$-polytope, in particular for a small cover. The proof is based on analysis of results in this direction obtained before by different authors.