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

The geometrization conjecture of Thurston (finally proved by Perelman) says that any oriented $3$-manifold can canonically be partitioned into pieces, which have a geometric structure modelled on one of the eight geometries: $S^3$, $\mathbb R^3$, $\mathbb H^3$, $S^2\times\mathbb R$, $\mathbb H^2\times \mathbb R$, the universal cover of $\mathrm{SL}(2,\mathbb{R})$, $\mathrm{Nil}$ and $\mathrm{Sol}$. In a seminal paper (1991) Davis and Januszkiewicz introduced a wide class of $n$-dimensional manifolds, small covers over simple $n$-polytopes. We give a complete answer to the following problem: build an explicit canonical decomposition of any orientable $3$-manifold defined by a vector colouring of a simple $3$-polytope, in particular, of a small cover. The proof is based on an analysis of results in this direction obtained previously by different authors.
Bibliography: 44 titles.