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

Abstract

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:Nil and 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.