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

TL;DR

This paper provides an explicit canonical decomposition of orientable 3-manifolds, especially small covers over simple 3-polytopes, aligning with Thurston's geometrization conjecture and building on prior research.

Contribution

It offers a complete method to construct canonical decompositions for orientable 3-manifolds defined by vector-colourings of 3-polytopes, including small covers.

Findings

Explicit canonical decomposition for orientable 3-manifolds

Application to small covers over simple 3-polytopes

Connection with Thurston's geometrization conjecture

Abstract

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.