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

## TL;DR

This paper explores the relationship between 4-colourings of certain simple 3-polytopes and the resulting manifolds, establishing conditions under which these manifolds are diffeomorphic based on colouring equivalence.

## Contribution

It introduces a classification linking 4-colourings of Pogorelov polytopes to the diffeomorphism types of associated manifolds, bridging combinatorics and geometric topology.

## Key findings

- Manifolds are diffeomorphic iff their 4-colourings are equivalent.
- Defines two families of manifolds: quasitoric and small covers.
- Establishes a correspondence between colourings and manifold diffeomorphism types.

## Abstract

Let $\mathcal{P}$ 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 $\mathbb{L}^3$ with right dihedral angles if and only if $P \in \mathcal{P}$. 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.