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.

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.

Equations

$$\lambda (F)=\{\begin{array}{ll}{\text{a}}_i& \text{if }\chi (F)=i,i=1,2,3,\\ {\epsilon }_1{\text{a}}_1+{\epsilon }_2{\text{a}}_2+{\epsilon }_3{\text{a}}_3& \text{if }\chi (F)=4.\end{array}$$

$${\lambda }_{\chi ,k}(F)=\{\begin{array}{ll}{\text{e}}_i& \text{if }\chi (F)=i,i=1,2,3,\\ {\text{e}}_1+{\text{e}}_2+k{\text{e}}_3& \text{if }\chi (F)=4.\end{array}$$

Full text

00footnotetext: The work was carried out at the Steklov Institute of Mathematics and supported by the Russian Science Foundation grant no. 14-11-00414.

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 [1] and Andreev [2], 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 polytopes $P\in\mathcal{P}$: six-dimensional quasitoric manifolds over $P$ and three-dimensional small covers of $P$; the latter are also known as three-dimensional hyperbolic manifolds of Löbell type [3]. We prove that two manifolds from either of the families are diffeomorphic if and only if the corresponding $4$-colourings are equivalent.

A quasitoric manifold (respectively, a small over) over a simple $n$-polytope $P$ is a $2n$-dimensional ($n$-dimensional) smooth manifold $M$ with a locally standard action of the torus $T^{n}$ (the group $\mathbb{Z}_{2}^{n}$) and a projection $M\to P$ whose fibres are the orbits of the action, see [4], [5].

Let $\mathcal{F}=\{F_{1},\ldots,F_{m}\}$ be the set of facets of a simple $3$-polytope $P$. A characteristic function over $\mathbb{Z}$ (over $\mathbb{Z}_{2}$) is a map $\lambda\colon\mathcal{F}\to\mathbb{Z}^{n}$ ($\lambda\colon\mathcal{F}\to\mathbb{Z}_{2}^{n}$) satisfying the condition: if $F_{i_{1}}\cap F_{i_{2}}\cap F_{i_{3}}$ is a vertex, then $\lambda(F_{i_{1}}),\lambda(F_{i_{2}}),\lambda(F_{i_{3}})$ is a basis of the lattice $\mathbb{Z}^{n}$ (of the space $\mathbb{Z}_{2}^{n}$). Characteristic functions $\lambda$ and $\lambda^{\prime}$ are equivalent ($\lambda\sim\lambda^{\prime}$) if one is obtained from the other by a change of basis in $\mathbb{Z}^{n}$ and changing the direction of some of the vectors $\lambda(F_{i})$ to the opposite (by a change of basis in $\mathbb{Z}_{2}^{n}$). Characteristic pairs $(P,\lambda)$ and $(P^{\prime},\lambda^{\prime})$ are equivalent ($(P,\lambda)\sim(P^{\prime},\lambda^{\prime})$) if $P$ and $P^{\prime}$ are combinatorially equivalent ($P\simeq P^{\prime}$) and $\lambda\sim\lambda^{\prime}$.

Every quasitoric manifold (small cover) $M$ over $P$ is defined by a characteristic pair $(P,\lambda)$; with two such manifolds $M=M(P,\lambda)$ and $M^{\prime}=M^{\prime}(P^{\prime},\lambda^{\prime})$ being equivariantly homeomorphic if and only if $(P,\lambda)\sim(P^{\prime},\lambda^{\prime})$ (see [4], [5, Prop. 7.3.11]). In general, there exist non-equivalent pairs $(P,\lambda)$ and $(P^{\prime},\lambda^{\prime})$ whose corresponding manifolds $M$ and $M^{\prime}$ are (non-equivariantly) diffeomorphic.

A (regular) $4$-colouring of a simple polytope $P$ is a map $\chi\colon\mathcal{F}\to\{1,2,3,4\}$ such that $\chi(F_{i})\neq\chi(F_{j})$ whenever $F_{i}\cap F_{j}\neq\varnothing$. Such a $4$-colouring always exists by the Four Colour Theorem. Two $4$-colourings $\chi$ and $\chi^{\prime}$ are equivalent ($\chi\sim\chi^{\prime}$) if $\chi^{\prime}=\sigma\chi$ for a permutation $\sigma\in S_{4}$.

Let $\chi$ be a $4$-colouring, ${\textbf{{a}}}_{1},{\textbf{{a}}}_{2},{\textbf{{a}}}_{3}$ a basis in $\mathbb{Z}^{3}$, and $\varepsilon_{i}=\pm 1$, $i=1,2,3$. These data define a characteristic function $\lambda=\lambda(\chi,{\textbf{{a}}}_{1},{\textbf{{a}}}_{2},{\textbf{{a}}}_{3},\varepsilon_{1},\varepsilon_{2},\varepsilon_{3})$ given by

[TABLE]

Denote by ${\textbf{{e}}}_{1},{\textbf{{e}}}_{2},{\textbf{{e}}}_{3}$ the standard basis $(1,0,0),(0,1,0),(0,0,1)$ in $\mathbb{Z}^{3}$ or $\mathbb{Z}_{2}^{3}$.

Note that the equivalence classes of characteristic functions are the orbits of the group $\mathrm{GL}_{3}(\mathbb{Z})$ or $\mathrm{GL}_{3}(\mathbb{Z}_{2})$, while the equivalence classes of $4$-colourings are the orbits of the symmetric group $S_{4}$. Nevertheless, we have

Conversely, assume $\lambda_{\chi}\sim\lambda_{\chi^{\prime}}$. By the definition of characteristic functions, we have $\lambda_{\chi^{\prime}}=\lambda(\chi,{\textbf{{a}}}_{1},{\textbf{{a}}}_{2},{\textbf{{a}}}_{3},\varepsilon_{1},\varepsilon_{2},\varepsilon_{3})$ for some basis ${\textbf{{a}}}_{1},{\textbf{{a}}}_{2},{\textbf{{a}}}_{3}$ and $\varepsilon_{i}=\pm 1$. The image of $\lambda_{\chi^{\prime}}$ is the set $\{{\textbf{{e}}}_{1},{\textbf{{e}}}_{2},{\textbf{{e}}}_{3},{\textbf{{e}}}_{1}+{\textbf{{e}}}_{2}+{\textbf{{e}}}_{3}\}$, while the image of $\lambda(\chi,{\textbf{{a}}}_{1},{\textbf{{a}}}_{2},{\textbf{{a}}}_{3},\varepsilon_{1},\varepsilon_{2},\varepsilon_{3})$ is the set $\{{\textbf{{a}}}_{1},{\textbf{{a}}}_{2},{\textbf{{a}}}_{3}$, $\varepsilon_{1}{\textbf{{a}}}_{1}+\varepsilon_{2}{\textbf{{a}}}_{2}+\varepsilon_{3}{\textbf{{a}}}_{3}\}$. Therefore, these two sets of $4$ vectors coincide, that is, ${\textbf{{a}}}_{1}={\textbf{{e}}}_{\sigma(1)}$, ${\textbf{{a}}}_{2}={\textbf{{e}}}_{\sigma(2)}$, ${\textbf{{a}}}_{3}={\textbf{{e}}}_{\sigma(3)}$ and $\varepsilon_{1}{\textbf{{a}}}_{1}+\varepsilon_{2}{\textbf{{a}}}_{2}+\varepsilon_{3}{\textbf{{a}}}_{3}={\textbf{{e}}}_{\sigma(4)}$ for some $\sigma\in S_{4}$. Thus, $\chi^{\prime}=\sigma\chi$ and $\chi\sim\chi^{\prime}$. ∎

Conversely, if $M$ and $M^{\prime}$ are diffeomorphic, then $P\simeq P^{\prime}$ and $\lambda_{\chi}\sim\lambda_{\chi^{\prime}}$, by Theorem 1. Therefore, $\chi\sim\chi^{\prime}$, by Proposition 2. ∎

We say that a characteristic function $\lambda\colon\mathcal{F}\to\mathbb{Z}^{3}$ is defined by a $4$-colouring $\chi$ if $\lambda(F)=\lambda(F^{\prime})$ whenever $\chi(F)=\chi(F^{\prime})$. The image of such a characteristic function consists of $4$ vectors. Examples are $\lambda(\chi,{\textbf{{a}}}_{1},{\textbf{{a}}}_{2},{\textbf{{a}}}_{3},\varepsilon_{1},\varepsilon_{2},\varepsilon_{3})$ and $\lambda_{\chi}$. A regular $4$-colouring of a simple $3$-polytope is complete if for any set of three different colours there exists a vertex whose incident facets have these colours.

Note that it is necessary that the $4$-colouring is complete. For example, assume that there is no vertex with the combination of colours $\{1,2,4\}$ for the incident facets. Then for each $k\in\mathbb{Z}$ consider the characteristic function $\lambda_{\chi,k}$ given by

[TABLE]

Then $\lambda_{\chi,k}\not\sim\lambda_{\chi}$ for $k\neq\pm 1$ (and $\lambda_{\chi,0}\not\sim\lambda_{\chi,1}=\lambda_{\chi}$ over $\mathbb{Z}_{2}$).

The class $\mathcal{P}$ contains all fullerenes, that is, simple $3$-polytopes with only pentagonal and hexagonal facets [7]. If a fullerene has two adjacent pentagons, then all four combinations of three colours are realised in the vertices of these two pentagons, so any $4$-colouring of such a fullerene is complete. For fullerenes without adjacent pentagons (IPR-fullerenes) there exist non-complete $4$-colourings $\chi$. It is easy to see that the corresponding hyperbolic $3$-manifolds $M(P,\lambda_{\chi,0})$ are non-orientable, unlike $M(P,\lambda_{\chi})$.