By a fullerene we mean a combinatorial simple 3-polytope with only pentagonal and hexagonal facets. This is a mathematical model for spherical shaped molecule of carbon with atoms linked into pentagonal and hexagonal rings (1996 Nobel Prize in chemistry to Robert Curl, Harold Kroto and Richard Smalley).

The fundamental construction of Toric Topology assigns to each simple n-polytope P with m facets a smooth (n+m)-manifold Z(P). There is a canonical action of compact m-torus T^m on Z(P), whose orbit space Z(P) / T^m is identified with the polytope P. There is a combinatorial invariant b(P) defined as the maximal dimension of a subtorus in T^m acting freely on Z(P). This invariant takes values between 1 to m-n.

In the case b(P)=m-n the orbit space M(P)=Z(P) / T^{m-n} is a quasitoric 2n-manifold.    It follows from the Four Color Theorem that for any simple 3-polytope P there exists a quasitoric 6-manifold M(P) over P (M.Davis, T.Januszkiewicz).

A Pogorelov polytope is a simple convex 3-polytope whose facets do not form 3- and 4-belts. It can be proved that each fullerene is a Pogorelov polytope. For every Pogorelov 3-polytope P together with a regular 4-coloring of its facets, there is an associated 3-dimensional hyperbolic manifold of Loebell type (A.Yu.Vesnin). It is an aspherical manifold realized as the fixed point set for the canonical involution on the quasitoric manifold M(P) over the Pogorelov polytope P.

Two Loebell manifolds are isometric if and only if their Z/2-cohomology rings are isomorphic (V.M.Buchstaber, N.Yu.Erochovets, M.Masuda, T.E.Panov, S.Park). The corollary of this result is: two Loebell manifolds are isometric if and only if the corresponding 4-colorings are equivalent (V.M.Buchstaber, T.E.Panov). 

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