A fullerene is a spherical-shaped molecule of carbon such that any atom belongs to exactly three carbon rings, which are pentagons or hexagons. Fullerenes have been the subject of intense research, both for their unique quantum physics and chemistry, and for their technological applications, especially in nanotechnology.
A convex 3-polytope is simple if every vertex of it is contained in exactly 3 facets.
A (mathematical) fullerene is a simple convex 3-polytope with all facets pentagons and hexagons. Each fullerene has exactly 12 pentagons and the number p6 of hexagons can be arbitrary except for 1. The number of combinatorial types of fullerenes as a function of p6 grows as p96.
Toric topology  assigns to each fullerene P a smooth (p6+15)-dimensional moment-angle manifold ZP with a canonical action of a compact torus Tm, where m=p6+12. The solution of the famous 4-color problem provides the existence of an integer matrix S of sizes m×(m−3) defining an (m−3)-dimensional toric subgroup in Tm acting freely on ZP. The orbit space of this action is called a quasitoric manifold M6(P,S). We have ZP/Tm=M6/T3=P.
In the talk we focus on the following recent results.
Two fullerenes P and Q are combinatorially equivalent if and only if there is a graded isomorphism of cohomology rings H∗(ZP,Z)?H∗(ZQ,Z) (see  and ).
A graded isomorphism H∗(M6(P,SP),Z)?H∗(M6(Q,SQ),Z) implies a graded isomorphism H∗(ZP,Z)?H∗(ZQ,Z) (see ).
Using results formulated above, we obtain:
Manifolds M6(P,SP) and M6(Q,SQ) are diffeomorphic if and only if there is a graded isomorphism H∗(M6(P,SP),Z)?H∗(