ИСТИНА

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 any its vertex 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 p_6 of hexagons can be arbitrary except for 1. A surface of the dual polytope is a triangulated sphere and has a locally flat metric with 12 cone singularities of angular defects equal to pi/3. W.Thurston proved that non-negatively curved triangulations of two-sphere with the prescribed angular defects and fixed number of triangles are in one-to-one correspondence with lattice points on certain special complex hyperbolic orbifold of complex dimension 9. This implies that the number of combinatorial types of fullerenes as \linebreak a function of $p_6$ grows as p_6^9. Toric topology assigns to each fullerene P a smooth (p_6+15)-dimensional moment-angle manifold Z_P with a canonical action of a compact torus T^m, where m=p_6+12. The solution of the famous 4-color problem provides the existence of an integer matrix S of sizes (m x (m-3)) defining an (m-3)-dimensional toric subgroup in T^m acting freely on Z_P. The orbit space of this action is called a quasitoric manifold M^6(P,S) We have Z_P/T^m=M^6/T^3=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^*(Z_P,Z)=H^*(Z_Q,Z) . A graded isomorphism H^*(M^6(P,S_P)Z)=H^*(M^6(Q,S_Q),Z) implies a graded isomorphism H^*(Z_P,Z)=H^*(Z_Q, Z). Using well-known result on classification of smooth simple-connected 6-dimensional manifolds and results formulated above, we obtain: Manifolds M^6(P,S_P) and M^6(Q,S_Q) are diffeomorphic if and only if there is a graded isomorphism H^*(M^6(P,S_P),Z)=H^*(M^6(Q,S_Q), Z). Corollary: Two fullerenes P and Q are combinatorially equivalent if and only if the manifolds M^6(P,S_P) and M^6(Q,S_Q) are diffeomorphic. Two manifolds M^6(P,S_P) and M^6(Q,S_Q) are diffeomorphic if and only if they are homotopy equivalent.