Combinatorics of fullerenes and toric topology

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 $p_6$ of hexagons can be arbitrary except for $1$. The number of combinatorial types of fullerenes as a function of $p_6$ grows as $p^9_6$.
Toric topology [1] assigns to each fullerene $P$ a smooth $(p_6+15)$-dimensional moment-angle manifold $\mathcal{Z}_P$ 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 $m\times (m-3)$ defining an $(m-3)$-dimensional toric subgroup in $T^m$ acting freely on $\mathcal{Z}_P$. The orbit space of this action is called a quasitoric manifold $M^6(P,S)$. We have $\mathcal{Z}_P/T^m=M^6/T^3=P$.
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 $H^*(\mathcal{Z}_P,\mathbb Z)\simeq H^*(\mathcal{Z}_Q,\mathbb Z)$ (see [2] and [3]).
A graded isomorphism $H^*(M^6(P,S_P),\mathbb Z)\simeq H^*(M^6(Q,S_Q),\mathbb Z)$ implies a graded isomorphism $H^*(\mathcal{Z}_P,\mathbb Z)\simeq H^*(\mathcal{Z}_Q,\mathbb Z)$ (see [4]).
Using 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 $H^*(M^6(P,S_P),\mathbb Z)\simeq H^*(M^6(Q,S_Q),\mathbb Z)$ (see [4]).
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.
In the end of the talk we will describe operations of construction of fullerenes [5].