ИСТИНА

The talk is based on the results obtained jointly with V.M.Buchstaber, M.Masuda, T.E.Panov and S.Park. A {\it Pogorelov polytope} is a combinatorial simple $3$-polytope realizable in the Lobachevsky (hyperbolic) space as a bounded right-angled polytope \cite{V17,BEMPP17,BE17}. It has no $3$- and $4$-gons. \begin{thm}[\cite{BEMPP17, BE17,E18}] A Pogorelov polytope may have any prescribed numbers of $k$-gons, $k\ge 7$. Any simple $3$-polytope with only $5$-, $6$- and at most one $7$-gon is a Pogorelov polytope. For any other prescribed numbers of $k$-gons, $k\ge 7$, there is an explicit construction of a Pogorelov and a non-Pogorelov polytopes. \end{thm} For any mapping $\Lambda$ from the set of faces of a Pogorelov polytope $P$ to $\mathbb Z_2$ (or $\mathbb Z$) satisfying the condition that for any vertex $v=F_i\cap F_j\cap F_k$ the vectors $\Lambda(F_i),\Lambda(F_j),\Lambda(F_k)$ form a basis in $\mathbb Z_2$ (respectively $\mathbb Z$) toric topology associates a $3$-dimensional manifold $R(P,\Lambda)$ with an action of $\mathbb Z_2^3$ called a {\it small cover} (respectively a $6$-dimensional manifold $M(P,\Lambda)$ with an action of the compact torus $T^3$ called a {\it quasitoric manifold}). Small covers over Pogorelov polytopes are also known in hyperbolic geometry (see \cite{V17}), since they admit a hyperbolic structure. \begin{thm}[\cite{BEMPP17}] The $\mathbb Z_2$-cohomology ring of $R(P,\Lambda)$ (respectively the $\mathbb Z$-cohomology ring of $M(P,\Lambda)$) uniquely defines the pair $(P,\Lambda)$ up to a natural equivalence of pairs. \end{thm} An example of Pogorelov polytopes is given by any {\it (mathematical) fullerene} -- a simple convex $3$-polytope with all facets $5$- and $6$-gons. Another example is given by a {\it $k$-barrel} (also called a {\it L\"{o}bell polytope})-- a~polytope with surface glued from two patches, each consisting of~a~$k$-gon surrounded by $5$-gons. Results by T.~Inoue \cite{I08} imply that~any~Pogorelov polytope can be~combinatorially obtained from $k$-barrels by~a~sequence of~{\it $(s,k)$-truncations} (cutting off $s$~subsequent edges of~a~$k$-gon by~a~single plane), $2\le s\le k-4$, and {\it connected sums along $k$-gonal faces} (combinatorial analog of~gluing two polytopes along $k$-gons orthogonal to~adjacent facets). $k$-barrels are irreducible with respect to these operations. \begin{thm}[\cite{BE17}]Any~Pogorelov polytope except for~$k$-barrels can be~obtained from the~$5$- or~the~$6$-barrel by~a~sequence of~$(2,k)$-truncations, $k\geqslant 6$, and~connected sums with $5$-barrels along $5$-gons. \end{thm} In~the~case of~fullerenes we~prove a~stronger result. Let $(2,k;m_1,m_2)$-truncation be~a~$(2,k)$-truncation that cuts~off two edges intersecting an~$m_1$-gon and~an~$m_2$-gon by~vertices different from~the~common vertex. There~is~an~infinite family of~connected sums of~$5$-barrels along $5$-gons surrounded by~$5$-gons called {\it $(5,0)$-nanotubes}. \begin{thm}[\cite{BE17}] Any fullerene except for~the~$5$-barrel and~the~$(5,0)$-nanotubes can be~obtained from~the~$6$-barrel by~a~sequence of~$(2,6;5,5)$-, $(2,6;5,6)$-, $(2,7;5,6)$-, $(2,7;5,5)$-truncations such that all~intermediate polytopes are either fullerenes or Pogorelov polytopes with facets $5$-, $6$- and at most one additional $7$-gon adjacent to a $5$-gon \end{thm} This result can not be literally extended to the latter class of polytopes. We prove that it becomes valid if we additionally allow connected sums with the $5$-barrel and $3$ new operations, which are compositions of $(2,6;5,6)$-, $(2,7;5,6)$-, and $(2,7;5,5)$-truncations. We generalize this result to the case when the $7$-gon may be isolated from $5$-gons \cite{E18}. This work is supported by the Russian Science Foundation under grant no. 14-11-00414. %The research is~partially supported by~the~Young Russian Mathematics award %and the~RFBR~grants 17-01-00671 and 16-51-55017. \begin{thebibliography}{5} \bibitem{V17} A.Yu.~Vesnin,\,\emph{Right-angled polyhedra and hyperbolic 3-manifolds}, Russian Mathematical Surveys, 2017, 72:2, 335-374. \bibitem{BEMPP17} V.M.~Buchstaber, N.Yu.~Erokhovets, M.~Masuda, T.E.~Panov, S.~Park,\,\emph{Cohomological rigidity of manifolds defined by right-angled $3$-dimensional polytopes}, Russian Math. Surveys, {\bf 72}:2 (2017), arXiv:1610.07575v3. \bibitem{BE17} V.M.~Buchstaber, N.Yu.~Erokhovets, \,\emph{Construction of~families of~three-dimensional polytopes, characteristic patches of fullerenes and Pogorelov polytopes}, Izvestiya: Mathematics, {\bf 81}:5 (2017). \bibitem{I08} T.~Inoue,\, \emph{Organizing volumes of right-angled hyperbolic polyhedra}, {\it Algebraic \&Geometric Topology}, {\bf 8} (2008), 1523--1565. \bibitem{E18} N.Yu.~Erokhovets, \,\emph{Construction of fullerenes and Pogorelov polytopes with 5-, 6- and one 7-gonal face}, Preprints 2018, 2018010289 (doi: 10.20944/preprints201801.0289.v1).