# Combinatorics, toric topology, and hyperbolic geometry of families of $3$-dimensional polytopes

We study families of combinatorial 3-dimensional simple convex polytopes defined by their cyclic k-edge-connectivity. A *k-belt* is a cyclic sequence of k faces such that faces are adjacent if and only if they follow each other, and no three faces have a common vertex. A simple polytope different from the * simplex* $\Delta^3$ is *cyclically k-edge-connected (ck-connected)*, if it has no s-belts for s less than k, and *strongly ck-connected (c*k-connected)*, if in addition any its k-belt surrounds a face. By definition $\Delta^3$ is c*3-connected but not c4-connected. Any simple polytope (family $P_s$) is c3-connected and at most c*5-connected. We obtain a chain of nested families:

\begin{equation}P_s\supset P_{aflag}\supset {P}_{flag}\supset P_{aPog}\supset P_{Pog}\supset P_{Pog*}\end{equation}

The family of c4-connected polytopes coincides with the family $P_{flag}$ of *flag* polytopes defined by the property that any set of pairwise adjacent faces has a non-empty intersection. The family of c*3-connected polytopes we call *almost flag* polytopes and denote $P_{aflag}$. Results by A.V.Pogorelov (1967) and E.M.Andreev (1970) imply that c5-connected polytopes (family $P_{Pog}$ of *Pogorelov polytopes*) are exactly polytopes realizable in the Lobachevsky space $L^3$ as bounded polytopes with right dihedral angles, and the realization is unique up to isometries. Andreev's result implies that flag polytopes are exactly polytopes realizable in $L^3$ as polytopes with equal non-obtuse dihedral angles. Results by T.Doslic (1998, 2003) imply that the family $P_{Pog}$ contains *fullerenes*, that is simple polytopes with only pentagonal and hexagonal faces. The family $P_{aPog}$ of c*4-connected polytopes we call *almost Pogorelov* polytopes, and the family $P_{Pog*}$ of c*5-connected polytopes -- * strongly Pogorelov*. G.D.Birkhoff (1913) reduced the 4-colour problem to the family $P_{Pog*}$.

Our research has three directions.

Fist direction is the connection to hyperbolic geometry. T.E.Panov remarked that Andreev's results should imply that almost Pogorelov polytopes correspond to right-angled polytopes of finite volume in $L^3$. Such polytopes may have $4$-valent vertices on the absolute, while all proper vertices have valency $3$.

**Theorem 1.** [E19] Cutting off 4-valent vertices defines a bijection between classes of combinatorial equivalence of right-angled polytopes of finite volume in $L^3$ and almost Pogorelov polytopes different from the cube and the pentagonal prism.

The second direction is toric topology. It follows from results by F.Fan, J.Ma and X.Wang [FMW15] that two Pogorelov polytopes P and Q are combinatorially equivalent, if the graded cohomology rings $H^*(Z_P)$ and $H^*(Z_Q)$ of moment-angle manifolds are isomorphic. Later V.M. Buchstaber, N.Yu.Erokhovets, M.Masuda, T.E.Panov and S.Park [BEMPP17] proved that for Pogorelov polytopes characteristic pairs $(P_1,\Lambda_1)$ and $(P_2,\Lambda_2)$ are equivalent, if the graded cohomology rings $H^*(M(P_1,\Lambda_1))$ and $H^*(M(P_2,\Lambda_2))$ of quasitoric manifolds are isomorphic. At that moment our research is focused on the class of almost Pogorelov polytopes.

**Proposition 2.** Let P and Q be almost Pogorelov polytopes different from the cube and the pentagonal prism. Then any isomorphism of graded cohomology rings $H^*(Z_P)$ and $H^*(Z_Q)$ sends the classes corresponding to 4-belts of P to the corresponding classes for Q .

The third direction is combinatorics. We improve known results by D.Barnette (1974, 1977), A.Kotzig (1969), J.W.Butler (1974), T.Inoue (2008), V.D.Volodin (2015) on construction of families by given sets of operations from given initial sets of polytopes and arrange them into a closed theory. Our main result in this direction is construction of the family of fullerenes by $5$ operations from two initial fullerenes in such a way that intermediate polytopes are either fullerenes or simple 3-polytopes with one heptagon adjacent to a pentagon and all the other faces pentagons and hexagons [BE17] (see also [E18]). Recently we proved the following result. Let $P_8$ be the cube with two nonadjacent orthogonal edges cut.

**Theorem 3.** [E19] Any almost Pogorelov polytope P different from the cube and the pentagonal prism can be obtained by cutting off a set of disjoint edges of a polytope in $P_{aPog}\sqcup \{P_8\}$ producing all the quadrangles.

The research is partially supported by the RFBR grants 18-51-50005 and 20-01-00675.

**References.**

[FMW15] F.Fan, J.Ma, X.Wang, B-Rigidity of flag 2-spheres without 4-belt, arXiv:1511.03624.

[BEMPP17] V.M.Buchstaber, N.Yu.Erokhovets, M.Masuda, T.E.Panov, S.Park, Cohomological rigidity of manifolds defined by 3-dimensional polytopes, Russian Math. Surveys, 72:2 (2017), 199-256, arXiv:1610.07575v3.

[BE17] V.M.Buchstaber, N.Yu.Erokhovets, Construction of families of three-dimensional polytopes, characteristic patches of fullerenes and Pogorelov polytopes, Izvestiya: Mathematics, 81:5 (2017), 901-972.

[E18] N.Erokhovets, Construction of Fullerenes and Pogorelov Polytopes with 5-, 6-and one 7-Gonal Face, Symmetry, 10:3 (2018), 67, 28 pp.

[E19] N.Yu.Erokhovets, Three-Dimensional Right-Angled Polytopes of Finite Volume in the Lobachevsky Space: Combinatorics and Constructions, Proc. Steklov Inst. Math., 305 (2019), 78-134.