Almost Pogorelov polytopes

Nikolai Erokhovets
Moscow State University
erochovetsn@hotmail.com
International Conference«Toric Topology 2019 in Okayama»
November 18-21, 2019
Okayama University of Sicences

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Polytopes
By a polytope P we mean a combinatorial convex
3-dimensional polytope.

A polytope P is simple, if any its vertex belongs to exactly 3
faces.

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k -belts
A k -belt (k > 3) 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.

3-belt4-belt5-beltProposition
Any simple 3-polytope P 6= ∆3 has a 3-, 4-, or 5-belt.

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Flag polytopes
A simple polytope is called flag if any its set of pairwiseintersecting faces has a nonempty intersection.
Proposition
A simple polytope P is flag iff P 6= ∆3 and P has no 3-belts;
A flag polytope with the smallest number of faces is the cube.

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Pogorelov polytopes
Problem (A.V. Pogorelov, 1967)
To characterize polytopes realizable in the Lobachevsky space
L3 as bounded polytopes with right dihedral angles.
We call such polytopes Pogorelov polytopes.
Motivation
Such polytopes produce «regular» partitions of L3 into equal
polytopes.

(figures by Ya.V. Kucherinenko)

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Pogorelov polytopes
Theorem (A.V. Pogorelov, 1967, E.M. Andreev, 1970)
A polytope P is a Pogorelov polytope iff it is a flag polytope
without 4-belts. The realization is unique up to isometries.
A Pog-polytope with the smallest number of faces is thedodecahedron.

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Cohomological rigidity

A family of manifolds is called cohomologically rigid over the
ring R, if for any two manifolds from the family a gradedisomorphism of cohomology rings over R implies
a diffeomorphism of manifolds.
Pogorelov polytopes give rise to cohomologically rigid families:(m + 3)-dimensional moment-angle manifolds ZP over Z,
where m is the number of faces of P
(F. Fan, J. Ma, X. Wang, 2015);
6-dimensional quasitoric manifolds M(P, Λ) over Z, and3-dimensional hyperbolic manifolds R(P, Λ2 ) over Z2(V. M. Buchstaber, N. Yu. Erokhovets, M. Masuda,
T. E. Panov, S. Park, 2017)7 / 41

Example of Pogorelov polytopes: k -barrels

k-gon

A k -barrel is a Pogorelov polytope for k > 5;
In 1931 F. Löbell glued 8 copies of the 6-barrel to construct thefirst example of a closed three-dimensional hyperbolic manifold.
In 1987 A. Yu. Vesnin constructed hyperbolic manifolds of«Löbell type» for all k -barrels, k > 5.

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Fullerenes
A (mathematical) fullerene is a simple polytope withall faces pentagons and hexagons.

Buckminsterfullerene C60

Truncated icosahedron

Theorem (T. Došlić, 1998, 2003)
Any fullerene is a Porogelov polytope.

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Cyclic k -edge-connectivity (ck -connectivity)

Definition
A simple polytope P 6= ∆3 is ck -connected, if it has nol-belts, l < k .
A simple polytope P 6= ∆3 is strongly ck -connected(c ∗ k -connected), if it is ck -connected and any k -beltsurrounds a face (is trivial).
By definition ∆3 is c ∗ 3-connected, but not c4-connected.

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Families of ck -connected polytopes
Any simple polytope is c3-connected, but at mostc ∗ 5-connected.
We obtain a chain of nested families of polytopes:
Ps ⊃ Paflag ⊃ Pflag ⊃ PaPog ⊃ PPog ⊃ PPog ∗c3-connected Ps – all simple polytopes;c ∗ 3-connected Paflag – almost flag polytopes;
c4-connected Pflag – flag polytopes;c ∗ 4-connected PaPog – almost Pogorelov polytopes ;c5-connected PPog – Pogorelov polytopes;c ∗ 5-connected PPog ∗ – strongly Pogorelov polytopes.
Theorem (G.D. Birkhoff, 1913)
The Four Colour Problem for planar graphs can be reduced
only to Pog ∗ polytopes.

11 / 41n-disk-fullerenes
Definition (M. Deza, M. Dutour Sikirić and M. I. Shtogrin, 2013)
An n-disk-fullerene is a simple polytope with marked n-gonal
face such that all other faces are pentagons and hexagons.

A unique 7-disk-fullerene with the minimal number of faces
Theorem (V. M. Buchstaber, N. Yu. Erokhovets, 2015-2018)
Any 3-disk fullerene is almost flag.
Any 4-disk-fullerene is almost Pogorelov.
Any 7-disk-fullerene is Pogorelov.
For any n > 8 there exist n-disk-fullerenes P and Q, where
P is not almost flag, and Q is Pog ∗ .

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Andreev’s theorem I

Theorem (E.M. Andreev, 1970)
A polytope P 6= ∆3 can be realized as a bounded polytope in L3
with dihedral angles ϕi,j ∈ (0, π2 ] at edges Fi ∩ Fj if and only if
P is simple;
ϕi,j + ϕj,k + ϕk ,i > π for any vertex Fi ∩ Fj ∩ Fk ;ϕi,j + ϕj,k + ϕk ,i < π for any 3-belt (Fi , Fj , Fk );ϕi,j + ϕj,k + ϕk ,l + ϕl,i < 2π for any 4-belt (Fi , Fj , Fk , Fl );if P = ∆2 × I, then there is an edge at a base with thedihedral angle < π2 .
The realization is unique up to isometries.

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Corollaries of Andreev’s theorem I
Corollary 1
A simple polytope P 6= ∆3 can be realized in L3 as a boundedpolytope with equal non-obtuse dihedral angles (∈ ( π3 , π2 ]) ⇔ P
is flag.
Corollary 2
A simple polytope P 6= ∆3 can be realized in L3 as a boundedpolytope with right dihedral angles ⇔ P is flag and has no4-belts.
Idea (T.E. Panov, 2018)
Andreev’s result imply that almost Pogorelov polytopes ≈right-angled polytopes of finite volume in L3 .

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Andreev’s theorem II (1970)
A polytope P 6= ∆3 can be realized as a polytope of finite
volume in L3 with dihedral angles ϕi,j ∈ (0, π2 ] if and only if
P has vertices of valency of 3 and 4;
ϕi,j + ϕj,k + ϕk ,i > π for any 3-valent vertex Fi ∩ Fj ∩ Fk ;ϕi,j = π2 for each edge at a 4-valent vertex;
ϕi,j + ϕj,k + ϕk ,i < π for any 3-belt (Fi , Fj , Fk );ϕi,j + ϕj,k + ϕk ,l + ϕl,i < 2π for any 4-belt (Fi , Fj , Fk , Fl );if P = ∆2 × I, then there is an edge at a base with thedihedral angle < π2 ;
ϕj,k + ϕk ,i < π, if faces Fi and Fj intersect at a 4-valentvertex and Fk is adjacent to both of them and does notcontain their common vertex.
The intersection with the absolute consists of the 4-valentvertices and the 3-valent vertices with the sum of dihedral
angles equal to π.

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Corollaries of Andreev’s theorem
There is nothing about a uniqueness of the realization.
A polytope P can be realized as a polytope of finite volume in
L3 with right dihedral angles ⇔ P
has vertices of valency 3 and 4;
has no 3- and 4-belts;
has no pair of faces intersecting at a 4-valent vertex andadjacent simultaneously to a face not containing it.
The intersection with the absolute consists of 4-valent vertices.
Strong (Mostow) rigidity ⇒ uniqueness of realization.
Theorem (N.Yu. Erokhovets, 2018)
Cutting off 4-valent vertices gives a bijection between
right-angled polytopes of finite volume in L3 and almost
Pogorelov polytopes different from the cube I 3 and thepentagonal prism M5 × I.

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Ideal right-angled polytopes

A medial graph of a plane graph G is another graph M(G) that
represents the adjacencies between edges in the faces of G.
For any polytope P its medial graph G(P) is the graph ofan ideal right-angled polytope;
The graph of any ideal right-angled polytope is the medial
graph for exactly two (possibly equal) polytopes. Moreover,these polytopes are dual to each other.

17 / 41k -antiprisms
The graph of the ideal octahedron is the medial graph ofthe tetrahedron.
The medial graph of a k -gonal pyramid is the graph of ak -antiprism.

k-gon

a)

b)

a) a k -gonal pyramid and its medial graph; b) a k -antiprism

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The Koebe-Andreev-Thurston theorem

The correspondence between ideal right-angled polytopes andmedial graphs plays a fundamental role in the well-known
theorem.
Any polytope P has a geometric realization in R3 such that allits edges are tangent to a sphere.

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Construction of simple polytopes
Theorem (V. Eberhard, 1891)
A polytope P is simple iff it is can be obtained from the simplex∆3 by a sequence of operations of cutting off a vertex, an edge,or two adjacent edges ((2, k )-truncations) by one hyperplane.

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Construction of almost flag polytopes

Proposition (N.Yu. Erokhovets, 2018)
A polytope P is almost flag if and only if one of the equivalentconditions holds
P can be obtained from P = ∆3 with at most two vertices
cut by a sequence of operations of cutting off a vertex, anedge, or a pair of adjacent edges not equivalent to a
cutting off a vertex of a triangle.
P is obtained by a simultaneous cutting off a disjoint set ofvertices of ∆3 or a flag polytope.

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Construction of flag polytopes

Theorem (A. Kotzig, 1967; V. Volodin, 2012+V.M. Buchstaber,
N.Yu. Erokhovets, 2015)
A polytope is flag iff it can be obtained from the cube I 3 by asequence of edge-truncations and (2, k )-truncations, k > 6.

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Construction of almost Pogorelov polytopes
Theorem (follows from the paper by D. Barnette, 1974)
A simple polytope P is almost Pogorelov iff either P is the cube,or the 5-gonal prism, or it can be obtained from the
3-dimensional associahedron (Stasheff polytope) by cuttings offedges not lying in 4-gons, and (2, k )-truncations, k > 6.

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«Resolution of singularities»
For any quadrangle of a flag polytope P there is a flag
polytope Q such that P is obtained from Q by cutting off anedge producing the prescribed quadrangle.
For almost Pogorelov polytopes analogous fact is not true.
Theorem (N.Yu. Erokhovets, 2018)
Any almost Pogorelov polytope P 6= I 3 , M5 × I is obtained by
cutting off a disjoint set of edges (a matching) of an almost
Pogorelov polytope Q or the polytope P8 , producing all its
quadrangles.

Polytope P8

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«Resolution of singularities»
Proposition
Any ideal right-angled polytope is obtained by a contraction ofedges of a perfect matching of an almost Pogorelov polytope or
the polytope P8 containing exactly one edge of eachquadrangle.
Problem
To characterize almost Pogorelov polytopes obtained by cutting
off matchings of Pogorelov polytopes.
Necessary condition
Each quadrangle is adjacent by a pair of opposite edges tofaces with at least six sides.

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Connected sum along k -gonal faces
A connected sum of two simple polytopes P and Q
along k -gonal faces F and G is a combinatorial analog
of glueing of two polytopes along congruent faces orthogonal toadjacent faces.

Connected sum with the dodecahedron along 5-gons.

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Construction of Pogorelov and Pogolelov∗ polytopes

Theorem (D. Barnette, 1977+V.M. Buchstaber-E., 2017)
A polytope P is Pog iff either P is a q-barrel, q > 5, or it can beconstructed from the 5- or the 6-barrel by a sequence of(2, k )-truncations, k > 6, and connected sums with the 5-barrel.
Theorem (D. Barnette+V. M. Buchstaber, N. Yu. Erokhovets)
A polytope P is Pog ∗ iff either P is a q-barrel, q > 5, or it canbe constructed from the 6-barrel by a sequence of(2, k )-truncations, k > 6.

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Non-Pog ∗ fullerenes=(5, 0)-nanotubes

1
2
3

Take patch C of the dodecahedron drawn on the left;
add k > 0 five-belts of hexagons;glue up by the patch C again to obtain the fullerene D5k .

Proposition
A fullerene has the form D5k iff it contains a patch C.
Th. (F. Kardoš, R. Škrekovski vs K. Kutnar, D. Marušič, 2008)
A fullerene is not Pog ∗ if and only if it is D5k , k > 1.

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Construction of fullerenes

Theorem (V.M. Buchstaber, N.Yu. Erokhovets, 2017)
Any Pogorelov∗ fullerene either is the dodecahedron or can beobtained from the 6-barrel by a sequence of (2, 6)- and(2, 7)-truncations such that intermediate polytopes are
fullerenes or 7-disk-fullerenes with the heptagon adjacent to apentagon.

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Construction of ideal right-angled polytopes
In the survey [Right-angled polyhedra and hyperbolic
3-manifolds, Russian Math. Surveys, 72:2 (2017), 335–374]
A.Yu. Vesnin comparing results by
I. Rivin (1996) on ideal polytopes, and
G. Brinkmann, S. Greenberg, C. Greenhill, B.D. McKay,
R. Thomas, and P. Wollan (2005) on graph theory
formulated theorem
Any ideal right-angled polytope can be obtained from somek -antiprism, k > 3, by operations of edge-twist.

Edge-twist. The edges belong to one face and are not adjacent.

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Construction of ideal right-angled polytopes

Theorem (N.Yu. Erokhovets, 2019)
A polytope P is ideal right-angled if and only if either P is a
k -antiprism, k > 3, or P can be obtained from the 4-antiprism
by a sequence of restricted edge-twists.

Restricted edge-twist. Edges are adjacent to the same edge.

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Rigid properties
Definition
A property is rigid for the family of manifolds, if any isomorphismof graded rings ϕ : H ∗ (M1 ) → H ∗ (M2 ), M1 , M2 ∈ F implies thatboth manifolds either have or do not have this property.
We say that a property is rigid for the class of polytopes, if itrigid for the corresponding family of moment-angle manifolds.
Proposition (F. Fan, J. Ma, X. Wang, 2015)
A property to be a flag polytope is rigid in the class of simple
3-polytopes.
Proof: The polytope P 6= ∆3 is flag if and only ife ∗ (ZP ))2 .
H m−2 (ZP ) ⊂ (H32 / 41

Rigid properties
Proposition (F. Fan, J. Ma, X. Wang, 2015)
A property to be Pogorelov polytope is rigid in the class of
simple 3-polytopes.
Proof: The flag polytope P is Pogorelov if and only if themultiplication
H 3 (ZP ) ⊗ H 3 (ZP ) → H 6 (ZP )is trivial.
Conjecture
The property to be almost Pogorelov polytope is rigid it theclass of simple 3-polytopes.

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Rigid sets
Let P be a simple 3-polytope. Then the cohomology ringsH ∗ (ZP ) and H ∗ (M(P, Λ)) have no torsion.
Assume that for any manifold M from a family F a set
SM ⊂ H ∗ (M) is given.
Definition
A set SM is rigid for the family F if ϕ(SM1 ) = SM2 for any
isomorphism of graded rings ϕ : H ∗ (M1 ) → H ∗ (M2 ),M1 , M2 ∈ F.
The group H 3 (ZP ) is a group with the basis {ai,j }corresponding to pairs of non-adjacent faces Fi and Fj .
Lemma (F. Fan, J. Ma, X. Wang, 2015)
The set {±ai,j } is rigid for the class of Pogorelov polytopes.

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Rigidity for belts (F. Fan, J. Ma, X. Wang, 2015)
Each k -belt corresponds to an element H k +2 (ZP ).
The free abelian subgroup in H k +2 (ZP ) with the basis
corresponding to k -belts is rigid for the class of all simple
3-polytopes.
The subset in H k +2 (ZP ) of ±elements corresponding to k -belts
is rigid for the class of Pogorelov polytopes.
The subset in H k +2 (ZP ) of ±elements corresponding to k -belts
around faces is rigid for the class of Pogorelov polytopes.
Thus, any isomorphism of graded rings ϕ : H ∗ (ZP ) → H ∗ (ZQ )for Pogorelov polytopes P and Q defines a bijection between
sets of faces.
This bijection sends adjacent faces to adjacent faces.

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Rigidity for quasitoric manifolds(V.M. Buchstaber, N.Yu. Erokhovets, M. Masuda,
T.E. Panov, S. Park, 2016)
Each face Fi corresponds to the element vi in H 2 (M(P, Λ)).
The set of elements {±vi : Fi is a face} is rigid for the class ofPogorelov polytopes.
Thus, any isomorphism of graded rings
ϕ : H ∗ (M(P, ΛP )) → H ∗ (M(Q, ΛQ )) for Pogorelov polytopes P
and Q defines a bijection between sets of faces.
This bijection sends adjacent faces to adjacent faces.

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Toric topology of almost Pogorelov polytopes
The image of H 3 (ZP ) ⊗ H 3 (ZP ) → H 6 (ZP ) is the subgroup withthe basis consisting of elements corresponding to 4-belts.
Proposition
The subset in H 6 (ZP ) of ± elements corresponding to 4-belts is
rigid for the class of almost Pogorelov polytopes different fromthe cube I 3 and the pentagonal prism M5 × I.
ProblemIs the set of1±ai,j ⊂ H 3 (ZP );2± elements corresponding to belts;

3± elements corresponding to belts around faces;

4±vi in H 2 (M(P, Λ))rigid for the class of almost Pogorelov polytopes 6= I 3 , M5 × I?

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References I
V.M. Buchstaber, N.Yu. Erokhovets,
Combinatorics and toric topology of fullerenes and relatedfamilies of polytopes
A book assumed to be published in AMS in 2020.
V.M. Buchstaber, N.Yu. Erokhovets,
Fullerenes, Polytopes and Toric Topology
Lecture Note Series, IMS, NUS, Singapore, 2017, 67–178,
arXiv:math.CO/160902949.
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.

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References II
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
N.Yu. Erokhovets,
Three-dimensional right-angled polytopes of finite volume
in the Lobachevsky space: combinatorics andconstructions
Proceedings of the Steklov Institute of Mathematics, 305(2019), 86–147 (in press).
V.M. Buchstaber, T.E. Panov,
Toric Topology
AMS Math. Surv. and monographs, vol. 204, 2015. 518 pp.

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References III

V.M. Buchstaber and T.E. Panov,
On manifolds defined by 4-colourings of simple 3-polytopes
Russian Math. Surveys, 71:6 (2016), 1137–1139.
F. Fan, J. Ma, X. Wang,
B-Rigidity of flag 2-spheres without 4-belt
arXiv:1511.03624.
A.Yu. Vesnin,
Right-angled polyhedra and hyperbolic 3-manifolds
Russian Math. Surveys, 72:2 (2017), 335–374.

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Thank You for the Attention!

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