Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra,...

51
7 th EACAT at Mohali, December 1–6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology and knot theory meet Andrei Vesnin December 6, 2017 Sobolev Institute of Mathematics, Novosibirsk, Russia

Transcript of Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra,...

Page 1: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

7thEACAT at Mohali, December 1–6, 2017

Pogorelov polyhedra, Coxeter groups

and hyperbolic 3-manifolds

When hyperbolic geometry, algebraic topology

and knot theory meet

Andrei Vesnin

December 6, 2017

Sobolev Institute of Mathematics, Novosibirsk, Russia

Page 2: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Content

1. Hyperbolic 3-manifolds. The Lobell example

2. Simple polytopes as orbit spaces

3. Right-angled hyperbolic polyhedra

4. Hyperbolic 3-manifolds from Pogorelov polytopes

5. Intersection of quadrics and Lobell type manifolds

6. Distinguishing of manifolds

1

Page 3: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Hyperbolic 3-manifolds. The

Lobell example

Page 4: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Clifford–Klein space forms

Klein, 1890: the problem of describing all connected compact

Riemannian manifolds with constant curvature.

By Killing – Hopf theorem such a 3-manifold with constant negative

curvature is isometric to the quotient space H3/Γ, where H3 is the

Lobachevsky (hyperbolic) space and Γ is a group of its isometries which

acts freely and properly discontinuous.

Following Killing, such manifolds are called Clifford–Klein space forms.

Klein, 1928: “Regarding hyperbolic geometry, we can only underscore

that a 3-dimensional closed hyperbolic space form with finite volume has

apparently not yet been found ”.

2

Page 5: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Hyperbolic 3-space and hyperbolic 3-manifolds

Let H3 denote a 3-dimensional hyperbolic space (Lobachevsky space) .

H3 can be modelled in R3+ with ds2 = dx2 + dy2 + dt2

t2 .

Recall that Isom+(H3) ∼= PSL(2,C).

An element g =

(a b

c d

)∈ PSL(2,C) acts in

H3 = {(z , t) | z ∈ C, t ∈ R+} by the rule:

g(z , t) =

((az + b)(cz + d) + act2

|cz + d |2 + |c |2t2 ,t

|cz + d |2 + |c |2t2

).

Let Γ be a discrete subgroup of Isom(H3) acting without fixed points.

The quotient space H3/Γ is a hyperbolic 3-manifold.

3

Page 6: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

First example

• Lobell, 1931: a closed orientable hyperbolic 3-manifold, glued from 8

copies of the 14-hedron with all dihedral angles π/2.

German mathematician Frank Richard Lobell [1893–1964].

1F. Lobell, Beispiele geschlossene dreidimensionaler Clifford–Kleinischer Raume

negative Krummung, Ber. Verh. Sachs. Akad. Lpz., Math.-Phys. Kl. 83 (1931),

168–174.

4

Page 7: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Right-angled Coxeter groups

An algebraic description of the Lobell example can be done in the

following way.1

• Let P be a bounded π/2–polyhedron in H3 with the set of faces

F = {F1, . . . ,Fn}.• Let G be the Coxeter group generated by reflections in faces of P:

• generators: g1, g2, . . . , gn, where gi is a reflection in Fi .

• relations: gigj = gjgi if faces Fi and Fj are adjacent.

It is known that G is a discrete subgroup of Iso(H3).

How to find a torsion-free subgroup of G?

1A. Vesnin, Three-dimensional hyperbolic manifolds of Lobell type. Siberian Math. J.

28(5) (1987), 50–53.

5

Page 8: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

The stabilizer

For each vertex v ∈ P its stabilizer in G is generated by three reflections

g1, g2, g3 in pairwise adjacent faces and is isomorphic to the

eight-element abelian group Z2 ⊕ Z2 ⊕ Z2.

The group Z2 ⊕ Z2 ⊕ Z2 can be regarded as the finite vector space over

the field GF (2) with a basis {(1, 0, 0), (0, 1, 0), (0, 0, 1)}.

6

Page 9: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Local linear independence

Al-Jubouri, 1980: The kernel Γ = Kerϕ of an epimorphism

ϕ : G → Z2 ⊕ Z2 ⊕ Z2

is torsion-free if and only if for any vertex v of P images of reflections in

faces incident to v are linearly independent in Z2 ⊕ Z2 ⊕ Z2.

The proof was done for a dodecahedron, but can be easy generalized for

arbitrary bounded right-angled polyhedron in H3.

Thus, if ϕ satisfies this local linear independence property then

M = H3/Kerϕ is a closed hyperbolic 3-manifold (orientable or

non-orientable) constructed from eight copies of P.

1N.K. Al-Jubouri, On nonorientable hyperbolic 3-manifolds, Quart. J. Math. 31

(1980), 9–18.

7

Page 10: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Four colors

Elements α = (1, 0, 0), β = (0, 1, 0), γ = (0, 0, 1) and δ = (1, 1, 1) are

such that any three of them are linearly independent in Z2 ⊕ Z2 ⊕ Z2.

Vesnin, 1987: If ϕ : G → Z2 ⊕ Z2 ⊕ Z2 is such that ϕ(g) ∈ {α, β, γ, δ}for any generator g ∈ G , then Kerϕ consists of orientation-preserving

isometries.

Cor. If an epimorphism ϕ : G → Z32 is such that

• for any generator g of G its image ϕ(g) belongs to {α, β, γ, δ},• for any two adjacent faces their images are different,

then M = H3/Kerϕ is a closed orientable hyperbolic 3-manifold, which

we called Lobell type manifold.

Cor. Any 4-coloring of faces of a bounded π/2-angled polyhedron

P ⊂ H3 defines a closed orientable hyperbolic 3-manifold.

8

Page 11: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Example: the Lobell manifold

Consider R = L(6) as in the figure. It can be realized as a bounded

π/2-angled polyhedron in H3.

Let G be the group, generated by reflections in faces of L(6), and

ϕ : G → Z2 ⊕ Z2 ⊕ Z2 be given by the 4-coloring as in the figure,

where α = (1, 0, 0), β = (0, 1, 0), γ = (0, 0, 1), δ = (1, 1, 1).

��HH

��HH

JJ��

��

AA

XX

��JJ

AA

��

XX

TTTTT����� T

TTTT�����

JJ

JJ

β

γ

δβ

γ

δγ

δ

β

γ

δ

β

α

α

Then H3/Kerϕ is the classical Lobell manifold, constructed in 1931.

9

Page 12: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Simple polytopes as orbit spaces

Page 13: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Small covers and quasitoric manifolds

An n-dimensional convex polytope is simple or of simple combinatorial

type if the number of codimension–one faces meeting at each vertex is n.

Davis, Januszkeviewicz, 1991: considered certain group actions on

manifolds, which have a simple convex polytope as orbit space.

Let Pn be an n–dimensional polytope of simple combinatorial type.

• The group is Zn2, Mn is n-dimensional and Mn/Zn

2 = Pn. In this case

we say that Mn is a small cover of Pn.

• The group is T n, M2n is 2n-dimensional and M2n/T n = Pn. In this

case we say that M2n is a toric manifold over Pn.

Davis, Januszkeviewicz: “The algebraic topology of these manifolds is

very beautiful”.

1M. Davis, T. Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions,

Duke Math. J. 62(2) (1991), 417–451.

10

Page 14: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Standard action

As d = 1 or 2 let

Fd =

{R if d = 1,

C if d = 2,and Gd =

{Z2 if d = 1,

S1 if d = 2.

Gd is regarded as the group of elements of Fd of norm 1.

The natural action of Gd of Fd is called the standard one-dimensional

representation.

The orbit space of that action is naturally identified with R+.

By a G nd –manifold over Pn we mean

• a G nd -action on a manifold Mdn locally isomorphic to the standard

representation,

• the projection map π : Mdn → Pn such that the fibers of π are the

G n2 orbits.

11

Page 15: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Right-angled Coxeter orbifolds

When d = 2, G n2 = T n, and an interesting class of examples arises in

algebraic geometry under the name of “nonsingular toric varieties”2 and

“quasitoric manifolds”3.

When d = 1, G nd is Zn

2 and Pn inherits the structure of an orbifold.

It is a right-angled (π/2-angled) Coxeter orbifold, which means nothing

more than that is it locally isomorphic to the orbifold Rn/Zn2.

We will interested in the case when π/2-angled Coxeter orbifold is a

hyperbolic geometrical orbifold and a small cover Mn is a hyperbolic

n-dimensional manifold.

2V.I. Danilov, The geometry of toric varieties. Russian Math. Surveys 33 (1978),

97–154.3S. Choi, M. Masuda, D.Y. Suh, Rigidity problems in toric topology: a survey. Proc.

Steklov Inst. Math. 275(1) (2011), 177–190.

12

Page 16: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Right-angled hyperbolic

polyhedra

Page 17: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Uniqueness of acute-angled polyhedra in Hn

Let Hn denote an n-dimensional hyperbolic space.

Andreev, 1970: Any bounded acute-angled (all dihedral angles are at

most π/2) polyhedron in Hn is uniquely determined by its combinatorial

type and dihedral angles.

• Right-angled polytopes (with all dihedral angles π/2) are uniquely

determined by the combinatorial type.

13

Page 18: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Existence of bounded right-angled polygons in H2

By Gauss – Bonnet theorem:

If P is a right-angled k-gon in the hyperbolic plane H2 then k ≥ 5.

Construction from Coxeter simplices.

Let T 2 be a triangle with angles πk , π

2 and π4 , where π

k + π2 + π

4 < π.

Let F be the edge opposite to πk .

Coxeter diagram of the group generated by reflection in sides of T 2:

πk

π2

π4F r r r

F

T 2 (with Dk)

k

“Black” subdiagram corresponds to the dihedral group Dk of order 2k.

Then the union

P2 =⋃

g∈Dk

g(T 2)

is a right-angled k-gon in H2 with sides ∂Pn = ∪g∈Dkg(F ). 14

Page 19: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Existence of bounded right-angled polytopes in H3 and H4

Lanner, 1950: There are exactly 9 bounded Coxeter simplices in H3 and 5

bounded Coxeter simplices in H4.

Among them are simplices T 3 ⊂ H3 and T 4 ⊂ H4 as below:

r r r rF

T 3 (with H3) r r r r rF

T 4 (with H4)

“Black” subdiagrams correspond to finite Coxeter groups H3 and H4 of

order 120 and 14400, respectively. For n = 3, 4 the union

Pn =⋃

g∈Hn

g(T n)

is a right-angled polytope in Hn with faces ∂Pn = ∪g∈Hng(F ).

P3 is a dodecahedron and P4 is a 120-cell.

15

Page 20: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

120-cell

16

Page 21: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Absence of bounded right-angled polytopes in Hn, n > 4

For a combinatorial n-dimensional polytope P let ak(P) be the number

of its k-dimensional faces and

a`k =1

ak

∑dim F=k

a`(F )

be the average number of `-faces in a k-dimensional subpolytope.

Nikulin, 1981: For n ≥ 4 and ` < k 6[n2

]we have

a`k <

(n − `n − k

) [ n2 ]`

+

[ n+12

]`

[ n2 ]

k

+

[ n+12

]k

.

Use this result for a12: ` = 1, k = 2. Then

a12 <

{4 + 4

n−2 , if n is even,

4 + 4n−1 , if n is odd.

But 5 ≤ a12.

Cor. No bounded right-angled polytopes in Hn for n > 4. 17

Page 22: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Pogorelov polytopes

Def. A combinatorial polytope is Pogorelov polytope if

• any vertex is incident to 3 edges (simple polytope);

• any prismatic circuit intersects at least 5 edges.

Russian and Ukrainian academician Aleksei Vasil’evich Pogorelov

[1919–2002]

18

Page 23: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

A dodecahedron

Combinatorially simplest Pogorelov polytope is a dodecahedron.

19

Page 24: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Fullerenes are Pogorelov polytopes

If simple polytope with only 5- and 6-gonal faces, it is called fullerene.

Doslic, 2003: If P is a fullerene, then any prismatic circuit intersects at

least 5 edges.

Cor. Fullerenes are Pogorelov polytopes.20

Page 25: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Right-angled polyhedra in H3

Pogorelov, 1967; Andreev, 1970: A polyhedron P can be realized in H3

as a bounded right-angled polyhedron if and only if

(1) any vertex is incident to 3 edges (simple polyhedron);

(2) any face has at least 5 sides;

(3) if a simple closed circuit on the surface of the polyhedron separates

two faces (prismatic circuit), then it intersects at least 5 edges.

That is if and only if P is Pogorelov polytope.

Conditions (1) and (3) imply (2).

21

Page 26: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

One subclass of the class of Pogorelov polytopes

Vesnin, 1987: For any integer n > 5 define a right-angled hyperbolic

(2n + 2)-hedron L(n). Polyhedra L(5) and L(6) look as following:

��HH

��HH6

6

JJ

��

AA

JJ

AA

��

��

XX

XX

�� TTTT���� T

TTT����

JJ

JJ

HH�� CC��

AA��

��CC

HH��

��

@@��

XX ��

JJ

PP ��

JJBBBBB####cc

cc�����

5

5

Polyhedra L(n) are said to be Lobelll polyhedra.

22

Page 27: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Two moves for bounded right-angled polyhedra, I

Let R be the set of all bounded right-angled polyhedra in H3.

Inoue, 2008: Two moves on R.

• Composition / Decomposition: Consider two combinatorial

polyhedra R1,R2 with k-gonal faces F1 ⊂ R1 and F2 ⊂ R2. Then

their composition is a union R = R1 ∪F1=F2 R2.

If R1,R2 ∈ R, then R ∈ R.

23

Page 28: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Two moves for bounded right-angled polyhedra, II

• Removing / adding edge: move from R to R − e and inverse:

n2

n1

n3 n4e

polyhedron R

n2 − 1

n1 − 1

n3 + n4 − 4

polyhedron R − e

If R ∈ R and e is such that faces F1 and F2 have at least 6 sides

each and e is not a part of prismatic 5-circuit, then R − e ∈ R.

Adding edge is known as a Endo–Kroto move for fullerenes. In the case

of fullerenes n1 = n2 = 6 and n3 = n4 = 5.

∗ Sir Harold W. Kroto – Nobel Prize in Chemistry in 1996

24

Page 29: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Reducing to Lobell polyhedra

Inoue, 2008: For any P0 ∈ R there exists a sequence of unions of

right-angled hyperbolic polyhedra P1, . . . ,Pk such that:

• each set Pi is obtained from Pi−1 by decomposition or edge

removing,

• any union Pk consists of Lobell polyhedra.

Moreover,

vol(P0) > vol(P1) > vol(P2) > . . . > vol(Pk).

25

Page 30: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Hyperbolic 3-manifolds from

Pogorelov polytopes

Page 31: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Tiling around a vertex, I

P

26

Page 32: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Tiling around a vertex, II

P ∪ g1(P) ∪ g2(P) ∪ g3(P)

27

Page 33: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Tiling around a vertex, III

P ∪ g1(P) ∪ g2(P) ∪ g3(P) ∪ g1g2(P) ∪ g1g3(P) ∪ g2g3(P)

28

Page 34: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Tiling around a vertex, IV

Manifold M = H3/Kerϕ is obtained from 8 copies of P:

P ∪ g1(P) ∪ g2(P) ∪ g3(P) ∪ g1g2(P) ∪ g1g3(P) ∪ g2g3(P) ∪ g1g2g3(P)

29

Page 35: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Example: manifold M(5)

Polyhedron L(5) is a dodecahedron D.

Up to symmetry it has only one 4-coloring:

HH�� C

C��

AA��

��

CC

HH��

��

@@ ��

XX ��

JJ

PP ��

JJBBBBBB#####c

cccc������

γ

α

αα

γ βδ

γδ β

δβ

Denote by M(5) the closed orientable hyperbolic 3-manifold

corresponding to this coloring.

30

Page 36: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Hamiltonian cycles of Pogorelov polyhedra

An n-dimensional manifold Mn is said to be hyperelliptic if it has an

involution τ such that Mn/〈τ〉 is homeomorphic to Sn. The involution τ

is called hyperelliptic.

Mednykh – Vesnin: Let P be a bounded right-angled polyhedron in H3

and G be the group generated by reflections in faces of P. Assume that

P has a Hamiltonian cycle. Then there exists a torsion-free subgroup

Γ / G of index |G : Γ| = 8 acting on H3 such that H3/Γ is a hyperelliptic

manifold.

Kardos, 2014: Fullerene polyhedra are Hamiltonian.

Cor. For each fullerene P there exists a hyperelliptic 3-manifold

constructed from eight copies of P.

31

Page 37: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Example: the Lobell manifold sister

Consider a Hamiltonian cycle C on the Lobell polyhedron L(6).

The closed curve C divides the surface of L(6) into two regions. Let us

colour faces of the first region by colours α and β, and faces of the

second region by colours γ and δ.

β

δ

βδ

β

δγ

α

γ

α

γ

α

α

δcolors of faces:

α = (1, 0, 0), β = (0, 1, 0),

γ = (0, 0, 1), δ = (1, 1, 1);

colors of edges:

a = α + γ = β + δ = (1, 0, 1)

b = α + δ = β + γ = (0, 1, 1)

c = α + β = γ + δ = (1, 1, 0)

32

Page 38: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

The branching set

M is the 2-fold branched covering of S3 branched over the 12-component

link formed by preimages of edges coloured in “c”.

33

Page 39: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Intersection of quadrics and

Lobell type manifolds

Page 40: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

From intersection of quadrics to 3-manifolds

Let A be a k × n matrix of rank k with entires in R and let Ai ∈ Rk be

its columns. Denote by V = V (A) be the intersection of the quadrics in

Rn given by the equationsn∑

i=1

AiX2i = 0.

Let R(A) be the intersection of V with the unit sphere∑n

i=1 X2i = 1.

Consider affine subspace Π = Π(A) of Rn given byn∑

i=1

AiXi = 0,n∑

i=1

Xi = 1.

Let P = P(A) be the convex polytope Π(A) ∩ (R≥0)n.

Then d = dimR = dimP = n − k − 1.

R = RP is called a real moment-angle complex associated to P. It is a

manifold if P is simple.

34

Page 41: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Example: Regular dodecahedron D

Bartolo, de Medrano, Lozano, 2016: P = D has 12 faces F1, . . . ,F12

HH�� C

C��

AA��

��CC

HH��

��

@@��

XX ��

JJ

PP ��

JJBBBBB####c

ccc�����

1

12

64

11 105

8

3 297

9 equations give D → R12≥0 such that face Fi ⊂ {Xi = 0}:

12∑i=1

Xi = 12,

∑k∼i

X 2k −√

5X 2i =

∑`∼j

X 2` −√

5X 2j | {k , `} ∈ E

.

Here a ∼ b means that faces a and b are adjacent.

35

Page 42: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Real moment-angle manifolds and Lobell manifolds

Let P be a Pogorelov polytope with k faces.

Another description of the moment-angle manifold: RP = H3/G ′, where

G ′ is the commutator of the groups G of reflections faces of in P.

1 −→ G ′ −→ G −→ G ab(= Zk2) −→ 1.

Lobell type manifold M corresponding to 4-coloring of P:

1 −→ π1(M) −→ G −→ Z32 −→ 1.

Since Z32 is Abelian, we get ϕ : G

ab−→ Zk2

χ−→ Z32, where ϕ is 4-coloring

and χ is a characteristic map.

Hence RP → M is (k − 3)-fold covering and RP is hyperbolic 3-manifold.

In particular, in the Bartolo – de Medrano – Lozano example M =M(5).

36

Page 43: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Distinguishing of manifolds

Page 44: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

When two 4-colourings induce the same manifolds?

Hyperbolic geometry’s point of views

(for non-arithmetic groups)

vice versa

algebraic topology’s point of views

(cohomological rigidity)

37

Page 45: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Hyperbolic geometry approach

Let P be a bounded right-angled hyperbolic polyhedron. Let G be

generated by reflections in faces of P, and Σ be the symmetry group of P.

A group G is said to be naturally maximal if 〈G ,Σ〉 is maximal discrete

group, i.e. is not a proper subgroup of any discrete group of Isom(H3).

Vesnin, 1991: Let G be non-arithmetic and naturally maximal. Let

ϕ1, ϕ2 : G → Z32 be epimorphisms induced by two 4-colourings.

Manifolds H3/Ker(ϕ1) and H3/Ker(ϕ2) are isometric if and only if

4-coloruings are equivalent.

38

Page 46: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Example: Lobell polyhedra

Let L(n), n ≥ 5, be the Lobell polyhedron and G (n) be the group

generated by reflections in faces of it.

1. Vesnin, Roeder: if n 6= 5, 6, 8 then group G (n) is non-arithmetic;

2. Mednykh: if n ≥ 6 then G (n) is naturally maximal.

Hence for n = 7 and n ≥ 9 manifolds are distinguished by colorings.

39

Page 47: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Toric topology methods for hyperbolic 3-manifolds

Let k be a commutative ring with unit.

Def. We say that a family of closed manifolds is cohomologically rigid

over k if manifolds in the family are distinguished up to homeomorphism

by their cohomology rings with coefficients in k . That is, a family is

cohomologically rigid if a graded ring isomorphism

H∗(M1; k) ∼= H∗(M2; k)

implies a homeomorphism M1∼= M2 whenever M1 and M2 are in the

family.

See Danilov – Jurkiewicz theorem for generators of H∗(M,Z).

40

Page 48: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Cohomological rigidity of Lobell manifolds

Buchstaber, Erochovets, Masuda, Panov, Park, 2017:

Let M = M(P,Λ) and M ′ = M(P ′,Λ′) be small covers of Pogorelov

polyhedra P and P ′. So, M and M ′ are hyperbolic Lobell type manifolds.

Then the following conditions are equivalent:

(a) there is a cohomology ring isomorphism

ϕ : H∗(M;Z2)→ H∗(M ′;Z2);

(b) there is a diffeomorphism M → M ′;

(c) there is an equivalence of Z2-characteristic pairs (P,Λ) ∼ (P ′,Λ′).

41

Page 49: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Further development we need

• In H3 also exist right-angle polyhedra P with some (or all) vertices

at infinity. In this case some vertices are 3-valence and some are

4-valence. There exists a method to construct hyperbolic 3-manifold

with cusps from 4-colorings of P. But instead of real moment-angle

manifolds we get real moment-angle complexes. Does similar

cohomological rigidity hold?

• What about rigidity in dimension 4 for simple combinatorial type

polytopes and in dimensions 4,5,6,7,8 for almost simple

combinatorial type polytopes?

42

Page 50: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

References

Actual surveys:

• 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.

• A. Vesnin, Right-angled polyhedra and hyperbolic 3-manifolds,

Russian Math. Surveys, 72:2 (2017), 335–374.

43

Page 51: Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra, Coxeter groups and hyperbolic 3-manifolds When hyperbolic geometry, algebraic topology

Thank you!

43