Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra,...
Transcript of Pogorelov polyhedra, Coxeter groups and … at Mohali, December 1{6, 2017 Pogorelov polyhedra,...
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
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
Hyperbolic 3-manifolds. The
Lobell example
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
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
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
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
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
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
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
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
Simple polytopes as orbit spaces
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
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
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
Right-angled hyperbolic
polyhedra
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
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
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
120-cell
16
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
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
A dodecahedron
Combinatorially simplest Pogorelov polytope is a dodecahedron.
19
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
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
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
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
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
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
Hyperbolic 3-manifolds from
Pogorelov polytopes
Tiling around a vertex, I
P
26
Tiling around a vertex, II
P ∪ g1(P) ∪ g2(P) ∪ g3(P)
27
Tiling around a vertex, III
P ∪ g1(P) ∪ g2(P) ∪ g3(P) ∪ g1g2(P) ∪ g1g3(P) ∪ g2g3(P)
28
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
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
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
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
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
Intersection of quadrics and
Lobell type manifolds
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
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
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
Distinguishing of manifolds
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
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
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
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
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
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
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
Thank you!
43