hantzsche-wendt flat manifoldsaszczepa/olsztyn.pdf · crystallographic groups let rn be...

Definitions and examples Hantzsche-Wendt groups Some properties Relations with quadratic forms over Z2 Complex GHW

Hantzsche-Wendt flat manifolds

Andrzej Szczepanski

Uniwersytet Gdanski ( Gdansk, Poland)

LausanneDecember 4, 2008

Hantzsche-Wendt flat manifolds A. Szczepanski


Definitions and examples

Hantzsche-Wendt groups

Some properties

Relations with quadratic forms over Z2

Complex GHW



Crystallographic groups

Let Rn be n-dimensional Euclidean space, with isometry groupE(n) = O(n) n Rn.

DefinitionΓ is a crystallographic group of rank n iff it is a discrete andcocompact subgroup of E(n).

A Bieberbach group is a torsion free crystallographic group.



Basic properties

Theorem(Bieberbach, 1910)

I If Γ is a crystallographic group of dimension n, then the setof all translations of Γ is a maximal abelian subgroup of afinite index.

I There is only a finite number of isomorphic classes ofcrystallographic groups of dimension n.

I Two crystallographic groups of dimension n are isomorphicif and only if there are conjugate in the group affinetransformations A(n) = GL(n,R) n Rn.



Pure abstract point of view

Theorem(Zassenhaus, 1947) A group Γ is a crystallographic group ofdimension n if and only if, it has a normal maximal abeliansubgroup Zn of a finite index.



Holonomy representation

DefinitionLet Γ be a crystallographic group of dimension n withtranslations subgroup A ' Zn. A finite group Γ/A = G we shallcall a holonomy group of Γ.

Let (A, a) ∈ E(n) and x ∈ Rn. Γ acts on Rn in the following way:

(A, a)(x) = Ax + a.



DefinitionLet Γ be n-dimensional Bieberbach group. We have thefollowing short exact sequence of groups.

0→ Zn → Γp→ Γ/Zn = H → 0.

Let us define a homomorphism hΓ : H → GL(n,Z). Put

∀h ∈ H, hΓ(h)(ei) = h−1eih,

where p(h) = h and ei ∈ Zn is a standard basis. hΓ is called aholonomy representation of a group Γ.



Flat manifold

Let Γ ⊂ E(n) be a torsion free crystallographic group. Since Γ iscocompact and discrete subgroup, then the orbit space Rn/Γ isa manifold. If Γ is not torsion free then the orbit space Rn/Γ isan orbifold.

DefinitionThe above manifolds (orbifolds) we shall call "flat".From elementary covering theory any compact Riemannianmanifold (orbifold) with sectional curvature equal to zero is flat.



ExampleFlat surfaces:

I torus S1 × S1,I Klein bottle S1 × S1/Z2

We shall see that many properties of the Bieberbach Groupscorrespond to properties of flat manifolds.



Definition

DefinitionAny Bieberbach group of dimension n with holonomy group(Z2)n−1 we shall call Generalized Hantzsche-Wendt (GHW forshort) Bieberbach group of rank n. An oriented GHW we shallcall Hantzsche-Wendt group (HW for short).A flat manifold Rn/Γ is GHW (HW) if and only if the Bieberbachgroup Γ is GHW (HW).Exercise: Let Mn be a flat Hantzsche-Wendt manifold ofdimension n. Shaw that n is an odd natural number.



ExampleKlein Bottle is GHW, 3-dimensional flat oriented manifold M3

with non-cyclic holonomy Z2 × Z2 is HW manifold. J.Conwaycalled it "didicosm".It can be proved that π1(M3) is a Fibonacci group F(2, 6), where

F(2, 6) = {x1, ..., x6 | x1x2 = x3, x2x3 = x4, ..., x6x1 = x2}.

It is clear, that any n-dimensional HW group is related to anelement of the second cohomology group H2(Zn−1

2 ,Zn). Hencethe number of non-isomorphic GHW groups of given dimensiongrowth exponentialy.



TheoremLet Γ be a n-dimensional GHW group. Then

hΓ((Z2)n−1) ⊂ GL(n,Z)

is a set of the diagonal matrices with ±1 on diagonal.

The proof is by induction and used the following lemmas.

LemmaLet ρ : Zn−1

2 → GL(n,Z) be a diagonal faithful integralrepresentation with -Id /∈ Im(ρ). Then there is g ∈ Zn−1

2 suchthat ρ(g) = diag(-1,-1,...,-1,1,-1,...,-1).Moreover, if furthermore Im(ρ) 6⊂ SL(n,Z), then there isg ∈ Zn−1

2 such that ρ(g) = diag(1,...,1,-1,1,...,1).



LemmaLet Γ be a n-rank Bieberbach group with translation lattice Λ.Suppose that (B, b) ∈ Γ and B has eigenvalues 1 and −1, withcorresponding eigenspaces V+ and V− of dimension 1 andn− 1 respectively.Then Λ = (Λ∩ V+)⊕ (Λ∩ V−), and the orthogonal projection ofb onto V+ lies in 1/2(Λ ∩ V+) \ (Λ ∩ V+).



Rational homology sphere

TheoremIf M = Rn/Γ, where Γ is HW, then M is a rational homologysphere.We calculate rational homology of M from the following formula

Hi(M,Q) = (Λi(Zn))Zn−12 .

From definition of the holonomy representation they are zero forall i 6= 0, n. Since HW is connected and oriented, then M is arational homology sphere.



Anosov relation

Let M be a flat manifold and f : M → M be a continous map.Define the Lefschetz number by

L(f ) = Σi0(−1)iTr(f∗ | Hi(M,Q)).

Moreover, let N(f ) be the Nielsen number of a map f .

Theorem(Anosov, 1985) If f : M → M is a continous map of nilmanifolds,then

N(f ) =| L(f ) | .

It is known, that for any non-oriented flat manifold the abovetheorem is not true.



Theorem(Bram DE ROCK, 2006) Let M be any HW flat manifold, thenfor any continous map f : M → M

N(f ) =| L(f ) | .



Abelianization

Theorem(B.Putrycz, 2006) Let M be any HW flat manifold of dimensionn > 3, then

H1(M,Z) = (Z2)n−1.

In the proof are used methods proposed by R. Miatello and J. P.Rossetti.



Homology

In 2008 K. Dekimpe and N. Petrosyan calculate, byconstructing a free resolution, homology of low dimensionaloriented GHW groups/manifolds. In dimension five there arethis same (for 2 groups). But in dimension 7 there are fourclasses on 62 groups. They find an algorithm.



Let us present the table for dimension five (2 groups):

Hk(Γ) k = 0 k = 1 k = 2 k = 3 k = 4 k = 5Z Z4

2 Z22 Z4

2 0 Z ,



for dimension seven (62 groups):

Hk(Γ) k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7I Z Z6

2 Z82 Z6

4 Z82 Z6

2 0 ZII Z Z6

2 Z92 Z10

2 ⊕ Z24 Z9

2 Z62 0 Z

III Z Z62 Z8

2 Z42 ⊕ Z4

4 Z82 Z6

2 0 ZIV Z Z6

2 Z82 Z8

2 ⊕ Z24 Z8

2 Z62 0 Z

.



TheoremFor any generalized Hantzsche-Wendt group Γ of dimensionn 3 and any subgroup A of index two of the maximal abeliansubgroup Zn we can associate the quadratic function

QΓA : (Z2)n−1 → Z2

and its associated alternating, bilinear quadratic form

BΓA : (Z2)n−1 × (Z2)n−1 → Z2.

Moreover, the function QΓA and form BΓ

A corresponds to thefollowing short exact sequence of finite groups

0→ Z2 = Zn/A→ Γ/A→ (Z2)n−1 → 0.



The proof follows from the lemma.

LemmaAny subgroup A ⊂ Zn of index two of maximal abelian subgroupof Γ is a normal subgroup of Γ.

Proof: We have Γ ⊂ E(n) = O(n) n Rn. For any a ∈ A and(G, g) ∈ Γ we have (G, g)(I, a)(G, g)−1 = (I,G(a)). Since(G− I)x ∈ 2(Zn), for any x ∈ Zn, then G(a) ∈ A.



We define the form B in language of crystallographic group Γ.Let (X, x), (Y, y) ∈ Γ are mapped by p to X,Y ∈ V = (Z2)n−1. Wehave

B(X,Y) = (X − I)y− (Y − I)x ∈ Zn

andQ(X) = (X, x)2 = (X + I)x ∈ Zn.

It is easy to see that B and Q are well defined. It means doesnot depend from the choice of an element (X, x) ∈ Γ. By thissame argument the bilinear form B is alternating. However, itdepends from an index 2 subgroup A ⊂ Zn.



We shall present an example. Let D ⊂ Zn be generated by thefollowing elements:

2e1, e1 − e2, e2 − e3, ..., en−1 − en,

where ei, i = 1, 2, ..., n is a standard basis of Zn. D is a subgroupof index two in Zn. For n 2, let Γn be the subgroup of E(n)generated by the set {(Bi, s(Bi)) | 1 ¬ i ¬ n− 1}. Recall, B′is arethe n× n diagonal matrices:

Bi = diag(−1, . . . ,−1, 1︸︷︷︸i

,−1, . . . ,−1)

and

s(Bi) = ei/2 + ei+1/2 for 1 ¬ i ¬ n− 1.



From definition we have

BΓnD (Bi,Bj) =

0 if i = j1 if i = j + 10 if i j + 2

and a matrix

X =

0 1 0 . . . 01 0 1 0 . . . 0. . . . . . . . . . . .0 . . . 0 1 0 10 . . . 0 1 0

.

Moreover for any Bi ∈ (Z2)n−1, where 1 ¬ i ¬ n− 1

QΓnD (Bi) = (Bi + I)(ei/2 + ei+1/2) = ei /∈ D.



We want to calculate the Arf invariant c of QΓnD . For this, we have

to transform X to a symplectic matrix. Let us introduce a newbasis f1, f2, . . . , fk, fk+1, . . . , f2k :

fi = B2i−1, 1 ¬ i ¬ k

andfk+i = B2i + B2i+2 + B2i+4 + · · ·+ B2k, 1 ¬ i ¬ k.

It is easy to see that the matrix X with respect to the new basisis a symplectic, [

0 II 0

].



Finally we have

PropositionFor any odd number n = 2k + 1 the group Γn/D is anextraspecial group with quadratic form of the quaternion type,(for k = 2l− 1, 2l and l-even) or the real type (for k = 2l− 1, 2land l-odd).



Let M = Rn/Γ be a flat manifold.

DefinitionA holonomy representation ΨΓ : H → GL(n,Z) is essentiallycomplex if there exists a matrix A ∈ GL(n,R), such that,

∀h ∈ H,AΨΓ(h)A−1 ∈ GL(12

n,C).



Theorem(F.E.A. Johnson, E. Rees, 1991) The following conditions areequivalent:

I M is a flat Kähler manifold,I ΨΓ is essentially complex,I Γ is a discrete cocompact torsion-free subgroup of

U(12 n) n C

12 n.



In the same paper is given the following characterization of anessentially complex representation.

ΨΓ : H → GL(n,Z)

is essentially complex if and only if n is an even number andeach R-irreducible summand of ΨΓ which is also C-irreducibleoccurs with even multiplicity.



DefinitionA flat manifold has a Complex structure if and only if theirholonomy representation is essentially complex.In Algebraic geometry the flat Kähler manifolds are calledhyperelliptic varieties.



2-dimensional hyperelliptic varieties :

holonomy CARAT notations1 15.1.1Z2 18.1.1; 18.1.2Z3 35.1.1; 35.1.2Z4 25.1.2; 27.1.1Z6 70.1.1

There is also a list of all 3-dimensional complex flat manifolds.There are 174 such objects. If a holonomy group of a flatmanifold M is a subgroup of SU(n), then M is called Calabi-Yaumanifold.



Theorem(Hodge, 1941) Let M be n-dimensional complex Kählermanifold. We have:

I Hr(M,C) = Σp+q=rHp,q(M),I if hp,q(M) = dimCHp,q(M), then hp,q(M) = hq,p(M),I number br(M) = Σp+q=rhp,q(M) is even if r is odd.

The table of numbers {hp,q(M), 0 ¬ p, q ¬ n} is called theHodge diamond of M.



DefinitionA flat Kähler manifold of complex - dimension n with Zn−1

2holonomy group is called a complex Hantzsche-Wendtmanifold.They are Calabi-Yau manifolds. Here we present their Hodgediamond:



(n0

)0 0

0

(n1

)0

0 0 0 0

0 0

(n2

)0 0

· · · · · · · · · · · · · · · · · ·(n0

) (n1

) (n2

)· · · · · ·

(n

n− 1

) (nn

)· · · · · · · · · · · · · · · · · ·

0 0

(n

n− 2

)0 0

0 0 0 0

0

(n

n− 1

)0

0 0(n0

)Hantzsche-Wendt flat manifolds A. Szczepanski


Thank You.


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