cohomology and stiefel-whitney classes of flat manifolds...

Cohomology andStiefel-Whitney Classes

of Flat Manifolds

Sergio Console

Compact flat manifolds& Bieberbach groupsFlat manifolds and spectra

Bieberbach groups

Bieberbach groups

Cohomology of B’bach grps

The LHS SpectralSequenceThe Charlap-Vasquezmethod

Use of the LHS SpectralSequence

2. Stiefel-Whitney class

Topology and spectraCohomology and spectralproperties

Stiefel-Whitney classes andspectral properties

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Cohomology and Stiefel-WhitneyClasses of Flat Manifolds

joint work with Roberto Miatello and Juan Pablo Rossetti,Cordoba, Argentina

CLAM Córdoba – July 2012

Sergio ConsoleDipartimento di Matematica

Università di Torino


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Bieberbach groups

Bieberbach groups







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1 Compact flat manifolds and Bieberbach groupsFlat manifolds and spectraBieberbach groupsBieberbach groupsCohomology of Bieberbach groups

2 The Lyndon-Hochschild-Serre Spectral SequenceThe Charlap-Vasquez methodUse of the LHS Spectral SequenceSecond Stiefel-Whitney class

3 Topology and spectraCohomology and spectral propertiesStiefel-Whitney classes and spectral properties


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Bieberbach groups

Bieberbach groups







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Isospectral manifolds

The original examples of isospectral but not isometricmanifolds were found by Milnor – these are flat tori.

Since then, effort was:finding manifolds which are isospectral but not isometric(“one cannot hear the shape of a drum”)and which even have different topologies.

These kind of problems have been investigated in the contextof nilmanifolds, solvmanifolds and compact flat manifolds.

The latter turn out to be a rich family where one can rather explicitlycompute the multiplicities of eigenvalues of Laplace type operators,the real cohomology and the lengths of closed geodesics.


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Bieberbach groups

Bieberbach groups







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Flat manifolds and spectra

M, M ′ are p-isospectral⇐⇒ have the same spectrumwith respect to the Hodge Laplacian ∆p acting on p-forms.

M, M ′ p-isospectral =⇒ bp(M) = bp(M ′)(the (Z)-Betti number bp equals the multiplicity of the 0 eigenvalue of ∆p)

So, the torsionfree part cannot be distinguished =⇒it is not so easy to exhibit p-isospectral manifolds for all phaving different cohomological properties.

We construct for instance M,M ′, p-isospectral for all p with• H1(M,Z2) ∼= H1(M ′,Z2) but H2(M,Z2) 6∼= H2(M ′,Z2)

• same (Z2)-cohomology but such thatw2(M) 6= 0 and w2(M ′) = 0


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Bieberbach groups

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Bieberbach groups

A crystallographic group is a discrete, cocompact subgroup Γof the isometry group of Rn, I(Rn) ∼= O(n) nRn.

If Γ is also torsion-free, then Γ is a Bieberbach group.

Such Γ acts properly discontinuously and freely on Rn, thusMΓ = Γ\Rn is a compact flat Riemannian manifold withfundamental group Γ.

Any compact flat manifold arises in this way.

MΓ = Γ\Rn is an Eilenberg-MacLane space =⇒the cohomology of M = the group cohomology of Γ


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Bieberbach groups

Bieberbach groups







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Bieberbach groups

Any γ ∈ I(Rn) can be written uniquely

γ = BLbwhere B ∈ O(n) and Lb is a translation by b ∈ Rn.

The restriction to Γ of r : I(Rn)→ O(n), r(BLb) = B,is a homomorphism with kernel Λ=lattice,r(Γ) ∼= F is a finite subgroup of O(n): the holonomy group (orpoint group) of Γ.

Algebraically, Γ is an extension of F by Λ, i.e., there is an exactsequence 0→ Λ→ Γ

r→ F → 1

Conjugation by BLb induces an action of F on Λ which is givenby λ ∈ Λ 7→ Bλ and is called the holonomy representation.

If the holonomy representation diagonalizes the correspondingmanifolds are called compact flat manifolds of diagonal type.


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Bieberbach groups

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Examples in column notation

Hantzsche-Wendt 3-manifold or didicosm

MΓ: flat manifold of dimension n = 3,

Holonomy group of Γ = Z22,

generated by B1 = diag(1,−1,−1), B2 = diag(−1,1,−1),

b1 =e1 + e3

2, b2 =

e1 + e2

2; i.e., Γ = 〈B1Lb1 , B2Lb2 ; LZ3〉.

MΓ is orientable since det B = 1 for every BLb ∈ Γ.

In column notation

B1 B2

1 12−1

−1 12

1 12

−1 −1 12

or

B1 B2 B1B2

1 12−1 −1 1

2

−1 12

1 12−1

−1 −1 12

1 12

.


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Bieberbach groups

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Z2-class polynomial

Fact: H∗(Zk2,Z2) ∼= Z2[x1, . . . , xk ] , with gen. x1, . . . , xk in dim. 1

Γ of diagonal type with holonomy group Zk2 = 〈B1, . . . ,Bk 〉.

β̄ ∈ H2(Zk2,Λ∗ ⊗ Z2) ∼=

(H2(Zk

2,Z2))n

The components β̄` of β̄ are homogeneous polynomials ofdegree two called the Z2-class polynomials of Γ

Proposition

β̄` =∑

i : Bie` = e`

bi` = 12

x2i +

∑i : bi` = 1

2

∑j 6= i

Bje` = −e`

xixj ,

where e1, . . . ,en is the standard basis of Rn.


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Bieberbach groups

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Lyndon-Hochschild-Serre spectral sequence

Γ is an extension of F by Λ, i.e., 0→ Λ→ Γr→ F → 1

Ep,qr =⇒ Hp+q(Γ,R)

withEp,q

2∼= Hp(F ,Hq(Λ,R)) ,

(the coefficient ring R is regarded as a trivial Γ-module and p, q ≥ 0)


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Bieberbach groups

Bieberbach groups







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LHS spectral sequence for Bieberbach groups

For Γ Bieberbach group of diag. type with holonomy group Zk2

Ep,q2 = Hp(Zk

2,Z2)⊗∧q(Zn

2)∗

and their dimensions are given by

Fact: the Z2-class polynomials determine the differential d2


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Bieberbach groups

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The differential dp,q2 in low dimensions

Let ε1, . . . , εn be a basis ofΛ∗ ⊗ Z2 ∼= (Zn

2)∗,

d0,12 : E0,1

2∼= H1(Zn,Z2) ∼= (Zn

2)∗ −→ E2,0

2∼= H2(Zk

2,Z2)

d0,12 εi = β̄i i = 1, . . . ,n

d1,12 : E1,1

2∼= H1(Zk

2,Z2)⊗ Zn2 −→ E3,0

2∼= H3(Zk

2,Z2)

d1,12 (xi ⊗ εj ) = xi ∪ β̄j i = 1, . . . , k , j = 1, . . . ,n

d0,22 : E0,2

2∼=

∧2(Zn2)

∗ −→ E2,12∼= H2(Zk

2, (Zn2)

∗)

d0,22 (εi ∧ εj )(εk ) = δik β̄j + δjk β̄i , i , j , k = 1, . . . ,n

where {εi ∧ εj} (i < j) is a basis of∧2(Zn

2)∗.


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Bieberbach groups

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Examples of application of the Charlap-Vasquez method

Computation of H1(Γ,Z2):

Proposition

Let M be an n-dimensional compact flat manifold with diagonalholonomy Zk

2. Then

dim H1(M,Z2) = n − rank d0,12 + k .

Note: rank d0,12 = # linearly indep. Z2-class polynomials β̄`, ` = 1, . . . , n.


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Bieberbach groups

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Examples of application of the Charlap-Vasquez method

Computation of H2(Γ,Z2):

Proposition

If there are n − 1 linearly independent Z2-class polynomials,then

dim H2(M,Z2) =

(k + 1

2

)− rank d0,1

2 + kn − rank d1,12


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Bieberbach groups

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Second Stiefel-Whitney class

The characteristic classes of a compact Riemannian manifoldare closely related to its geometry.

Fact: Pontrjagin classes for any compact flat manifold vanish,since they can be expressed in terms of the curvature tensor(Chern-Weyl theorem),

Surprising: the Stiefel-Whitney classes need not vanish(Auslander and Szczarba)

Recall: w2(M) ∈ H2(M,Z2) and an oriented Riemannianmanifold M admits a spin structure if and only if w2(M) = 0.

Theorem (Tool to compute the 2nd SW class)

MΓ n-dim compact flat manifold with diagonal holonomy Zk2.

Then,

w2 6= 0⇐⇒ σ2(ω1, . . . , ωn) is not a sum of Z2-class polynomials


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Bieberbach groups

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More details on the computation of SW classesand proof of the above Theorem

Since M has diagonal holonomy Zk2, we have the inclusion

Zk2

i↪→ D(n)

D(n) ∼= Zk2 = diagonal matrices in O(n).

Let x1, . . . , xk be a basis of H1(Zk2,Z2) and let x ′1, . . . , x

′n be the

standard basis of H1(D(n),Z2). The classes

ω` = i∗(x ′`) =∑

m

am`xm ,

are called the 2-weights of the map i .

Observe that ω` =∑

m am`xm where am` = 1 if the entry (`,m)of Γ in column notation is −1∗ and it is zero if it is 1∗.


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Letσ2(ω1, . . . , ωn) := 2-nd elem. symmetric function in ω1, . . . , ωn.

Proposition

Let Γr→ F = Zk

2 be the projection map. Then

w2(M) = r∗σ2(ω1, . . . , ωn).

By the LHS Exact Sequence

· · · → H1(Λ,Z2)d0,1

2→ H2(F ,Z2)r∗→ H2(Γ,Z2)

=⇒ ker r∗ = Im d0,12 =sums of Z2-class polynomials

the above Theorem (tool to compute the 2nd SW class)


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Bieberbach groups

Bieberbach groups







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Cohomology and spectral properties

• We consider all 4-dimensional flat manifolds of diagonaltype with F ≡ Z2

2 or F ≡ Z32 and we show several

isospectral or p-isospectral pairs, with 1 ≤ p ≤ 3, havingdifferent Z2-cohomology groups and where some of themhave different lengths of closed geodesics.

• We find, for n = 5, many isospectral pairs with F ≡ Z42

having different H2(MΓ,Z2) and having the sameH1(MΓ,Z2) and the same holonomy representations. Suchexamples are not possible to obtain in dimension 4.

Example (#g1,#g4 in the CARAT (Aachen) list)


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Bieberbach groups

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Cohomology and spectral properties

• We determine the Z2-cohomology of all GHW manifolds indimensions 3, 4 and 5, listing all isospectral classes.

GHW=generalized Hantzsche-Wendt manifolds:dimension n flat manifolds having holonomy group Zn−1

2

HW = orientable GHWthey are all rational homology spheres

Table: Cohomology classes of GHW manifolds in dimension 5.

bettiZ21 bettiZ2

2 List of manifolds4 5 7, 14, 16, 21, 58, 61, 67, 69, 74, 77, 84, 85, 104, 105, 106,

107, 112, 115, 117,118, 121, 1224 6 2, 3, 4, 6, 8, 9, 10, 11, 13, 17, 18, 20, 22, 23, 36, 37, 38, 39

40, 41, 44, 45, 50, 51, 52, 53, 57, 59, 60, 62, 65, 66, 6871, 73, 75, 76, 78, 80, 81, 82, 83, 90, 91, 92, 93, 96, 97, 98,100, 101, 102, 109, 111, 113, 114, 116, 119

4 7 1, 5, 12, 15, 19, 28, 29, 30, 31, 42, 43, 46, 47, 48, 49, 54,55, 56, 63, 64, 70, 72, 79, 86, 87, 88, 89, 94, 95, 99,103, 108, 110, 120, 123

5 10 24, 25, 26, 27, 32, 33, 34, 35

some isospectral manifolds have the same colors


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Bieberbach groups

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Stiefel-Whitney classes and spectral properties

• For n = 4, we exhibit p-isospectral pairs for all p,M,M ′, that have the same (Z2)-cohomology but such thatw2(M) 6= 0 and w2(M ′) = 0

See manifolds labelled (1,1,0) and (1,0,1)in the family K4:

B1 B2 B3

−1 1 x2

1 y2

1 12−1 1 z

2

1 1 12−1

1 1 1 12


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Table: Family K4

(x,y,z) bettiZ21 = betti

Z23 betti

Z22 w2 Sunada n. isospectral pairs

(0, 0, 0) 4 6 6= 0( 1 0 0

2 1 03 0 0

)(1, 0, 0) 3 4 6= 0

( 1 0 01 2 02 1 0

)♣

(1, 1, 0) 3 4 0( 1 0 0

2 1 01 2 0

)♥

(0, 1, 0) 3 4 6= 0( 1 0 0

1 2 02 1 0

)♣

(0, 0, 1) 4 6 6= 0( 1 0 0

3 0 02 1 0

)(1, 0, 1) 3 4 6= 0

( 1 0 02 1 01 2 0

)♥

(0, 1, 1) 3 4 6= 0( 1 0 0

2 1 02 0 1

)(1, 1, 1) 3 4 0

( 1 0 03 0 01 1 1

)

Sunada numbers: cs,t= number of elements in the holonomy F of MΓ,having exactly s 1’s in the diagonal (or column) and

t 12 ’s coming with those 1’s, 0 ≤ t ≤ s ≤ n.


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Bieberbach groups

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S. Console, R. Miatello and J.P. Rossetti:Z2-cohomology and spectral properties of flat manifolds ofdiagonal type,Journal of Geometry and Physics 60 (2010), 760-781

S. Console, R. Miatello and J.P. Rossetti:Second Stiefel-Whitney class and spin structures on flatmanifolds of diagonal type,AIP Conference Proceedings Volume 1360,XIX International Fall Workshop on Geometry and PhysicsPorto, (Portugal), 6-9 September 2010

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