Cohomology and Stiefel-Whitney Classes of Flat Manifolds Sergio Console Compact flat manifolds & Bieberbach groups Flat manifolds and spectra Bieberbach groups Bieberbach groups Cohomology of B’bach grps The LHS Spectral Sequence The Charlap-Vasquez method Use of the LHS Spectral Sequence 2. Stiefel-Whitney class Topology and spectra Cohomology and spectral properties Stiefel-Whitney classes and spectral properties 1 Cohomology and Stiefel-Whitney Classes of Flat Manifolds joint work with Roberto Miatello and Juan Pablo Rossetti, Cordoba, Argentina Porto – 9.09.10 Sergio Console Dipartimento di Matematica Università di Torino
Transcript
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
1
Cohomology and Stiefel-WhitneyClasses of Flat Manifolds
joint work with Roberto Miatello and Juan Pablo Rossetti,Cordoba, Argentina
Porto – 9.09.10
Sergio ConsoleDipartimento di Matematica
Università di Torino
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
2
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
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
3
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.
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
3
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.
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
4
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
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
4
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
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
4
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
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
5
Bieberbach groups
A crystallographic group is a discrete, cocompact subgroup Γof the isometry group of Rn, I(Rn) ∼= O(n) n Rn.
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
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
5
Bieberbach groups
A crystallographic group is a discrete, cocompact subgroup Γof the isometry group of Rn, I(Rn) ∼= O(n) n Rn.
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
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
6
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.
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
6
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.
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
6
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.
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
7
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
.
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
8
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.
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
8
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.
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
9
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)
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
10
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
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
11
Examples of application of the Charlap-Vasquez method
Computation of H1(Γ,Z2):
Theorem
Let M be an n-dimensional compact flat manifold with diagonalholonomy Zk
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
12
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
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
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
12
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
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
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
13
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)
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
13
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)
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
13
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)
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
14
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,
