Cohomology and Stiefel-Whitney Classes of Flat Manifolds...
Transcript of 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
1
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
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
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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.
Cohomology andStiefel-Whitney Classes
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Compact flat manifolds& Bieberbach groupsFlat manifolds and spectra
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2. Stiefel-Whitney class
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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
Cohomology andStiefel-Whitney Classes
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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
Cohomology andStiefel-Whitney Classes
of Flat Manifolds
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Compact flat manifolds& Bieberbach groupsFlat manifolds and spectra
Bieberbach groups
Bieberbach groups
Cohomology of B’bach grps
The LHS SpectralSequenceThe Charlap-Vasquezmethod
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2. Stiefel-Whitney class
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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.
Cohomology andStiefel-Whitney Classes
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Compact flat manifolds& Bieberbach groupsFlat manifolds and spectra
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The LHS SpectralSequenceThe Charlap-Vasquezmethod
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2. Stiefel-Whitney class
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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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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
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Compact flat manifolds& Bieberbach groupsFlat manifolds and spectra
Bieberbach groups
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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)
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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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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)∗.
Cohomology andStiefel-Whitney Classes
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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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2. Stiefel-Whitney class
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Stiefel-Whitney classes andspectral properties
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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
Cohomology andStiefel-Whitney Classes
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14
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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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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17
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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18
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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19
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.
Cohomology andStiefel-Whitney Classes
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21
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