Lecture 2: L2-Betti numbers - Department of...
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Lecture 2: L2-Betti numbersL2 algebraic topology
L2-Betti numbers and Euler characteristicsThe Hopf Conjecture and the Singer Conjecture
Lecture 2: L2-Betti numbers
Mike Davis
July 4, 2006
Mike Davis Lecture 2: L2-Betti numbers
Lecture 2: L2-Betti numbersL2 algebraic topology
L2-Betti numbers and Euler characteristicsThe Hopf Conjecture and the Singer Conjecture
1 L2 algebraic topologyProperties
2 L2-Betti numbers and Euler characteristicsAtiyah’s FormulaExamplesPoincare duality
3 The Hopf Conjecture and the Singer ConjectureThe Euler Characteristic ConjectureSinger Conjecture
Mike Davis Lecture 2: L2-Betti numbers
Lecture 2: L2-Betti numbersL2 algebraic topology
L2-Betti numbers and Euler characteristicsThe Hopf Conjecture and the Singer Conjecture
Properties
Last time
L2Ci (X ) := {ϕ : {i-cells} → R |∑
ϕ(e)2 < ∞}
L2Hi (X ) := Zi (X )/Bi (X )
L2Hi (X ) := Z i (X )/B i (X ).
Mike Davis Lecture 2: L2-Betti numbers
Lecture 2: L2-Betti numbersL2 algebraic topology
L2-Betti numbers and Euler characteristicsThe Hopf Conjecture and the Singer Conjecture
Properties
A harmonic 1-cycle
1/41/4
1/2
1/2
1/4
1/4
1/2
1/4
1/4
1
1/4
1/4
1/2
1 + 4
(1
2
)2
+ 8
(1
4
)2
+ · · · = 1 +∑
2n+1
(1
2
)2n
= 1 +∑ (
1
2
)n−1
< ∞
Mike Davis Lecture 2: L2-Betti numbers
Lecture 2: L2-Betti numbersL2 algebraic topology
L2-Betti numbers and Euler characteristicsThe Hopf Conjecture and the Singer Conjecture
Properties
L2 algebraic topology
(X ,Y ) a pair of CW-complexes. Γ acts properly and cellularly onX . Y is a Γ-stable subcx.Reduced L2-(co)homology groups L2Hi (X ,Y ) are defined in theusual manner by completing of Ci (X ,Y ). Versions of most of theEilenberg-Steenrod Axioms hold for L2H∗(X ,Y ).Some standard properties.
Functorality
f : (X1,Y1) → (X2,Y2) a Γ-map. There is an induced mapf∗ : L2Hi (X1,Y1) → L2Hi (X2,Y2) giving a functor from pairs ofΓ-complexes and Γ-homotopy classes of maps to Hilbert Γ-modules.
Mike Davis Lecture 2: L2-Betti numbers
Lecture 2: L2-Betti numbersL2 algebraic topology
L2-Betti numbers and Euler characteristicsThe Hopf Conjecture and the Singer Conjecture
Properties
Properties
Exact sequence of a pair
The sequence,
→ L2Hi (Y ) → L2Hi (X ) → L2Hi (X ,Y ) →
is weakly exact.
Excision
U is a Γ-stable subset of Y s.t. Y − U is a subcx. Then(X − U,Y − U) ↪→ (X ,Y ) induces an iso:
L2Hi (X − U,Y − U) ∼= L2Hi (X ,Y ).
Mike Davis Lecture 2: L2-Betti numbers
Lecture 2: L2-Betti numbersL2 algebraic topology
L2-Betti numbers and Euler characteristicsThe Hopf Conjecture and the Singer Conjecture
Properties
Mayer-Vietoris sequence
X = X1 ∪ X2, with X1, X2 Γ-stable subcxes. The M-V sequence,
→ L2Hi (X1 ∩ X2) → L2Hi (X1)⊕ L2Hi (X2) → L2Hi (X ) →
is weakly exact.
Mike Davis Lecture 2: L2-Betti numbers
Lecture 2: L2-Betti numbersL2 algebraic topology
L2-Betti numbers and Euler characteristicsThe Hopf Conjecture and the Singer Conjecture
Properties
Twisted products
H a subgp of Γ and Y is a space with H-action.The twisted product:
Γ×H Y := (Γ× Y )/H
where the H-action is defined by h · (g , y) = (gh−1, hy).
It is a left Γ-space and a Γ-bundle over Γ/H. Since Γ/H is discrete,Γ×H Y is a disjoint union of copies of Y , one for each element ofΓ/H. If Y is an H-CW-complex, then Γ×H Y is a Γ-CW-complex.
Mike Davis Lecture 2: L2-Betti numbers
Lecture 2: L2-Betti numbersL2 algebraic topology
L2-Betti numbers and Euler characteristicsThe Hopf Conjecture and the Singer Conjecture
Properties
More properties
Twisted products and the induced representation
L2Hi (Γ×H Y ) ∼= IndΓH(L2Hi (Y )).
Kunneth Formula
Γ = Γ1 × Γ2 and Xj is a Γj -CW-cx , j = 1, 2. Then X1 × X2 is aΓ-CW-cx and
L2Hk(X1 × X2) ∼=∑
i+j=k
L2Hi (X1)⊗L2Hj(X2),
where ⊗ denotes the completed tensor product.
Mike Davis Lecture 2: L2-Betti numbers
Lecture 2: L2-Betti numbersL2 algebraic topology
L2-Betti numbers and Euler characteristicsThe Hopf Conjecture and the Singer Conjecture
Properties
Reduced homology of Euclidean space
Example
We know for X = E1 (= R) with standard action of Γ = Z that
L2Hk(E1) = 0 for k = 0, 1.
By the Kunneth Formula,
L2Hk(En) = 0, ∀k.
Mike Davis Lecture 2: L2-Betti numbers
Lecture 2: L2-Betti numbersL2 algebraic topology
L2-Betti numbers and Euler characteristicsThe Hopf Conjecture and the Singer Conjecture
Atiyah’s FormulaExamplesPoincare duality
Review of dimΓ( )
The von Neumann dimension of V (or its Γ-dimension) is definedby
dimΓ(V ) := trΓ(pV ).
Properties
dimΓ(V ) ∈ [0,∞) and dimΓ(V ) = 0 iff V = 0.
Γ = {1} =⇒ dimΓ(V ) = dimR(V ).
dimΓ(L2(Γ)) = 1.
dimΓ(V ⊕W ) = dimΓ(V ) + dimΓ(W ).
Mike Davis Lecture 2: L2-Betti numbers
Lecture 2: L2-Betti numbersL2 algebraic topology
L2-Betti numbers and Euler characteristicsThe Hopf Conjecture and the Singer Conjecture
Atiyah’s FormulaExamplesPoincare duality
More properties of dimΓ( )
f : V → W a map of Hilbert Γ-modules, then
dimΓ(V ) = dimΓ(Ker f ) + dimΓ(Im f )
= dimΓ(Ker f ) + dimΓ(Im f ∗).
H ⊂ Γ index m =⇒ dimH(V ) = m dimΓ(V ).
Γ finite =⇒ dimΓ(V ) = 1|Γ| dim(V ).
H ⊂ Γ, W then
dimΓ(IndΓH(W )) = dimH(W ).
dimΓ1×Γ2(V1⊗V2) = dimΓ1(V1) dimΓ2(V2).
Mike Davis Lecture 2: L2-Betti numbers
Lecture 2: L2-Betti numbersL2 algebraic topology
L2-Betti numbers and Euler characteristicsThe Hopf Conjecture and the Singer Conjecture
Atiyah’s FormulaExamplesPoincare duality
Definition
The i th L2-Betti number of X is:
L2bi (X ; Γ) := dimΓ L2Hi (X ).
If X is contractible (and the Γ-action is proper andcocompact), then L2bi (X ; Γ) is an invariant of Γ.
Denote it L2bi (Γ) and call it the L2-Betti number of Γ.
Mike Davis Lecture 2: L2-Betti numbers
Lecture 2: L2-Betti numbersL2 algebraic topology
L2-Betti numbers and Euler characteristicsThe Hopf Conjecture and the Singer Conjecture
Atiyah’s FormulaExamplesPoincare duality
Properties of L2-Betti numbers
L2bi (X ; Γ) = 0 =⇒ L2Hi (X ) = 0.
H ⊂ Γ index m =⇒ L2bi (X ;H) = m(L2bi (X ; Γ)).
Kunneth Formula:
L2bk(X1 × X2; Γ1 × Γ2) =∑
i+j=k
L2bi (X1; Γ1)L2bj(X2; Γ2)
Suppose Γ1, Γ2 both infinite. Then
L2bi (Γ1 ∗ Γ2) =
{L2bi (Γ1) + L2bi (Γ2), if i > 1,
L2b1(Γ1) + L2b1(Γ2)− 1 if i = 1
(Mayer-Vietoris sequence).
Mike Davis Lecture 2: L2-Betti numbers
Lecture 2: L2-Betti numbersL2 algebraic topology
L2-Betti numbers and Euler characteristicsThe Hopf Conjecture and the Singer Conjecture
Atiyah’s FormulaExamplesPoincare duality
Orbihedral Euler characteristic
χorb(X/Γ) :=∑
orbits of cells
(−1)dim c
|Γc |∈ Q,
where |Γc | is the order of the stabilizer of the cell c .
If Γacts freely, then χorb(X/Γ) is the ordinary Eulercharacteristic χ(X/Γ).
If H ⊂ Γ is index m, then χorb(X/H) = mχorb(X/Γ).
χorb(X1/Γ1 × X2/Γ2) = χorb(X1/Γ1)χorb(X2/Γ2)
Mike Davis Lecture 2: L2-Betti numbers
Lecture 2: L2-Betti numbersL2 algebraic topology
L2-Betti numbers and Euler characteristicsThe Hopf Conjecture and the Singer Conjecture
Atiyah’s FormulaExamplesPoincare duality
Atiyah’s Formula
The L2-Euler characteristic
L2χ(X ; Γ) :=∞∑i=0
(−1)iL2bi (X ; Γ).
Theorem (Atiyah)
χorb(X/Γ) = L2χ(X ; Γ).
Lemma
C∗ a chain complex of Hilbert Γ-modules. Hi (C∗) = reducedhomology. Then∑
i
(−1)i dimΓ Ci =∑
i
(−1)i dimΓHi (C∗).
Mike Davis Lecture 2: L2-Betti numbers
Lecture 2: L2-Betti numbersL2 algebraic topology
L2-Betti numbers and Euler characteristicsThe Hopf Conjecture and the Singer Conjecture
Atiyah’s FormulaExamplesPoincare duality
Proof of Lemma
Proof. Put Zi := Ker(Ci → Ci−1), Bi := Im(Ci+1 → Ci ) and
ci := dimΓ(Ci ), hi := dimΓ(Hi (C∗))
zi := dimΓ(Zi ), bi := dimΓ(Bi ).
Weak short exact sequences:
0 →Zi → Ci → Bi−1 → 0
0 →Bi → Zi → Hi → 0.
So, ci = zi + bi−1 and zi = hi + bi .
Mike Davis Lecture 2: L2-Betti numbers
Lecture 2: L2-Betti numbersL2 algebraic topology
L2-Betti numbers and Euler characteristicsThe Hopf Conjecture and the Singer Conjecture
Atiyah’s FormulaExamplesPoincare duality
∑(−1)ici =
∑(−1)i (zi + bi−1) =
∑(−1)i (hi + bi + bi−1)
=∑
(−1)ihi .
Proof of Atiyah’s Formula.
ci := dimΓ(Ci (X )) =∑
orbits of i-cells
dimΓ(L2(Γ/Γc))
=∑ 1
|Γc |.
So,∑
(−1)ici = χorb(X/Γ) and Lemma =⇒ Formula.
Mike Davis Lecture 2: L2-Betti numbers
Lecture 2: L2-Betti numbersL2 algebraic topology
L2-Betti numbers and Euler characteristicsThe Hopf Conjecture and the Singer Conjecture
Atiyah’s FormulaExamplesPoincare duality
Free groups
Example
Y = a figure 8. T its universal cover (a regular 4-valent tree).F2 = free group of rank 2.L2b0(T ;F2) = 0 (because F2 is infinite). So,
L2b1(T ;F2) = −L2χ(T ;F2) = −χ(Y ) = 1.
s-1 1
t-1
s
t
Mike Davis Lecture 2: L2-Betti numbers
Lecture 2: L2-Betti numbersL2 algebraic topology
L2-Betti numbers and Euler characteristicsThe Hopf Conjecture and the Singer Conjecture
Atiyah’s FormulaExamplesPoincare duality
Surface groups
Example
Y = closed surface of genus g (> 0), X its univ cover, Γ = π1(Y ).Showed previously L2b0 = 0 = L2b2. So,
L2b1(X ; Γ) = −L2χ(X : Γ) = −χ(Y ) = 2g − 2
Notation
BΓ := K (Γ, 1) and EΓ := its univ cover.
Mike Davis Lecture 2: L2-Betti numbers
Lecture 2: L2-Betti numbersL2 algebraic topology
L2-Betti numbers and Euler characteristicsThe Hopf Conjecture and the Singer Conjecture
Atiyah’s FormulaExamplesPoincare duality
2-dimensional groups
Example
Suppose BΓ is a finite 2-dim cx (e.g., Γ is a small cancellation gp).
g = #{1-cells} = #{generators}r = #{2-cells} = #{relations}
χ(Γ) = 1− g + r and L2χ(Γ) = L2b2(Γ)− L2b1(Γ). So,
r ≥ g =⇒ χ(Γ) > 0 =⇒ L2b2(Γ) > 0
r < g − 1 =⇒ χ(Γ) < 0 =⇒ L2b1(Γ) > 0.
Mike Davis Lecture 2: L2-Betti numbers
Lecture 2: L2-Betti numbersL2 algebraic topology
L2-Betti numbers and Euler characteristicsThe Hopf Conjecture and the Singer Conjecture
Atiyah’s FormulaExamplesPoincare duality
Deficiency of a finitely presented group
Definition
The deficiency of a presentation of Γ is g − r =#{generators} −#{relations}. The deficiency of a gp Γ, denoteddef(Γ), is the maximum of g − r over all presentations of Γ.
Let Y be presentation cx with χ(Y ) minimum. Since Y can becompleted to BΓ by attaching cells of dim ≥ 3, b1(Y ) = b1(Γ) andb2(Y ) ≥ b2(Γ).So, def(Γ) = 1− χ(Y ) = b1(Y ))− b2(Y ) ≤ b1(Γ)− b2(Γ).Similarly, def(Γ) ≤ L2b1(Γ)− L2b2(Γ) + 1. So, for example,L2b1(Γ) = 0 =⇒ def(Γ) ≤ 1.
Mike Davis Lecture 2: L2-Betti numbers
Lecture 2: L2-Betti numbersL2 algebraic topology
L2-Betti numbers and Euler characteristicsThe Hopf Conjecture and the Singer Conjecture
Atiyah’s FormulaExamplesPoincare duality
Poincare duality
Theorem
X n an n-mfld, then L2bi (Xn; Γ) = L2bn−i (X
n; Γ).
There is a nonsingular pairing:
L2Hi (X )⊗ L2Hn−i (X ) → R,
defined by α⊗ β → 〈α ∪ β, [X ]〉.Point is the cup product of 2 L2-classes is L1, [X ] is a boundedclass and you can evaluate an L1-cohomology class on a boundedhomology class.
Mike Davis Lecture 2: L2-Betti numbers
Lecture 2: L2-Betti numbersL2 algebraic topology
L2-Betti numbers and Euler characteristicsThe Hopf Conjecture and the Singer Conjecture
Atiyah’s FormulaExamplesPoincare duality
Remark
Suppose X is the univ cover of a Poincare duality cx. Sameargument shows L2Hi (X ) ∼= L2Hn−i (X ).
Example
Suppose Γ is a PD2-gp (i.e., the fund gp of a 2-dim PD cx whoseuniv cover X is contractible). This implies Γ is infinite. So,L2b0 = 0. By Poincare duality L2b2 = 0. So,χ(Γ) = χ(X/Γ) = −L2b1(X ; Γ) ≤ 0. So,
b1(Γ)− 2 = −b0(Γ) + b1(Γ)− b2(Γ) ≥ 0.
So, b1(Γ) = rk(Γab) ≥ 2. (This fact was important in proof thatPD2-gps are surface gps.)
Mike Davis Lecture 2: L2-Betti numbers
Lecture 2: L2-Betti numbersL2 algebraic topology
L2-Betti numbers and Euler characteristicsThe Hopf Conjecture and the Singer Conjecture
The Euler Characteristic ConjectureSinger Conjecture
The Euler Characteristic Conjecture
A space Y is aspherical if its univ cover is contractible.
Example
A complete Riemannian mfld M of nonpositive sectional curvatureis aspherical. (Pf: exp : TxM → M is a diffeomorphism.)
Conjecture
If M2k is a closed aspherical mfld, then (−1)kχ(M2k) ≥ 0.
In nonpositively curved context this is called the Chern–HopfConj or Hopf Conj.
Conj doesn’t follow from the Gauss–Bonnet Theorem.
Mike Davis Lecture 2: L2-Betti numbers
Lecture 2: L2-Betti numbersL2 algebraic topology
L2-Betti numbers and Euler characteristicsThe Hopf Conjecture and the Singer Conjecture
The Euler Characteristic ConjectureSinger Conjecture
Euler Char Conj
For odd-dimensional mflds, χ = 0. (Pf: Poincare duality).
Conj true for surfaces: M2 is aspherical iff χ(M2) ≤ 0. (Pf:χ = 0 ⇐⇒ univ cover = E2. χ < 0 ⇐⇒ univ cover = H2.)
Conj true for product of surfaces: if M2k is product of ksurfaces of nonpositive Euler char, then (−1)kM2k ≥ 0(because χ is multiplicative for products).
True for closed hyperbolic mflds and other locally symmetricmflds.
Mike Davis Lecture 2: L2-Betti numbers
Lecture 2: L2-Betti numbersL2 algebraic topology
L2-Betti numbers and Euler characteristicsThe Hopf Conjecture and the Singer Conjecture
The Euler Characteristic ConjectureSinger Conjecture
Other versions
Conjecture
Suppose Γ acts properly and cocompactly on contractible M2k
(i.e., M2k/Γ is an aspherical orbifold). Then
(−1)kχorb(M2k/Γ) ≥ 0.
Conjecture
Suppose Γ is a PD2k -gp. Then (−1)kχ(Γ) ≥ 0.
Mike Davis Lecture 2: L2-Betti numbers
Lecture 2: L2-Betti numbersL2 algebraic topology
L2-Betti numbers and Euler characteristicsThe Hopf Conjecture and the Singer Conjecture
The Euler Characteristic ConjectureSinger Conjecture
The Dodziuk–Singer Conjecture
Conjecture
Mn a contractible mfld with cocompact proper Γ-action. Then
L2bi (Mn; Γ) = 0, ∀i 6= n
2.
If n is odd, this means all L2-Betti numbers are 0.
Theorem
Singer Conj. =⇒ Euler Char. Conj.
Mike Davis Lecture 2: L2-Betti numbers
Lecture 2: L2-Betti numbersL2 algebraic topology
L2-Betti numbers and Euler characteristicsThe Hopf Conjecture and the Singer Conjecture
The Euler Characteristic ConjectureSinger Conjecture
Proof.
Suppose n = 2k, Γ = π1(Mn). Singer Conj =⇒ only L2bk 6= 0.
Atiyah’s Formula gives:
(−1)kL2bk(M2k ; Γ) = χorb(M2k/Γ).
So, (−1)kχorb(M2k/Γ) ≥ 0.
Mike Davis Lecture 2: L2-Betti numbers