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Lecture 2: L 2 -Betti numbers L 2 algebraic topology L 2 -Betti numbers and Euler characteristics The Hopf Conjecture and the Singer Conjecture Lecture 2: L 2 -Betti numbers Mike Davis July 4, 2006 Mike Davis Lecture 2: L 2 -Betti numbers
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Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology L2-Betti numbers

Oct 16, 2020

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Page 1: Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology L2-Betti numbers

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

Page 2: Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology 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

Page 3: Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology 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

Page 4: Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology 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

Page 5: Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology 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

Page 6: Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology 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

Page 7: Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology 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

Page 8: Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology 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

Page 9: Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology 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

Page 10: Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology 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

Page 11: Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology 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

Page 12: Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology 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

Page 13: Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology 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

Page 14: Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology 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

Page 15: Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology 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

Page 16: Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology 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

Page 17: Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology 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

Page 18: Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology 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

Page 19: Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology 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

Page 20: Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology 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

Page 21: Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology 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

Page 22: Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology 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

Page 23: Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology 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

Page 24: Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology 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

Page 25: Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology 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

Page 26: Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology 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

Page 27: Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology 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

Page 28: Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology 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

Page 29: Lecture 2: L2-Betti numbers - Department of Mathematicsmath.osu.edu/~davis.12/talks/Montreal/Montreal-L2.pdf · Lecture 2: L2-Betti numbers L2 algebraic topology 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