Introduction to L

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Introduction to L

²

-Betti numbers

Wolfgang Lück Bonn Germany

email wolfgang.lueck@him.uni-bonn.de http://131.220.77.52/lueck/

Düsseldorf, November 2015

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Basic motivation

Given an invariant for finiteCW-complexes, one can get much more sophisticated versions by passing to the universal covering and defining an analogue taking the action of the fundamental group into account.

We want to apply this principle to (classical)Betti numbers b_n(X):=dim_C(H_n(X;C)).

We will use the following successful approach which is essentially due toAtiyah.

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Basic motivation

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Basic motivation

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Basic motivation

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Group von Neumann algebras

Definition

Define thegroup von Neumann algebra

N(G) :=B(L²(G),L²(G))^G =CG^weak

to be the algebra of boundedG-equivariant operatorsL²(G)→L²(G).

Thevon Neumann traceis defined by

tr_N_(G):N(G)→C, f 7→ hf(e),ei_L2(G).

Example (FiniteG)

IfGis finite, thenCG=L²(G) =N(G). The trace tr_N_(G)assigns to P

g∈Gλg·gthe coefficientλe.

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Group von Neumann algebras

Definition

Example (FiniteG)

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Group von Neumann algebras

Definition

Example (FiniteG)

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Example (G=Zⁿ)

LetGbeZⁿ. LetL²(Tⁿ)be the Hilbert space ofL²-integrable functions Tⁿ→C. Fourier transform yields an isometricZⁿ-equivariant

isomorphism

L²(Zⁿ)−^∼⁼→L²(Tⁿ).

LetL^∞(Tⁿ)be the Banach space of essentially bounded measurable functionsf:Tⁿ→C. We obtain an isomorphism

L^∞(Tⁿ)−→ N^∼⁼ (Zⁿ), f 7→M_f

whereM_f:L²(Tⁿ)→L²(Tⁿ)is the boundedZⁿ-operatorg 7→g·f. Under this identification the trace becomes

tr_N₍_Zⁿ₎:L^∞(Tⁿ)→C, f 7→

Z

Tⁿ

fdµ.

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von Neumann dimension

Definition (Finitely generated Hilbert module)

Afinitely generated HilbertN(G)-moduleV is a Hilbert spaceV together with a linear isometricG-action such that there exists an isometric linearG-embedding ofV intoL²(G)ⁿfor somen≥0.

Amap of finitely generated HilbertN(G)-modulesf:V →W is a boundedG-equivariant operator.

Definition (von Neumann dimension)

LetV be a finitely generated HilbertN(G)-module. Choose a G-equivariant projectionp:L²(G)ⁿ →L²(G)ⁿwith im(p)∼=N(G)V. Define thevon Neumann dimensionofV by

dim_N_(G)(V):=tr_N_(G)(p) :=

n

X

i=1

tr_N_(G)(p_i,i) ∈[0,∞).

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von Neumann dimension

dim_N_(G)(V):=tr_N_(G)(p) :=

n

X

i=1

tr_N_(G)(p_i,i) ∈[0,∞).

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von Neumann dimension

dim_N_(G)(V):=tr_N_(G)(p) :=

n

X

i=1

tr_N_(G)(p_i,i) ∈[0,∞).

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Example (FiniteG)

For finiteGa finitely generated HilbertN(G)-moduleV is the same as a unitary finite dimensionalG-representation and

dim_N_(G)(V) = 1

|G|·dim_C(V).

Example (G=Zⁿ)

LetGbeZⁿ. LetX ⊂Tⁿbe any measurable set with characteristic functionχ_X ∈L^∞(Tⁿ). LetMχ_X:L²(Tⁿ)→L²(Tⁿ)be the

Zⁿ-equivariant unitary projection given by multiplication withχ_X. Its imageV is a HilbertN(Zⁿ)-module with

dim_N_(Zⁿ₎(V) =vol(X).

In particular eachr ∈[0,∞)occurs asr =dim_N₍_Zⁿ₎(V).

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Example (FiniteG)

For finiteGa finitely generated HilbertN(G)-moduleV is the same as a unitary finite dimensionalG-representation and

dim_N_(G)(V) = 1

|G|·dim_C(V).

Example (G=Zⁿ)

LetGbeZⁿ. LetX ⊂Tⁿbe any measurable set with characteristic functionχ_X ∈L^∞(Tⁿ). LetMχ_X:L²(Tⁿ)→L²(Tⁿ)be the

Zⁿ-equivariant unitary projection given by multiplication withχ_X. Its imageV is a HilbertN(Zⁿ)-module with

dim_N_(Zⁿ₎(V) =vol(X).

In particular eachr ∈[0,∞)occurs asr =dim_N₍_Zⁿ₎(V).

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L

²

-homology and L

²

-Betti numbers

Definition (L²-homology andL²-Betti numbers)

LetX be a connectedCW-complex of finite type. LetXe be its universal covering andπ =π₁(M). Denote byC∗(Xe)itscellularZπ-chain

complex.

Define itscellularL²-chain complexto be the HilbertN(π)-chain complex

C_∗⁽²⁾(Xe):=L²(π)⊗_Z_πC_∗(Xe) =C_∗(Xe).

Define itsn-thL²-homologyto be the finitely generated Hilbert N(G)-module

H_n⁽²⁾(Xe):=ker(c_n⁽²⁾)/im(c⁽²⁾_n+1).

Define itsn-thL²-Betti number

b_n⁽²⁾(Xe) :=dim_N_(π) H_n⁽²⁾(Xe)

∈R^≥0.

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L

²

-homology and L

²

-Betti numbers

Definition (L²-homology andL²-Betti numbers)

LetX be a connectedCW-complex of finite type. LetXe be its universal covering andπ =π₁(M). Denote byC∗(Xe)itscellularZπ-chain

complex.

Define itscellularL²-chain complexto be the HilbertN(π)-chain complex

C_∗⁽²⁾(Xe):=L²(π)⊗_Z_πC_∗(Xe) =C_∗(Xe).

Define itsn-thL²-homologyto be the finitely generated Hilbert N(G)-module

H_n⁽²⁾(Xe):=ker(c_n⁽²⁾)/im(c⁽²⁾_n+1).

Define itsn-thL²-Betti number

b_n⁽²⁾(Xe) :=dim_N_(π) H_n⁽²⁾(Xe)

∈R^≥0.

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Theorem (Main properties of Betti numbers)

Let X and Y be connected CW -complexes of finite type.

Homotopy invariance

If X and Y are homotopy equivalent, then b_n(X) =b_n(Y);

Euler-Poincaré formula If X is finite, we have

χ(X) =X

n≥0

(−1)ⁿ·b_n(X);

Poincaré duality

Let M be an oriented closed manifold of dimension d . Then b_n(M) =b_d−n(M);

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Theorem (Main properties ofL²-Betti numbers) Let X and Y be connected CW -complexes of finite type.

Homotopy invariance

If X and Y are homotopy equivalent, then b⁽²⁾_n (Xe) =b_n⁽²⁾(Ye);

Euler-Poincaré formula If X is finite, we have

χ(X) =X

n≥0

(−1)ⁿ·b⁽²⁾_n (Xe);

Poincaré duality

Let M be an oriented closed manifold of dimension d . Then b_n⁽²⁾(M) =e b_d−n⁽²⁾ (M);e

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Theorem (Continued) Künneth formula

bn(X ×Y) = X

p+q=n

bp(X)·bq(Y);

Zero-th L²-Betti number We have

b₀(X) =1;

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b_n⁽²⁾(X^×Y) = X

p+q=n

b_p⁽²⁾(Xe)·b⁽²⁾_q (Ye);

b⁽²⁾₀ (Xe) = 1

|π|;

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b_n⁽²⁾(X^×Y) = X

p+q=n

b_p(X)·b_q(Y);

b⁽²⁾₀ (Xe) = 1

|π|; Finite coverings

If X →Y is a finite covering with d sheets, then b⁽²⁾_n (Xe) =d·b_n⁽²⁾(Ye).

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Some computations and results

Example (Finiteπ) Ifπis finite then

b⁽²⁾_n (Xe) = bn(Xe)

|π| .

Example (Finite self coverings)

We get for a connectedCW-complexX of finite type, for which there is a selfcoveringX →X withd-sheets for some integerd ≥2,

b_n⁽²⁾(Xe) =0 forn≥0.

This implies for each connectedCW-complexY of finite type b⁽²⁾_n (S^¹×Y) =0 forn≥0.

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Some computations and results

|π| .

b_n⁽²⁾(Xe) =0 forn≥0.

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Some computations and results

|π| .

b_n⁽²⁾(Xe) =0 forn≥0.

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Theorem (S¹-actions,Lück)

Let M be a connected compact manifold with S¹-action. Suppose that for one (and hence all) x ∈X the map S¹→M, z 7→zx isπ1-injective.

Then we get for all n≥0

b_n⁽²⁾(M) =e 0.

Theorem (S¹-actions on aspherical manifolds,Lück)

Let M be an aspherical closed manifold with non-trivial S¹-action.

Then

1 The action has no fixed points;

2 The map S¹→M, z 7→zx isπ1-injective for x ∈M;

3 b_n⁽²⁾(M) =e 0for n≥0andχ(M) =0.

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Theorem (S¹-actions,Lück)

Let M be a connected compact manifold with S¹-action. Suppose that for one (and hence all) x ∈X the map S¹→M, z 7→zx isπ1-injective.

Then we get for all n≥0

b_n⁽²⁾(M) =e 0.

Theorem (S¹-actions on aspherical manifolds,Lück)

Let M be an aspherical closed manifold with non-trivial S¹-action.

Then

1 The action has no fixed points;

2 The map S¹→M, z 7→zx isπ1-injective for x ∈M;

3 b_n⁽²⁾(M) =e 0for n≥0andχ(M) =0.

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Example (L²-Betti number of surfaces)

LetF_g be the orientable closed surface of genusg ≥1.

Then|π₁(Fg)|=∞and henceb₀⁽²⁾(Ffg) =0.

By Poincaré dualityb⁽²⁾₂ (fF_g) =0.

dim(F_g) =2, we getb⁽²⁾_n (fF_g) =0 forn≥3.

The Euler-Poincaré formula shows

b₁⁽²⁾(fFg) = −χ(F_g) = 2g−2;

b_n⁽²⁾(Ff₀) = 0 for n6=1.

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Theorem (Hodge - de Rham Theorem)

Let M be an oriented closed Riemannian manifold. Put Hⁿ(M) ={ω∈Ωⁿ(M)|∆n(ω) =0}

Then integration defines an isomorphism of real vector spaces Hⁿ(M)−→^∼⁼ Hⁿ(M;R).

Corollary (Betti numbers and heat kernels) b_n(M) = lim

t→∞

Z

M

tr_R(e^−t∆ⁿ(x,x))dvol. where e^−t∆ⁿ(x,y)is the heat kernel on M.

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Theorem (L²-Hodge - de Rham Theorem,Dodziuk) Let M be an oriented closed Riemannian manifold. Put

Hⁿ₍₂₎(M) =e {ωe∈Ωⁿ(M)e |∆en(ω) =e 0, ||ω||e _L2 <∞}

Then integration defines an isomorphism of finitely generated Hilbert N(π)-modules

H₍₂₎ⁿ (M)e −→^∼⁼ H₍₂₎ⁿ (M).e

Corollary (L²-Betti numbers and heat kernels) b_n⁽²⁾(M) =e lim

t→∞

Z

F

tr_R(e^−t^∆^eⁿ(˜x,˜x))dvol.

where e^−t^∆^eⁿ(˜x,y˜)is the heat kernel onM ande F is a fundamental domain for theπ-action.

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Theorem (Hyperbolic manifolds,Dodziuk)

Let M be a hyperbolic closed Riemannian manifold of dimension d . Then:

b⁽²⁾_n (M) =e

=0 , if2n6=d;

>0 , if2n=d. Proof.

A direct computation shows thatH₍₂₎^p (H^d)is not zero if and only if 2n=d. Notice thatM is hyperbolic if and only ifMe is isometrically diffeomorphic to the standard hyperbolic spaceH^d.

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Theorem (Hyperbolic manifolds,Dodziuk)

Let M be a hyperbolic closed Riemannian manifold of dimension d . Then:

b⁽²⁾_n (M) =e

=0 , if2n6=d;

>0 , if2n=d. Proof.

A direct computation shows thatH₍₂₎^p (H^d)is not zero if and only if 2n=d. Notice thatM is hyperbolic if and only ifMe is isometrically diffeomorphic to the standard hyperbolic spaceH^d.

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Corollary

Let M be a hyperbolic closed manifold of dimension d . Then

1 If d =2m is even, then

(−1)^m·χ(M)>0;

2 M carries no non-trivial S¹-action.

Proof.

(1) We get from the Euler-Poincaré formula and the last result (−1)^m·χ(M) =b⁽²⁾_m (M)e >0.

(2) We give the proof only ford =2meven. Thenb_m⁽²⁾(M)e >0. Since Me =H^d is contractible,M is aspherical. Now apply a previous result aboutS¹-actions.

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Corollary

Let M be a hyperbolic closed manifold of dimension d . Then

1 If d =2m is even, then

(−1)^m·χ(M)>0;

2 M carries no non-trivial S¹-action.

Proof.

(1) We get from the Euler-Poincaré formula and the last result (−1)^m·χ(M) =b⁽²⁾_m (M)e >0.

(2) We give the proof only ford =2meven. Thenb_m⁽²⁾(M)e >0. Since Me =H^d is contractible,M is aspherical. Now apply a previous result aboutS¹-actions.

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Theorem (3-manifolds,Lott-Lück)

Let the3-manifold M be the connected sum M₁] . . . ]M_r of (compact connected orientable) prime3-manifolds M_j. Assume thatπ1(M)is infinite. Then

b⁽²⁾₁ (M)e = (r −1)−

r

X

j=1

1

|π₁(M_j)|−χ(M) +

{C∈π₀(∂M)|C∼=S²} ; b⁽²⁾₂ (M)e = (r −1)−

r

X

j=1

1

|π₁(M_j)| +

{C∈π₀(∂M)|C∼=S²} ; b⁽²⁾_n (M)e = 0 for n 6=1,2.

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Theorem (Mapping tori,Lück)

Let f:X →X be a cellular selfhomotopy equivalence of a connected CW -complex X of finite type. Let T_f be the mapping torus. Then

b⁽²⁾_n (Te_f) =0 for n≥0.

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Proof:

AsT_fd →T_f is ad-sheeted covering (up to homotopy), we get b⁽²⁾_n (Te_f) = b⁽²⁾_n (Tf_fd)

d .

Ifβn(X)is the number ofn-cells, then there is aCW-structure on T_fd withβ_n(T_fd) =β_n(X) +β_n−1(X).

We have

b⁽²⁾_n (Tf_fd)≤β_n(T_fd).

This implies for alld ≥1

b⁽²⁾_n (Te_f) ≤ βn(X) +βn−1(X)

d .

Taking the limit ford → ∞yields the claim.

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The fundamental square and the Atiyah Conjecture

Conjecture (Atiyah Conjecture for torsionfree finitely presented groups)

Let G be a torsionfree finitely presented group. We say that G satisfies theAtiyah Conjectureif for any closed Riemannian manifold M with π₁(M)∼=G we have for every n≥0

b_n⁽²⁾(M)e ∈Z.

All computations presented above support the Atiyah Conjecture.

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The fundamental square and the Atiyah Conjecture

b_n⁽²⁾(M)e ∈Z.

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The fundamental square and the Atiyah Conjecture

b_n⁽²⁾(M)e ∈Z.

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Thefundamental squareis given by the following inclusions of rings

ZG ^//

N(G)

D(G) ^//U(G)

U(G)is thealgebra of affiliated operators. Algebraically it is just theOre localizationofN(G)with respect to the multiplicatively closed subset of non-zero divisors.

D(G)is thedivision closureofZGinU(G), i.e., the smallest subring ofU(G)containingZGsuch that every element inD(G), which is a unit inU(G), is already a unit inD(G)itself.

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Thefundamental squareis given by the following inclusions of rings

ZG ^//

N(G)

D(G) ^//U(G)

U(G)is thealgebra of affiliated operators. Algebraically it is just theOre localizationofN(G)with respect to the multiplicatively closed subset of non-zero divisors.

D(G)is thedivision closureofZGinU(G), i.e., the smallest subring ofU(G)containingZGsuch that every element inD(G), which is a unit inU(G), is already a unit inD(G)itself.

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IfGis finite, its is given by

ZG ^//

CG

id

QG ^//CG

IfG=Z, it is given by

Z[Z] ^//

L^∞(S¹)

Q[Z]⁽⁰⁾ ^//L(S¹)

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IfGis finite, its is given by

ZG ^//

CG

id

QG ^//CG

IfG=Z, it is given by

Z[Z] ^//

L^∞(S¹)

Q[Z]⁽⁰⁾ ^//L(S¹)

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Conjecture (Atiyah Conjecture for torsionfree groups)

Let G be a torsionfree group. It satisfies theAtiyah ConjectureifD(G) is a skew-field.

A torsionfree groupGsatisfies the Atiyah Conjecture if and only if for any matrixA∈M_m,n(ZG)the von Neumann dimension

dim_N_(G) ker r_A:N(G)^m → N(G)ⁿ is an integer. In this case this dimension agrees with

dim_D(G) r_A:D(G)^m→ D(G)ⁿ .

The general version above is equivalent to the one stated before if Gis finitely presented.

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The Atiyah Conjecture implies theZero-divisor Conjecturedue to Kaplanskysaying that for any torsionfree group and field of characteristic zeroF the group ringFGhas no non-trivial zero-divisors.

There is also a version of the Atiyah Conjecture for groups with a bound on the order of its finite subgroups.

However, there exist closed Riemannian manifolds whose universal coverings have anL²-Betti number which is irrational, seeAustin, Grabowski.

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Theorem (Linnell, Schick)

1 LetC be the smallest class of groups which contains all free groups and is closed under extensions with elementary amenable groups as quotients and directed unions. Then every torsionfree group G which belongs toCsatisfies the Atiyah Conjecture.

2 If G is residually torsionfree elementary amenable, then it satisfies the Atiyah Conjecture.

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Theorem (Linnell, Schick)

1 LetC be the smallest class of groups which contains all free groups and is closed under extensions with elementary amenable groups as quotients and directed unions. Then every torsionfree group G which belongs toCsatisfies the Atiyah Conjecture.

2 If G is residually torsionfree elementary amenable, then it satisfies the Atiyah Conjecture.

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Approximation

Theorem (Approximation Theorem,Lück)

Let X be a connected CW -complex of finite type. Suppose thatπis residually finite, i.e., there is a nested sequence

π=G₀⊃G₁⊃G₂⊃. . .

of normal subgroups of finite index with∩_i≥1G_i ={1}. Let X_i be the finite[π :G_i]-sheeted covering of X associated to G_i.

Then for any such sequence(G_i)i≥1

b_n⁽²⁾(X) =e lim

i→∞

bn(X_i) [G:G_i].

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Ordinary Betti numbers are not multiplicative under finite

coverings, whereas theL²-Betti numbers are. With the expression

i→∞lim

b_n(X_i) [G:G_i],

we try to force the Betti numbers to be multiplicative by a limit process.

The theorem above says thatL²-Betti numbers areasymptotic Betti numbers. It was conjectured byGromov.

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Applications to deficiency and signature

Definition (Deficiency)

LetGbe a finitely presented group. Define itsdeficiency defi(G):=max{g(P)−r(P)}

whereP runs over all presentationsP ofGandg(P)is the number of generators andr(P)is the number of relations of a presentationP.

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Applications to deficiency and signature

Definition (Deficiency)

LetGbe a finitely presented group. Define itsdeficiency defi(G):=max{g(P)−r(P)}

whereP runs over all presentationsP ofGandg(P)is the number of generators andr(P)is the number of relations of a presentationP.

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Lemma

Let G be a finitely presented group. Then

defi(G) ≤ 1− |G|⁻¹+b₁⁽²⁾(G)−b⁽²⁾₂ (G).

Proof.

We have to show for any presentationP that

g(P)−r(P) ≤ 1−b₀⁽²⁾(G) +b₁⁽²⁾(G)−b⁽²⁾₂ (G).

LetX be aCW-complex realizingP. Then

χ(X) =1−g(P) +r(P) =b⁽²⁾₀ (Xe) +b₁⁽²⁾(X)e −b⁽²⁾₂ (Xe).

Since the classifying mapX →BGis 2-connected, we get b⁽²⁾_n (Xe) = b⁽²⁾_n (G) forn=0,1;

b⁽²⁾₂ (Xe) ≥ b⁽²⁾₂ (G).

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Lemma

Let G be a finitely presented group. Then

defi(G) ≤ 1− |G|⁻¹+b₁⁽²⁾(G)−b⁽²⁾₂ (G).

Proof.

We have to show for any presentationP that

g(P)−r(P) ≤ 1−b₀⁽²⁾(G) +b₁⁽²⁾(G)−b⁽²⁾₂ (G).

LetX be aCW-complex realizingP. Then

χ(X) =1−g(P) +r(P) =b⁽²⁾₀ (Xe) +b₁⁽²⁾(X)e −b⁽²⁾₂ (Xe).

Since the classifying mapX →BGis 2-connected, we get b⁽²⁾_n (Xe) = b⁽²⁾_n (G) forn=0,1;

b⁽²⁾₂ (Xe) ≥ b⁽²⁾₂ (G).

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Theorem (Deficiency and extensions,Lück)

Let1→H −→ⁱ G−→^q K →1be an exact sequence of infinite groups.

Suppose that G is finitely presented H is finitely generated. Then:

1 b₁⁽²⁾(G) =0;

2 defi(G)≤1;

3 Let M be a closed oriented4-manifold with G as fundamental group. Then

|sign(M)| ≤χ(M).

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The Singer Conjecture

Conjecture (Singer Conjecture)

If M is an aspherical closed manifold, then

b⁽²⁾_n (M) =e 0 if2n6=dim(M).

If M is a closed Riemannian manifold with negative sectional curvature, then

b_n⁽²⁾(M)e

=0 if2n6=dim(M);

>0 if2n=dim(M).

The computations presented above do support the Singer Conjecture.

Under certain negative pinching conditions the Singer Conjecture has been proved byBallmann-Brüning, Donnelly-Xavier, Jost-Xin.

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The Singer Conjecture

b⁽²⁾_n (M) =e 0 if2n6=dim(M).

b_n⁽²⁾(M)e

=0 if2n6=dim(M);

>0 if2n=dim(M).

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The Singer Conjecture

b⁽²⁾_n (M) =e 0 if2n6=dim(M).

b_n⁽²⁾(M)e

=0 if2n6=dim(M);

>0 if2n=dim(M).

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Because of the Euler-Poincaré formula

χ(M) =X

n≥0

(−1)ⁿ·b⁽²⁾_n (M)e

the Singer Conjecture implies the following conjecture provided thatMhas non-positive sectional curvature.

Conjecture (Hopf Conjecture)

If M is a closed Riemannian manifold of even dimension with sectional curvaturesec(M), then

(−1)^dim(M)/2·χ(M) > 0 if sec(M) < 0;

(−1)^dim(M)/2·χ(M) ≥ 0 if sec(M) ≤ 0;

χ(M) = 0 if sec(M) = 0;

χ(M) ≥ 0 if sec(M) ≥ 0;

χ(M) > 0 if sec(M) > 0.