WolfgangLückBonnGermanyemailwolfgang.lueck@him.uni-bonn.dehttp://131.220.77.52/lueck/Bonn,24.&26.April2018 L -Bettinumbers

L

-Betti numbers

Wolfgang Lück Bonn Germany

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

Bonn, 24. & 26. April 2018

Outline

We introduceL²-Betti numbers.

We present their basic properties and tools for their computation.

We compute theL²-Betti numbers of all 3-manifolds.

We discuss theAtiyah Conjectureand theSinger Conjecture.

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.

Examples:

Classical notion generalized version Homology with coeffi-

cients inZ

Homology with coefficients in representations

Euler characteristic∈Z Walls finiteness obstruction in K₀(Zπ)

Lefschetz numbers∈Z Generalized Lefschetz invari- ants inZπ_φ

Signature∈Z Surgery invariants inL_∗(ZG)

— torsion invariants

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

Here are two naive attempts which fail:

dim_C(Hn(eX;C)) dim_Cπ(Hn(eX;C)),

where dim_Cπ(M)for aC[π]-module could be chosen for instance as dim_C(C⊗_CGM).

The problem is thatCπ is in general not Noetherian and dim_Cπ(M) is in general not additive under exact sequences.

We will use the following successful approach which is essentially due toAtiyah[1].

Group von Neumann algebras

Throughout these lectures letGbe a discrete group.

Given a ringRand a groupG, denote byRGorR[G]thegroup ring.

Elements are formal sumsP

g∈Grg·g, whererg∈Rand only finitely many of the coefficientsr_gare non-zero.

Addition is given by adding the coefficients.

Multiplication is given by the expressiong·h:=g·hforg,h∈G (with two different meanings of·).

In generalRGis a very complicated ring.

Denote byL²(G)the Hilbert space of (formal) sumsP

g∈Gλ_g·g such thatλg∈CandP

g∈G|λ_g|²<∞.

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.

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.

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) ∈R^≥0.

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). LetM_χX: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 ∈R^≥0occurs asr =dim_N(Zⁿ₎(V).

Definition (Weakly exact)

A sequence of HilbertN(G)-modulesU −→ⁱ V −→^p W isweakly exactat V if the kernel ker(p)ofpand the closure im(i)of the image im(i)ofi agree.

A map of HilbertN(G)-modulesf:V →W is aweak isomorphismif it is injective and has dense image.

Example

The morphism ofN(Z)-Hilbert modules

M_z−1:L²(Z) =L²(S¹)→L²(Z) =L²(S¹), u(z)7→(z−1)·u(z) is a weak isomorphism, but not an isomorphism.

Theorem (Main properties of the von Neumann dimension)

1 Faithfulness

We have for a finitely generated HilbertN(G)-module V V =0⇐⇒dim_N(G)(V) =0;

2 Additivity

If0→U →V →W →0is a weakly exact sequence of finitely generated HilbertN(G)-modules, then

dim_N(G)(U) +dim_N(G)(W) =dim_N(G)(V);

3 Cofinality

Let{V_i |i∈I}be a directed system of HilbertN(G)- submodules of V , directed by inclusion. Then

dim_N(G) [

i∈I

V_i

!

=sup{dim_N(G)(V_i)|i ∈I}.

L

-homology and L

-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)⊗_ZπC_∗(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.

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 We have

χ(X) =X

n≥0

(−1)ⁿ·b⁽²⁾_n (Xe);

Poincaré duality

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

Theorem (Continued) Künneth formula

b_n⁽²⁾(X^×Y) = X

p+q=n

b_p⁽²⁾(Xe)·b⁽²⁾_q (Ye);

Zero-th L²-Betti number We have

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).

Example (Finiteπ) Ifπis finite then

b⁽²⁾_n (Xe) = bn(Xe)

|π| .

Example (S¹)

Consider theZ-CW-complexSf¹. We get forC∗⁽²⁾(Sf¹) . . .→0→L²(Z)−−−→^Mz−1 L²(Z)→0→. . . and henceH_n⁽²⁾(Sf¹) =0 andb⁽²⁾_n (Sf¹) =0 for all≥0.

Example (π =Z^d)

LetX be a connectedCW-complex of finite type with fundamental groupZ^d. LetC[Z^d]⁽⁰⁾be the quotient field of the commutative integral domainC[Z^d]. Then

b⁽²⁾_n (Xe) =dim

C[Z^d]⁽⁰⁾

C[Z^d]⁽⁰⁾⊗_Z[Zd]H_n(Xe)

Obviously this implies

b_n⁽²⁾(Xe)∈Z.

For a discrete groupGwe can consider more generally any free finiteG-CW-complexX which is the same as aG-covering X →X over a finiteCW-complexX. (Actually proper finite G-CW-complex suffices.)

The universal coveringp:Xe →X over a connected finite CW-complex is a special case forG=π1(X).

Then one can apply the same construction to the finite free ZG-chain complexC_∗(X). Thus we obtain the finitely generated HilbertN(G)-module

H_n⁽²⁾(X;N(G)):=H_n⁽²⁾(L²(G)⊗_ZGC_∗(X)), and define

b⁽²⁾_n (X;N(G)):=dim_N(G) H_n⁽²⁾(X;N(G))

∈R^≥0.

Leti:H →Gbe an injective group homomorphism andC∗be a finite freeZH-chain complex.

Theni∗C∗ :=ZG⊗_ZHC∗ is a finite freeZG-chain complex.

We have the following formula

dim_N(G) H_n⁽²⁾(L²(G)⊗_ZGi∗C∗)

=dim_N(H) H_n⁽²⁾(L²(H)⊗_ZHC∗) .

Lemma

If X is a finite free H-CW -complex, then we get b_n⁽²⁾(i∗X;N(G)) =b_n⁽²⁾(X;N(H)).

The corresponding statement is wrong if we drop the condition thati is injective.

An example comes fromp:Z→ {1}andXe =Sf¹since then p∗Sf¹=S¹and we have forn=0,1

b⁽²⁾_n (Sf¹;N(Z)) =b_n⁽²⁾(Sf¹) =0, and

b⁽²⁾_n (p_∗Sf¹;N({1})) =b_n(S¹) =1.

The L

-Mayer Vietoris sequence

Lemma Let 0→C∗⁽²⁾

i∗⁽²⁾

−−→D∗⁽²⁾ p∗⁽²⁾

−−→E∗⁽²⁾ →0be a weakly exact sequence of finite HilbertN(G)-chain complexes.

Then there is a long weakly exact sequence of finitely generated HilbertN(G)-modules

· · · ^δ

(2)

−−→n+1 H_n⁽²⁾(C∗⁽²⁾) ^H

(2) n (i∗⁽²⁾)

−−−−−→H_n⁽²⁾(D∗⁽²⁾) ^H

(2) n (p⁽²⁾∗ )

−−−−−−→H_n⁽²⁾(E∗⁽²⁾)

δ⁽²⁾_n

−−→H_n−1⁽²⁾ (C_∗⁽²⁾) ^H

(2) n−1(i∗⁽²⁾)

−−−−−−→H_n−1⁽²⁾ (D_∗⁽²⁾)

H_n−1⁽²⁾ (p∗⁽²⁾)

−−−−−−→H_n−1⁽²⁾ (E_∗⁽²⁾) ^δ

(2)

−−−n−1→ · · ·.

Lemma Let

X₀ ^//

X₁

X₂ ^//X

be a cellular G-pushout of finite free G-CW -complexes, i.e., a

G-pushout, where the upper arrow is an inclusion of a pair of free finite G-CW -complexes and the left vertical arrow is cellular.

Then we obtain a long weakly exact sequence of finitely generated HilbertN(G)-modules

· · · →H_n⁽²⁾(X₀;N(G))→H_n⁽²⁾(X₁;N(G))⊕H_n⁽²⁾(X₂;N(G))

→H_n⁽²⁾(X;N(G))→H_n−1⁽²⁾ (X₀;N(G))

→H_n−1⁽²⁾ (X₁;N(G))⊕H_n−1⁽²⁾ (X₂;N(G))→H_n−1⁽²⁾ (X;N(G))→ · · ·.

Proof.

From the cellularG-pushout we obtain an exact sequence of ZG-chain complexes

0→C_∗(X₀)→C_∗(X₁)⊕C_∗(X₂)→C_∗(X)→0.

It induces an exact sequence of finite HilbertN(G)-chain complexes

0→L²(G)⊗_ZGC∗(X₀)→L²(G)⊗_ZGC∗(X₁)⊕L²(G)⊗_ZGC∗(X₂)

→L²(G)⊗_ZGC∗(X)→0.

Now apply the previous result.

Definition (L²-acyclic)

A finite (not necessarily connected)CW-complexX is called L²-acyclic, ifb_n⁽²⁾(C) =e 0 holds for everyC∈π₀(X)andn∈Z.

IfX is a finite (not necessarily connected)CW-complex, we define b⁽²⁾_n (Xe):= X

C∈π₀(X)

b_n⁽²⁾(C)e ∈R^≥0.

Definition (π1-injective)

A mapX →Y is calledπ₁-injective, if for every choice of base point in X the induced map on the fundamental groups is injective.

Consider a cellular pushout of finiteCW-complexes X₀ ^//

X₁

X₂ ^//X

such that each of the mapsX_i →X isπ₁-injective.

Lemma

We get under the assumptions above for any n ∈Z If X₀is L²-acyclic, then

b_n⁽²⁾(Xe) =b⁽²⁾_n (Xe₁) +b_n⁽²⁾(Xf₂).

If X₀, X₁and X₂are L²-cyclic, then X is L²-acyclic.

Proof.

Without loss of generality we can assume thatX is connected.

By pulling back the universal coveringXe →X toX_i, we obtain a cellularπ =π₁(X)-pushout

X₀ ^//

X₁

X₂ ^//Xe

Notice thatX_i is in general not the universal covering ofX_i.

Proof continued.

Because of the associated long exactL²-sequence and the weak exactness of the von Neumann dimension, it suffices to show for n∈Zandi=1,2

H_n⁽²⁾(X₀;N(π)) = 0;

b_n⁽²⁾(X_i;N(π)) = b⁽²⁾_n (Xe_i).

This follows fromπ₁-injectivity, the lemma above aboutL²-Betti numbers and induction, the assumption thatX₀isL²-acyclic, and the faithfulness of the von Neumann dimension.

Some computations and results

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 that S¹×Y isL²-acyclic.

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.

Since dim(F_g) =2, we getb_n⁽²⁾(fF_g) =0 forn≥3.

The Euler-Poincaré formula shows

b⁽²⁾₁ (Ffg) = −χ(F_g) =2g−2;

b⁽²⁾_n (fF₀) = 0 for n6=1.

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 M is L²-acyclic.

Proof.

Each of theS¹-orbitsS¹/H inMsatisfiesS¹/H ∼=S¹. Now use induction over the number of cellsS¹/H_i×Dⁿand a previous result usingπ1-injectivity and the vanishing of theL²-Betti numbers of spaces of the shapeS¹×X.

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.

Proof.

The hard part is to show that the second assertion holds, sinceMis aspherical. Then the first assertion is obvious and the third assertion follows from the previous theorem.

Theorem (L²-Hodge - de Rham Theorem,Dodziuk[2]) Let M be a 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.

Theorem (hyperbolic manifolds,Dodziuk[3])

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.

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.

Theorem (3-manifolds,Lott-Lück [7])

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.

Proof.

We have already explained why a closed hyperbolic 3-manifold is L²-acyclic.

One of the hard parts of the proof is to show that this is also true for any hyperbolic 3-manifold with incompressible toral boundary.

Recall that these have finite volume.

One has to introduce appropriate boundary conditions and Sobolev theory to write down the relevant analyticL²-deRham complexes andL²-Laplace operators.

A key ingredient is the decomposition of such a manifold into its core and a finite number of cusps.

Proof continued.

This can be used to write theL²-Betti number as an integral over a fundamental domainF of finite volume, where the integrand is given by data depending onIH³only:

b⁽²⁾_n (M) =e lim

t→∞

Z

F

tr_R(e^−t∆en(˜x,˜x))dvol.

SinceH³has a lot of symmetries, the integrand does not depend onx˜and is a constantCndepending only onIH³.

Hence we get

b_n⁽²⁾(M) =e C_n·vol(M).

From the closed case we deduceC_n=0.

Proof continued.

Next we show that any Seifert manifold with infinite fundamental group isL²-acyclic.

This follows from the fact that such a manifold is finitely covered by the total space of anS¹-bundleS¹→E →F over a surface with injectiveπ₁(S¹)→π₁(E)using previous results.

In the next step one shows that any irreducible 3-manifoldM with incompressible or empty boundary and infinite fundamental group isL²-acyclic.

Recall that by the Thurston Geometrization Conjecture we can find a family of incompressible tori which decomposeMinto hyperbolic and Seifert pieces. The tori and all these pieces areL²-acyclic.

Now the claim follows from theL²-Mayer Vietoris sequence.

Proof continued.

In the next step one shows that any irreducible 3-manifoldM with incompressible boundary and infinite fundamental group satisfies b₁⁽²⁾(M) =e −χ(M)andb⁽²⁾_n (M) =e 0 forn6=1.

This follows by consideringN =M∪_∂MM using the

L²-Mayer-Vietoris sequence, the already proved fact thatN is L²-acyclic and the previous computation of theL²-Betti numbers for surfaces.

In the next step one shows that any irreducible 3-manifoldM with infinite fundamental group satisfiesb⁽²⁾₁ (M) =e −χ(M)and

b_n⁽²⁾(M) =e 0 forn6=1.

Proof continued.

This is reduced by an iterated application of the Loop Theorem to the case where the boundary is incompressible. Namely, using the Loop Theorem one gets an embedded diskD²⊆Malong which one can decomposeM asM₁∪_D2 M₂or as

M₁∪_S0×D²D¹×D²depending on whetherD²is separating or not.

Since the only prime 3-manifold that is not irreducible isS¹×S², and every manifoldMwith finite fundamental group satisfies the result by a direct inspection of the Betti numbers of its universal covering, the claim is proved for all prime 3-manifolds.

Finally one uses theL²-Mayer Vietoris sequence to prove the claim in general using the prime decomposition.

Corollary

Let M be a3-manifold. Then M is L²-acyclic if and only if one of the following cases occur:

M is an irreducible3-manifold with infinite fundamental group whose boundary is empty or toral.

M is S¹×S²orRP³]RP³.

Corollary

Let M be a compact n-manifold such that n ≤3and its fundamental group is torsionfree.

Then all its L²-Betti numbers are integers.

Theorem (mapping tori,Lück[9])

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.

Proof.

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

d .

Proof continued.

Ifβn(X)is the number ofn-cells, then there is up to homotopy equivalence aCW-structure onT_fd with

β_n(T_fd) =β_n(X) +β_n−1(X). We have

b_n⁽²⁾(Tf_fd) =dim_N(G)

H_n⁽²⁾(C_n⁽²⁾(Tf_fd))

≤dim_N(G)

C_n⁽²⁾(Tf_fd)

=βn(T_fd).

This implies for alld ≥1

b_n⁽²⁾(Te_f)≤ βn(X) +βn−1(X)

d .

Taking the limit ford → ∞yields the claim.

LetM be an irreducible manifoldMwith infinite fundamental group and empty or incompressible toral boundary which is not a closed graph manifold.

Agolproved the Virtually Fibering Conjecture for suchM.

This implies by the result above thatMisL²-acyclic.

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.

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.

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)

IfGis elementary amenable torsionfree, thenD(G)can be identified with the Ore localization ofZGwith respect to the multiplicatively closed subset of non-zero elements.

In general the Ore localization does not exist and in these cases D(G)is the right replacement.

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) ker r_A:D(G)^m→ D(G)ⁿ .

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

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[4].

Theorem (Linnell[6],Schick[11])

1 LetC be the smallest class of groups which contains all free groups, 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.

Strategy to prove the Atiyah Conjecture

1 Show thatK₀(C)→K₀(CG)is surjective

(This is implied by theFarrell-Jones Conjecture)

2 Show thatK₀(CG)→K₀(D(G))is surjective.

3 Show thatD(G)is semisimple.

Approximation

In general there are no relations between the Betti numbersbn(X) and theL²-Betti numbersb⁽²⁾_n (Xe)for a connectedCW-complexX of finite type except for the Euler Poincaré formula

χ(X) =X

n≥0

(−1)ⁿ·b⁽²⁾_n (Xe) =X

n≥0

(−1)ⁿ·b_n(X).

Given an integerl ≥1 and a sequencer₁,r₂,. . .,r_l of

non-negative rational numbers, we can construct a groupGsuch thatBGis of finite type and

b⁽²⁾_n (BG) = r_n for 1≤n≤l;

b⁽²⁾_n (BG) = 0 forl+1≤n;

b_n(BG) = 0 forn≥1.

For any sequences₁,s₂,. . .of non-negative integers there is a CW-complexX of finite type such that forn≥1

bn(X) = sn; b⁽²⁾_n (Xe) = 0.

Theorem (Approximation Theorem,Lück[8])

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→∞

b_n(X_i) [G:G_i].

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.

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.

Example

The free groupF_ghas the obvious presentationhs₁,s₂, . . .s_g| ∅i and its deficiency is realized by this presentation, namely

defi(Fg) =g.

IfGis a finite group, defi(G)≤0.

The deficiency of a cyclic groupZ/nis 0, the obvious presentation hs |sⁿirealizes the deficiency.

The deficiency ofZ/n×Z/nis−1, the obvious presentation hs,t |sⁿ,tⁿ,[s,t]irealizes the deficiency.

Example (deficiency and free products)

The deficiency is not additive under free products by the following example due toHog-Lustig-Metzler. The group

(Z/2×Z/2)∗(Z/3×Z/3)

has the obvious presentation

hs₀,t₀,s₁,t₁|s²₀=t₀²= [s₀,t₀] =s³₁=t₁³= [s₁,t₁] =1i One may think that its deficiency is−2. However, it turns out that its deficiency is−1 realized by the following presentation

hs₀,t₀,s₁,t₁|s²₀=1,[s₀,t₀] =t₀²,s³₁=1,[s₁,t₁] =t₁³,t₀²=t₁³i.

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).

Theorem (Deficiency and extensions,Lück)

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

Suppose that G is finitely presented and 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).

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.

The Singer Conjecture gives also evidence for the Atiyah Conjecture.

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.

Definition (Kähler hyperbolic manifold)

AKähler hyperbolic manifoldis a closed connected Kähler manifoldM whose fundamental formωised(bounded), i.e. its liftωe∈Ω²(M)e to the universal covering can be written asd(η)holds for some bounded 1-formη∈Ω¹(M).e

Theorem (Gromov[5])

Let M be a closed Kähler hyperbolic manifold of complex dimension c.

Then

b⁽²⁾_n (M)e = 0 if n6=c;

b⁽²⁾_n (M)e > 0;

(−1)^m·χ(M) > 0;

LetM be a closed Kähler manifold. It is Kähler hyperbolic if it admits some Riemannian metric with negative sectional curvature, or, if, generallyπ₁(M)is word-hyperbolic andπ₂(M)is trivial.

A consequence of the theorem above is that any Kähler hyperbolic manifold is a projective algebraic variety.

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