WolfgangLückBonnGermanyemailwolfgang.lueck@him.uni-bonn.dehttp://www.him.uni-bonn.de/lueck/NotreDame,April2019 L -Bettinumbersandtheirapplications

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L

²

-Betti numbers and their applications

Wolfgang Lück Bonn Germany

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

Notre Dame, April 2019

Wolfgang Lück (MI, Bonn) L²-Betti numbers Notre Dame, April 2019 1 / 104

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Some motivation

We start with presenting some interesting and comprehensible results from different areas of mathematics.

Theorem (Euler characteristic of amenable groups, Cheeger-Gromov)

Let G be a group which contains a normal infinite amenable subgroup.

Suppose that there is a finite model for BG.

Then its Euler characteristic satisfies χ(BG) =0.

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Definition (Amenable group)

A groupGis calledamenableif there is a leftG-invariant linear operatorµ:L^∞(G,R)→Rwithµ(1) =1 which satisfies for all f ∈L^∞(G,R)

inf{f(g)|g∈G} ≤µ(f)≤sup{f(g)|g∈G}.

A group which containsF₂as a subgroup is never amenable.

There are non-amenable groups which do not containF₂as subgroup.

But they are hard to construct and a group which does not contain F₂is very likely to be amenable.

Solvable groups are amenable.

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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.

The deficiency is an important invariant in group theory and low-dimensional topology.

Lower bounds can be obtained by investigating specific presentations. The hard part is to find upper bounds.

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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 |sⁿ =1irealizes the deficiency.

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

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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|s₀²=t₀²= [s₀,t₀] =s₁³=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|s₀²=1,[s₀,t₀] =t₀²,s₁³=1,[s₁,t₁] =t₁³,t₀²=t₁³i.

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Theorem (Deficiency and group 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:

defi(G)≤1.

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An important invariant of a closed oriented 4k-dimensional manifoldMis itssignature

sign(M)∈Z

which is the signature of its intersection pairing.

We have the relation sign(M)≡χ(M) mod 2.

Ifk =1, we have by theHirzebruch signature formula sign(M) = 1

3 · hp₁(M),[M]i.

Theorem (Signatures of 4-manifolds and group extensions, Lück)

Let M be a closed oriented4-manifold. Suppose thatπ₁(M)contains an infinite normal finitely generated subgroup of infinite index.

Then

|sign(M)| ≤χ(M).

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LetR be a ring and letGbe a group. Thegroup ringRG, sometimes also denoted byR[G], is theR-algebra, whose underlyingR-module is the freeR-module generated byGand whose multiplication comes from the group structure.

An elementx ∈RGis a formal sumP

g∈Gr_g·g such that only finitely many of the coefficientsrg∈Rare different from zero.

The multiplication comes from the tautological formula g·h=g·h, more precisely



 X

g∈G

r_g·g



·



 X

g∈G

s_g·g



:=X

g∈G



 X

h,k∈G,hk=g

r_hs_k



·g.

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Group rings arise in representation theory and topology as follows.

AnRG-moduleP is the same asG-representation with

coefficients inR, i.e., aR-modulP together with aG-action by R-linear maps.

LetX →X be aG-covering of theCW-complexX, i.e., a principal G-bundle overX or, equivalently, a normal covering withGas group of deck transformations. An example for connectedX is the universal coveringXe →X withG=π1(X).

Then thecellularZ-chain complexC∗(X), which is a priori a free Z-chain complex, inherits from theG-action onX the structure of a freeZG-chain complex, where the set ofn-cells inX determines aZG-basis forC∗(X).

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Group rings are in general very complicated. For instance, there is the conjecture that the complex group ringCGis Noetherian if and only ifGis virtually poly-cyclic.

Let us figure out whether there areidempotentsx inRG, i.e., elements withx²=x.

Here is the only known construction of an idempotent inCG.

Consider an elementg∈Gwhich has finite ordern. Then we can take

x = 1 n ·

n−1

X

i=0

gⁱ.

In particular we know no idempotent inCGbesides 0 and 1 ifGis torsionfree.

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Conjecture (Idempotent Conjecture,(Kaplansky))

Let G be a torsionfree group. Then all idempotents ofCG are trivial, i.e., equal to0or1.

Conjecture (Zero-divisor Conjecture,(Kaplansky)) Let G be a torsionfree group. ThenCG has no zero-divisors.

Conjecture (Embedding Conjecture)

Let G be a torsionfree group. ThenCG embeds into a skew-field.

Embedding Conjecture =⇒ Zero-divisor Conjecture =⇒ Idempotent Conjecture.

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Definition (Projective class groupK0(R)) Theprojective class groupof a ringR

K₀(R)

is defined to be the abelian group whose generators are isomorphism classes[P]of finitely generated projectiveR-modulesP and whose relations are[P₀] + [P₂] = [P₁]for every exact sequence

0→P₀→P₁→P₂→0 of finitely generated projectiveR-modules.

Definition (G0(RG))

The abelian group of a ringR

G₀(R)

is defined as above but one replaces “finitely generated projective” by

“finitely generated” everywhere.

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Lemma

The element[CG]in K₀(CG)generates an infinite cyclic group.

Proof.

The augmentation homomorphism:CG→Cand the dimension of a complex vector space induce a homomorphism

K₀(CG)→Z, [P]7→dim_C(C⊗_C_GP) which sends[CG]to a generator ofZ.

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Lemma

Suppose that G contains F₂:=Z∗Zas subgroup. Then[CG] =0in G₀(CG).

Proof.

Leti:F₂→Gbe the inclusion andp:F₂→Zbe any surjective group homomorphism. Then induction withi and restriction withp induces group homomorphism

i∗:G₀(C[F₂]) → G₀(CG);

p^∗:G₀(C[Z]) → G₀(C[F₂]).

Considering the cellularC[Z]-chain complex of the universal covering ofS¹yields the exact sequence ofC[Z]-modules

0→C[Z]−−→^s−1 C[Z]→C→0 for a generators ∈Z. Hence we get inG₀(C[Z])

[C] = [C[Z]]−[C[Z]] =0.

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Continued.

Considering the cellularC[F₂]-chain complex of the universal covering ofS¹∨S¹yields the exact sequence ofC[F₂]-modules 0→C[F₂]⊕C[F₂]−−−−−−−→^(s¹^−1,s²⁻¹⁾ C[F₂]→C→0 for the standard generatorss₁,s₂∈F₂. Hence we get inG₀(C[F₂])

[C] =−[C[F₂]].

We compute inG₀(C[F₂])

[C[G]] =i∗([C[F₂]]) =−i_∗([C]) =−i_∗◦p^∗([C]) =−i_∗◦p^∗(0) =0.

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Theorem (OnG0(CG)for amenable groups G,Lück) Suppose that G is amenable. Then the element[CG]in G₀(CG) generates an infinite cyclic group.

Conjecture (Characterization of amenability byG-theory) The group G is amenable if and only if the element[CG]in G₀(CG)is non-trivial.

The group G is amenable if and only if G₀(CG)is non-trivial.

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Conjecture (Euler characteristic and sectional curvature,Hopf) Let M be a closed Riemannian manifold of even dimension2n. Then:

If its sectional curvature satisfiedsec(M)≤0, then (−1)ⁿ·χ(M)≥0;

If its sectional curvature satisfiedsec(M)<0, then (−1)ⁿ·χ(M)>0.

Theorem (S¹-actions and hyperbolic manifolds) Any S¹-action on a hyperbolic closed manifold is trivial.

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Theorem (Slice knots and Casson-Gordon invariants, Cochran-Orr-Teichner)

There are obstructions for knots to be slice which go far beyond the classical Casson-Gordon invariants.

Theorem (Kähler manifolds and projective algebraic varieties, Gromov)

Let M be a closed Kähler manifold, i.e., a complex manifold which comes with a so called Kähler Hermitian metric and Kähler2-form.

Suppose that it admits some Riemannian metric with negative

sectional curvature, or, more generally, thatπ₁(M)is hyperbolic (in the sense of Gromov) andπ₂(M)is trivial.

Then M is a projective algebraic variety.

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So far noL²-invariants have occurred in the talk and the audience may wonder why the title contains the wordL²-invariants at all.

The point is that the proofs of the results above or of the

conjectures in certain special cases do rely onL²-methods. The use ofL²-methods made a lot of progress possible although on the first glance they seem to be unrelated to the results and conjectures mentioned above.

Next we give an introduction to theL²-setting.

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Basic motivation for the passage to the L

²

-setting

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 Wall’s finiteness obstruction inK₀(Zπ)

Lefschetz numbers∈Z Generalized Lefschetz invariants inZπ_φ

Signature∈Z Surgery invariants inL_∗(ZG)

— torsion invariants

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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(H_n(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_C_π(M) is in general not additive under exact sequences.

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

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Group von Neumann algebras

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

g∈Gλg·g such thatλg∈CandP

g∈G|λ_g|²<∞.

Definition (Group von Neumann algebra and its trace) Define thegroup von Neumann algebra

N(G) :=B(L²(G),L²(G))^G=CG^weak to be the algebra of boundedG-equivariant operators L²(G)→L²(G).

Thevon Neumann traceis defined by

tr_N_(G):N(G)→C, f 7→ hf(e),ei_L2(G).

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Example (FiniteG) LetGbe a finite group.

ThenCG=L²(G) =N(G).

The trace tr_N_(G)assigns toP

g∈Gλg·g the coefficientλe. So the new approach is only interesting whenGis infinite.

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Example (G=Zⁿ)

LetGbeZⁿ. This is an easy but very illuminating example.

LetL²(Tⁿ)be the Hilbert space ofL²-integrable functionsTⁿ →C. TheFourier transformyields 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 (Murray-von Neumann dimension)

LetV be a finitely generated HilbertN(G)-module. Choose a

G-equivariant projectionp:L²(G)ⁿ→L²(G)ⁿsuch that there is a linear isometricG-equivariant isomorphism im(p)−^∼⁼→V. Define the

Murray-von Neumann dimensionofV by dim_N_(G)(V):=tr_N_(G)(p) :=

n

X

i=1

tr_N_(G)(p_i,i) ∈R^≥0.

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Example (FiniteG) LetGbe a finite group.

A finitely generated HilbertN(G)-moduleV is the same as a unitary finite dimensionalG-representation.

We get for its von Neumann dimension dim_N_(G)(V) = 1

|G|·dim_C(V).

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Example (G=Zⁿ) LetGbeZⁿ.

LetX ⊂Tⁿ be any measurable set with characteristic function χ_X ∈L^∞(Tⁿ). LetM_χ_X:L²(Tⁿ)→L²(Tⁿ)be theZⁿ-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).

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Definition (Weakly exact)

A sequence of HilbertN(G)-modulesU −→ⁱ V −→^p W isweakly exactatV 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 a weak isomorphismif it is injective and has dense image.

Example (FiniteG)

IfGis finite, weakly exact is the same as exact and weak isomorphism is the same as isomorphism.

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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.

Its kernel is trivial. Its image is dense but not equal toL²(S¹).

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Theorem (Properties of the Murray-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}.

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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)itscellular Zπ-chain complex.

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

C_∗⁽²⁾(Xe):=L²(π)⊗_Z_π C_∗(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_n+1⁽²⁾ ).

Define itsn-thL²-Betti number

b⁽²⁾_n (Xe) :=dim_N_(π) H_n⁽²⁾(Xe)

∈R^≥0.

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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 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

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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).

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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)−−−→^M^z−1 L²(Z)→0→. . . and henceH_n⁽²⁾(Sf¹) =0 andb⁽²⁾_n (Sf¹) =0 for all≥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 (Ffg) =0.

Since dim(Fg) =2, we getb_n⁽²⁾(Ffg) =0 forn≥3.

The Euler-Poincaré formula shows

b⁽²⁾₁ (Ff_g) = −χ(F_g) =2g−2;

b⁽²⁾_n (Ff_g) = 0 for n6=1.

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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_n⁽²⁾(Xe) =dim_C[Zd]⁽⁰⁾

C[Z^d]⁽⁰⁾⊗

Z[Z^d]Hn(Xe)

.

Obviously this implies

b⁽²⁾_n (Xe)∈Z.

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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)⊗_Z_GC_∗(X)), and define

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

∈R^≥0.

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

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

We have the following formula

dim_N_(G) H_n⁽²⁾(L²(G)⊗_Z_Gi∗C∗)

=dim_N_(H) H_n⁽²⁾(L²(H)⊗_Z_HC∗) .

Lemma

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

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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.

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The L

²

-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

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−−→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→ · · ·.

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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))→ · · ·.

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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)⊗_Z_GC∗(X₀)→L²(G)⊗_Z_GC∗(X₁)⊕L²(G)⊗_Z_GC∗(X₂)

→L²(G)⊗_Z_GC∗(X)→0.

Now apply the previous result.

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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.

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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.

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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-acyclic, then X is L²-acyclic.

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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.

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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.

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Some computations and results

Example (Finite self-coverings)

We get for a connectedCW-complexX of finite type, for which there is a self-coveringX →X withd-sheets for some integer d ≥2,

b⁽²⁾_n (Xe) =0 forn≥0.

This is certainlynottrue for the classical Betti numbersb_n(X).

This implies for each connectedCW-complexY of finite type that S¹×Y isL²-acyclic.

The latter equation follows also from the Künneth formula for L²-Betti numbers and the previous computation thatb⁽²⁾_n (Sf¹) vanishes for alln.

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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 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.

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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.

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

Hⁿ₍₂₎(M) =e {eω∈Ωⁿ(M)e |∆en(eω) =0, ||eω||_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.

Notice thatMis hyperbolic if and only ifMe is isometrically diffeomorphic to the standard hyperbolic spaceH^d.

A direct computation shows thatHⁿ₍₂₎(H^d)is not zero if and only if 2n=d.

Henceb⁽²⁾_n (M) =e dim_N_(π)(H^p₍₂₎(M))e is not zero if and only if 2n=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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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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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.

Notice 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.

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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.

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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.

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.

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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.

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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.

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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.

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Theorem (Mapping tori,Lück)

Let f:X →X be a cellular self-homotopy 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 .

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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.

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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 suchMsaying that a finite covering ofMis a mapping torus.

This implies by the result above thatMisL²-acyclic.

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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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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 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.

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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) ker 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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Obviously the Atiyah Conjecture implies theEmbedding Conjectureand hence theZero-divisor Conjectureand the Idempotent Conjecturedue toKaplansky.

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 under 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.

A group is calledlocally indicableif every non-trivial finitely generated subgroup admits an epimorphism ontoZ. Examples are one-relator-groups.

Theorem (Jaikin-Zapirain & Lopez-Alvarez)

If G is locally indicable, then it satisfies the Atiyah Conjecture.

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Theorem (Strategy to prove the Atiyah Conjecture I,Linnell) Let G be a torsionfree group. Then the Atiyah Conjecture holds if the following three assertions are true:

The map K₀(C)→K₀(CG)is surjective;

The map K₀(CG)→K₀(D(G))is surjective;

The ringD(G)is semisimple.

Theorem (Strategy to prove the Atiyah Conjecture II)

Let G be a torsionfree group. Then the Atiyah Conjecture holds if the following two assertions are true, whereR(G)⊆ U(G)is the smallest

∗-regular subring of the∗-regular ringU(G).

The map K₀(C)→K₀(CG)is surjective;

The map K₀(CG)→K₀(R(G))is surjective.

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Theorem

Let G be a torsionfree group. Then the following assertions are equivalent:

The Atiyah Conjecture holds for G;

D(G)is a skew-field;

R(G)is a skew-field.

If the torsionfree groupGsatisfies the Atiyah Conjecture, then D(G) =R(G).

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Recall that for a torsionfree group the Atiyah Conjecture predicts that for a closed Riemannian manifold withGas fundamental group the following integral is an integer

tlim→∞

Z

F

tr_R(e^−t^∆^eⁿ(˜x,˜x))dvol

Me.

By the last Theorems it is equivalent to theK-theoretic statement thatKe₀(R(G))vanishes and to the ring theoretic statement that D(G)is a skew-field.

The proof that these three facts imply the Atiyah Conjecture is rather involved. It is based on the fundamental square and the fact that the generalized dimension function forN(G)which we will introduce later, extends to an appropriate dimension function dim_U_(G)forU(G)-modules such that for anyN(G)-moduleM we have

dim_N_(G)(M) =dim_U(G)(U(G)⊗_N_(G)M).

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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).