Introduction to L

Introduction to L

-invariants

Wolfgang Lück Bonn Germany

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

Fort Worth, June, 2015

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.

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) ∈[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). 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).

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

Mz−1:L²(Z) =L²(S¹)→L²(Z) =L²(S¹), u(z)7→(z−1)·u 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) [ 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.

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⁽²⁾_n (Xe) =0 forn≥0.

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

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.

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.

Theorem (L²-Hodge - de Rham Theorem,Dodziuk) 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)

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)

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.

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.

Proof:

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

d .

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

βn(T_fd) =βn(X) +βn−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⁽²⁾_n (Te_f) ≤ βn(X) +βn−1(X)

d .

Taking the limit ford → ∞yields the claim.

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) 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.

Theorem (Linnell, Schick)

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)

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 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.

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)

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.