Introduction to L

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Introduction to L

²

-invariants

Wolfgang Lück Bonn Germany

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

Madrid, February 2018

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Some appetizers

Theorem

Let G be a group with finite classifying space BG. Suppose that G contains a normal infinite solvable subgroup. Then

χ(BG) =0.

Theorem

Let M be a closed hyperbolic manifold of even dimension n=2k . Then (−1)^k·χ(M)>0,

and every S¹-action on M is trivial.

Wolfgang Lück (HIM, Bonn) Introduction toL²-invariants Madrid, February 2018 2 / 37

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Theorem

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 defi(G)≤1;

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

|sign(M)| ≤χ(M).

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Conjecture (Zero-divisor Conjecture)

Let F be a field of characteristic zero and G be a torsionfree group.

Then the group ring FG has no non-trivial zero-divisors.

Theorem

Let M be a closed Kähler manifold. Suppose that it admits some Riemannian metric with negative sectional curvature.

Then M is a projective algebraic variety.

The point is that the statements of these theorems have nothing to do withL²-invariants, but their proofs have. This list can be

extended considerably.

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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 groupπ into account.

Examples:

Classical notion Generalized version Homology with coeffi-

cients inZ

Homology with coefficients in representations ofπ

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

Signature∈Z Surgery invariants inL∗(Zπ)

— 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(Hn(eX;C)),

where dim_Cπ(M)for aCπ-module could be chosen for instance as dim_C(C⊗_CπM).

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 toAtiyahand motivated byL²-index theory.

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Group von Neumann algebras

Given a ringRand a groupG, denote byRGthegroup 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.

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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.

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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) ∈R^≥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). LetM_χ_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 ∈R^≥0occurs asr =dim_N_(Zⁿ₎(V).

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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}.

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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 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 (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[Zd]⁽⁰⁾

C[Z^d]⁽⁰⁾⊗

Z[Z^d]Hn(Xe)

Obviously this implies

b_n⁽²⁾(Xe)∈Z.

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Some computations and results

Theorem (S¹-actions on aspherical manifolds,Lück)

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

Then we get for n≥0

b_n⁽²⁾(M)e = 0;

χ(M) = 0.

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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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.

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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.

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 Every S¹-action on M is trivial S¹.

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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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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.

This aspect has recently played an important role in the construction of new invariants for 3-manifolds such as the universalL²-torsion or theL²-polytope byFriedl-Lück.

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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(QG)the von Neumann dimension

dim_N_(G)

ker r_A:L²(G)^m →L(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.

An even stronger version allowsA∈Mm,n(CG).

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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)

LetCbe 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 toC satisfies the Atiyah Conjecture (overC).

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Strategy to prove the Atiyah Conjecture:

1 Show thatK₀(C)→K₀(CG)is surjective.

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

3 Show thatD(G)is semisimple.

Notice that the Atiyah Conjecture originally was statement about an invariant extracted from the heat kernel of the universal covering, namely about thethe analyticL²-Betti number.

However, the strategy described above is based on and requires K-theoreticandring theoretic input.

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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).

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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].

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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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Letpbe a prime andFp be the field withpelements.

Conjecture (Approximation Conjecture in characteristicp) Let X be a connectedasphericalCW -complex of finite type. Suppose thatπ is 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⁾(Xe) = lim

i→∞

b_n(X_i;Fp) [G:G_i] .

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Schickused approximation techniques to prove the Atiyah Conjecture for matrices overQGfor a large class of groups.

A lot of work has been done byJaikin-Zapirainto extend this from QGtoCGusing ring theoretic methods.

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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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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.

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Theorem (Gromov)

Let M be a closed Kähler manifold of complex dimension c. Suppose that it admits some Riemannian metric with negative sectional

curvature. Then

b⁽²⁾_n (M)e = 0 if n6=c;

b⁽²⁾_n (M)e > if n=c;

(−1)^m·χ(M) > 0;

Moreover, M is a projective algebraic variety.

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Further important problems or connections

Conjecture about the equality of the firstL²-Betti number,cost, andrank gradientsof groups, e.g., theFixed Prize Conjecture.

L²-invariants and measured and geometricgroup theory Zero-in-the-spectum Conjecture.

Determinant Conjecture.

The Conjecture ofBergeron-Venkateshfor the growth of the torsion part of the homology andL²-torsion.

Conjecture about the vanishing of allL²-invariants for closed aspherical manifolds with vanishingsimplical volume.

L²-invariants andK-theory.

L²-invariants andentropy.

L²-invariants andgraph theory.

TwistedL²-invariants and3-manifolds.

Applications tovon Neumann algebras.

and so on.