Universal torsion, L

Universal torsion, L

-invariants, polytopes and the Thurston norm

Wolfgang Lück Bonn Germany

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

Fort Worth, June, 2015

Review of classical L

-invariants

Given a groupG, itsgroup von Neumann algebrais defined to be N(G)=B(L²(G),L²(G))^G =CG^weak.

It comes with a naturaltrace

tr_N(G):N(G)→C, f 7→ hf(e),ei_L2(G). Forx =P

g∈Gλg·g∈CGwe have tr_N(G)(x) =λe. IfGis finite,N(G) =CG.

IfG=Z, we get identifications

N(Z) = L^∞(S¹);

tr_N(Z(f) = Z

S¹

f(z)dµ.

The trace yields via the Hattori-Stallings rank the dimension function with values in[0,∞)for finitely generated projective N(G)-modules due toMurray-von Neumann.

This has been extended to the dimension functiondim_N(G) with values in[0,∞]for arbitraryN(G)-modules, which has still nice properties such as additivity and cofinality, byLück.

Definition (L²-Betti number)

LetY be aG-space. Define itsnthL²-Betti number b_n⁽²⁾(Y;N(G)):=dim_N(G) H_n(N(G)⊗_ZGC∗^sing(Y))

∈[0,∞].

Sometimes we omitN(G)from the notation whenGis obvious.

This definition is the end of a long chain of generalizations of the original notion due toAtiyahwhich was motivated by index theory.

He defined for aG-coveringM →M of a closed Riemannian manifold

b⁽²⁾_n (M) := lim

t→∞

Z

F

tr e^−t·∆n(x,x)

dvol_M.

IfGis finite, we have

b_n⁽²⁾(X) = 1

|G|·bn(X).

IfG=Z, we have b⁽²⁾_n (X) =dim

C[Z]⁽⁰⁾ C[Z]⁽⁰⁾⊗_C[Z]H_n(X;C)

∈Z.

In the sequel3-manifoldmeans a prime connected compact orientable 3-manifold with infinite fundamental group whose boundary is empty or a union of tori and which is notS¹×D²or S¹×S².

Theorem (Lott-Lück)

For every3-manifold M all L²-Betti numbers b_n⁽²⁾(M)e vanish.

We are interested in case where allL²-Betti numbers vanish, since then a very powerful secondary invariant comes into play, the so calledL²-torsion.

L²-torsion can be defined analytical in terms of the spectrum of the Laplace operator, generalizing the notion ofanalytic

Ray-Singer torsion. It can also be defined in terms of the cellular ZG-chain complex, generalizing of theReidemeister torsion.

The definition ofL²-torsion is based on the notion of the Fuglede-Kadison determinantwhich is a generalization of the classical determinant to the infinite-dimensional setting. It is defined for an elementf ∈ N(G)to be the non-negative real number

det⁽²⁾(f)=exp 1

2 · Z

ln(λ)dν_f^∗_f

∈R^>0

whereν_f^∗_f is the spectral measure of the positive operatorf^∗f. IfGis finite, then det⁽²⁾(f) =|det(f)|^1/|G|.

Theorem (Lück-Schick)

Let M be a3-manifold. Let M₁, M₂, . . . , M_mbe the hyperbolic pieces in its Jaco-Shalen decomposition.

Then

ρ⁽²⁾(M) :=e − 1 6π ·

m

X

i=1

vol(M_i).

Universal L

-torsion

Definition (K₁^w(ZG))

LetK₁^w(ZG)be the abelian group given by:

generators

Iff:ZG^m →ZG^mis aZG-map such that the inducedN(G)-map N(G)^m→ N(G)^mmap is a weak isomorphism, i.e., the

dimensions of its kernel and cokernel are trivial, then it determines a generator[f]inK₁^w(ZG).

relations

f₁ ∗ 0 f₂

= [f₁] + [f₂];

[g◦f] = [f] + [g].

DefineWh^w(G):=K₁^w(ZG)/{±g |g∈G}.

Definition (UniversalL²-torsion)

LetG→X →X be aG-covering of a finiteCW-complex. Suppose thatX isL²-acyclic, i.e.,b⁽²⁾_n (X)vanishes for alln∈Z.

Then itsuniversalL²-torsionis defined as an element ρ⁽²⁾_u (X) =ρ⁽²⁾_u (X;N(G))∈K₁^w(ZG).

We do not present the actual definition but it is similar to the definition of Whitehead torsion in terms of chain contractions, one has now to use weak chain contractions.

It has indeed a universal property on the level of weakly acyclic finite based freeZG-chain complexes.

By stating its main properties and its relation to classical invariant we want to convince the audience that this is a very interesting and powerful invariant which is worthwhile to be explored further.

Homotopy invariance

Consider a pullback ofG-coverings of finiteCW-complexes withf a homotopy equivalence

X ^{f //}

Y

X ^{f //}Y Suppose thatY isL²-acyclic.

ThenX isL²-acyclic and the obvious homomorphism Wh(G)→Wh^w(G)

sends theWhitehead torsionτ(f:X →Y)to the difference of universalL²-torsionsρ⁽²⁾_u (Y)−ρ⁽²⁾_u (X).

Sum formula

LetG→X →X be aG-covering of a finiteCW-complexX. Let X_i ⊆X fori=0,1,2CW-subcomplexes withX =X₁∪X₂and X₀=X₁∩X₂. LetX_i →X_i be given by restriction. Suppose thatX_i isL²-acyclic fori =0,1,2.

ThenX isL²-acyclic and we get

ρ⁽²⁾_u (X;N(G)) =ρ⁽²⁾_u (X₁;N(G)) +ρ⁽²⁾_u (X₂;N(G))−ρ⁽²⁾_u (X₀;N(G)).

There areproduct formulasor more general formulas forfibrations and forproperS¹-actionswhich we will not state.

Whenever an invariant comes from the universal torsion by applying a group homomorphism, these formulas automatically extend to the other invariant.

Example (TrivialG)

Suppose thatGis trivial. Then we get isomorphisms K₁^w({1})−^∼=→K₁(Q)−−→^det Q^× Wh^w({1})−→ {r^∼= ∈Q|r >0}.

A finite based freeZ-chain complexC∗ isL²-acyclic if and only if each of its homology groups is finite. Under the isomorphism above the universal torsion ofC∗ is sent toQ

n≥0|H_n(C∗)|⁽⁻¹⁾ⁿ. For finiteCW-complexes we essentially have the same

information as theMilnor torsion.

IfGis finite, we rediscover essentially the classicalReidemeister torsion.

The fundamental square and 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, it 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 Atiyah Conjecture implies theKaplansky Conjecturesaying that for any torsionfree group and field of characteristic zeroF the group ringFGhas no non-trivial zero-divisors.

A torsionfreeGsatisfies the Atiyah Conjecture if and only if each of its finitely generated subgroups does.

Fix a natural numberd ≥5. Then a finitely generated torsionfree groupGsatisfies the Atiyah Conjecture if and only if for any G-coveringM→M of a closed Riemannian manifold of dimension d we haveb⁽²⁾_n (M;N(G))∈Zfor everyn≥0.

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.

This Theorem and results byWaldhausenshow for the

fundamental groupπof a 3-manifold that it satisfies the Atiyah Conjecture and that Wh(π)vanishes.

Identifying K

( Z G) and K

(D(G))

Theorem (Linnell-Lück)

If G belongs toC, then the natural map

K₁^w(ZG)−^∼=→K₁(D(G)) is an isomorphism.

Its proof is based on identifyingD(G)as an appropriate Cohn localization ofZGand the investigating localization sequences in algebraicK-theory.

There is aDieudonné determinantwhich induces an isomorphism det_D:K₁(D(G))−→ D(G)^{∼= ×}/[D(G)^×,D(G)^×].

In particular we get forG=Z

K₁^w(Z[Z])∼=Q[Z]⁽⁰⁾\ {0}.

It turns out that then the universal torsion is the same as the Alexander polynomialof an infinite cyclic covering, as it occurs for instance in knot theory.

Twisting L

-invariants

Consider aCW-complexX withπ=π₁(M). Fix an element φ∈H¹(X;Z) =hom(π;Z).

Fort∈(0,∞), letφ^∗Ct be the 1-dimensionalπ-representation given by

w·λ:=t^φ(w)·λ forw ∈π, λ∈C.

One cantwisttheL²-chain complex ofX with this representation, or, equivalently, apply the following ring homomorphism to the cellularZG-chain complex before passing to the Hilbert space completion

CG→CG, X

g∈G

λg·g 7→ X

g∈G

λ·t^φ(g)·g.

Notice that for irrationalt the relevant chain complexes do not have coefficients inQGanymore and theDeterminant Conjecture does not apply. Moreover, the Fuglede-Kadison determinant is in general not continuous.

Thus we obtain theφ-twistedL²-torsion function ρ(Xe;φ): (0,∞)→R sendingt to theCt-twistedL²-torsion.

Its value att =1 is just theL²-torsion.

On the analytic side this corresponds for closed Riemannian manifoldMto twisting with the flat line bundleMe ×_πCt →M. It is obvious that some work is necessary to show that this is a well-defined invariant since theπ-action onCt isnotisometric.

Theorem (Lück)

Suppose thatX is Le ²-acyclic.

1 The L²torsion functionρ⁽²⁾:=ρ⁽²⁾(Xe;φ) : (0,∞)→Ris well-defined.

2 The limitslim_t→∞ ^ρ_ln(t)^(2)(t) andlim_t→0^ρ_ln(t)^(2)(t) exist and we can define thedegree ofφ

deg(X;φ)∈R to be their difference.

3 There is aφ-twisted Fuglede-Kadison determinant det⁽²⁾_tw,φ:K₁^w(ZG)→map((0,∞),R) which sendsρ⁽²⁾_u (Xe)toρ⁽²⁾(Xe;φ).

Definition (Thurston norm)

LetMbe a 3-manifold andφ∈H¹(M;Z)be a class. Define its Thurston norm

x_M(φ)=min{χ−(F)|F embedded surface inM dual toφ}

where

χi(F) = X

C∈π₀(M)

max{−χ(C),0}.

Thurstonshowed that this definition extends to the real vector spaceH¹(M;R)and defines aseminormon it.

IfF →M −→^p S¹is a fiber bundle andφ=π₁(p), then x_M(φ) =χ(F).

Theorem (Friedl-Lück)

Let M be a3-manifold. Then for everyφ∈H¹(M;Z)we get the equality deg(M;φ) =x_M(φ).

Polytope homomorphism

Consider the projection

pr:G→H₁(G)_f :=H₁(G)/tors(H₁(G)).

LetK be its kernel.

After a choice of a set-theoretic section of pr we get isomorphisms ZK ∗H₁(G)_f −^∼=→ ZG;

S⁻¹ D(K)∗H₁(G)_f ^∼=

−→ D(G),

where here and in the sequelS⁻¹denotes Ore localization with respect to the multiplicative closed set of non-trivial elements.

Givenx =P

h∈H1(G)_f u_h·h∈ D(K)∗H₁(G)_f, define itssupport supp(x):={h∈H₁(G)_f |h∈H₁(G)_f),u_h 6=0}.

The convex hull of supp(x)defines apolytope P(x)⊆R⊗_ZH₁(G)_f =H¹(M;R).

This is the non-commutative analogon of theNewton polygon associated to a polynomial in several variables.

Recall theMinkowski sumof two polytopesP andQ P+Q={p+q|p∈P,q ∈Q}

It satisfiesP₀+Q=P₁+Q =⇒ P₀=P₁.

Definition (Polytope group)

LetP(R⊗_ZH₁(G)_f)be the Grothendieck group of the abelian monoid of integral polytopes under the Minkowski sum modulo translations by elements inH₁(G)_f, where integral means that all extreme points lie on the latticeH₁(G)_f.

We haveP(x ·y) =P(x) +P(y)forx,y ∈(D(K)∗H₁(G)_f. Hence we can define a homomorphism of abelian groups

P⁰:

S⁻¹ D(K)∗H₁(G)_f×

→ P(R⊗_ZH₁(G)_f), by sendingx ·y⁻¹to[P(x)]−[P(y)].

The composite

K₁^w(ZG)−→^∼= K₁(D(G))−→ D(G)^{∼= × ∼}−⁼→

S⁻¹ D(K)∗H₁(G)_f× P⁰

−→ P(R⊗_ZH₁(G)_f) factories to thepolytope homomorphism

P: Wh^w(G)→ P(R⊗_ZH₁(G)_f).

Definition (Thurston polytope)

LetMbe a 3-manifold. Define theThurston polytopeto be subset of H¹(M;R)

T(M) :={φ∈H¹(M;R)|x_M(φ)≤1}.

Theorem (Friedl-Lück)

Let M be a3-manifold. Then the image of the universal L²-torsion ρ⁽²⁾_u (M)e under the polytope homomorphism

P: Wh^w(π₁(M))→ P(R⊗_ZH₁(π₁(M))_f)

is represented by the dual of the Thurston polytope, which is an integral polytope inR⊗_ZH₁(π₁(M))_f =H₁(M;R) =H¹(M;R)^∗.

Summary

We can assign to a finiteCW-complexX itsuniversalL²-torsion ρ⁽²⁾(Xe)∈Wh^w(π),

provided thatXe isL²-acyclic andπ satisfies the Atiyah Conjecture.

These assumptions are always satisfied for 3-manifolds.

The Alexander polynomial is a special case.

One can read of from the universalL²-torsion apolytopewhich for a 3-manifold is the dual of theThurston polytope.

One can twist theL²-torsion by a cycleφ∈H¹(M)and obtain a L²-torsion functionfrom which one can read of theThurston norm.

L

-Euler characteristic

Definition (L²-Euler characteristic) LetY be aG-space. Suppose that

h⁽²⁾(Y;N(G)) :=X

n≥0

b_n⁽²⁾(Y;N(G))<∞.

Then we define itsL²-Euler characteristic χ⁽²⁾(Y;N(G)) :=X

n≥0

(−1)ⁿ·b_n⁽²⁾(Y;N(G)) ∈R.

Definition (φ-L²-Euler characteristic)

LetX be a connectedCW-complex. Suppose thatXe isL²-acyclic.

Consider an epimorphismφ:π =π1(M)→Z. LetK be its kernel.

Suppose thatGis torsionfree and satisfies the Atiyah Conjecture.

Define theφ-L²-Euler characteristic

χ⁽²⁾(Xe;φ) :=χ⁽²⁾(Xe;N(K))∈R.

Notice thatXe/K is not a finiteCW-complex. Hence it is not obvious but true thath⁽²⁾(X;e N(K))<∞andχ⁽²⁾(Xe;φ)is a well-defined real number.

Theφ-L²-Euler characteristic has a bunch of good properties, it satisfies for instance asum formula,product formulaand is multiplicativeunder finite coverings.

Letf:X →X be a selfhomotopy equivalence of a connected finite CW-complex. LetT_f be its mapping torus. The projectionT_f →S¹ induces an epimorphismφ:π₁(T_f)→Z=π₁(S¹).

ThenTe_f isL²-acyclic and we get

χ⁽²⁾(Te_f;φ) =χ(X).

Theorem (Friedl-Lück)

Let M be a3-manifold andφ:π₁(M)→Zbe an epimorphism. Then

−χ⁽²⁾(M;e φ) =x_M(φ).

Higher order Alexander polynomials

Higher order Alexander polynomialswere introduced for a coveringG→M→M of a 3-manifold byHarveyandCochran, provided thatGoccurs in the rational derived series ofπ₁(M).

At least thedegreeof these polynomials is a well-defined invariant ofM andG.

We can identify the degree with theφ-L²-Euler characteristic.

Thus we can extend this notion of degree also to the universal covering ofM and can prove the conjecture that the degree coincides with the Thurston norm.

Group automorphisms

Theorem (Lück)

Let f:X →X be a self homotopy equivalence of a finite connected CW -complex. Let T_f be its mapping torus.

Then all L²-Betti numbers b_n⁽²⁾(Te_f)vanish.

Definition (Universal torsion for group automorphisms)

Letf:G→Gbe a group automorphism of the groupG. Suppose that there is a finite model forBGandGsatisfies the Atiyah Conjecture.

Then we can define theuniversalL²-torsionoff by

ρ⁽²⁾_u (f) :=ρ⁽²⁾(Te_f;N(Gof Z))∈Wh^w(Gof Z)

This seems to be a very powerful invariant which needs to be investigated further.

It has nice properties, e.g., it depends only on the conjugacy class off, satisfies asum formulaand a formula forexact sequences.

IfGis amenable, it vanishes.

IfGis the fundamental group of a compact surfaceF andf comes from an automorphisma:F →F, thenT_f is a 3-manifold and a lot of the material above applies.

For instance, ifais irreducible,ρ⁽²⁾_u (f)detects whetherais pseudo-Anosovsince we can read off the sum of the volumes of the hyperbolic pieces in the Jaco-Shalen decomposition ofT_f.

Suppose thatH₁(f) =id. Then there is an obvious projection pr:H₁(Gof Z)_f =H₁(G)_f ×Z→H₁(G)_f. Let

P(f)∈ P(R⊗_ZH₁(G)_f) be the image ofρ⁽²⁾_u (f)under the composite

Wh^w(Go Z)−→ P(^P R⊗_ZH₁(Gof Z))−−−→ P(^P(pr) R⊗_ZH₁(G)_f)

What are the main properties of this polytope? In which situations can it be explicitly computed? The case, whereF is a finitely generated free group, is of particular interest.

Summary (continued)

The Thurston norm can also be read of from anL²-Euler characteristic.

Thehigher order Alexander polynomialsdue toHarveyand

Cochraneare special cases of the universalL²-torsion and we can prove the conjecture that their degree is the Thurston norm.

The universalL²-torsion seems to give an interesting invariant for elements inOut(Fn)andmapping class groups.