Classical torsion and L

Classical torsion and L

-torsion

Wolfgang Lück Bonn Germany

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

Bonn, 29. May & 12. June

Outline

The first algebraicK-group and the Whitehead group.

Torsion invariants for chain complex.

Whitehead torsion.

The Alexander polynomial.

L²-torsion and its applications.

2

The first algebraic K -group and the Whitehead group

Definition (K1-groupK1(R))

Define theK₁-group of a ringR to be the abelian groupK₁(R), whose generators are conjugacy classes[f]of automorphismsf:P →Pof finitely generated projectiveR-modules with the following relations:

Given an exact sequence 0→(P₀,f₀)→(P₁,f₁)→(P₂,f₂)→0 of automorphisms of finitely generated projectiveR-modules, we get

[f₁] = [f₀] + [f₂];

Given two automorphismsf,gof the same finitely generated projectiveR-module, we get

[g◦f] = [f] + [g].

K₁(R)is isomorphic toGL(R)/[GL(R),GL(R)].

An invertible matrixA∈GL(R)can be reduced byelementary row and column operationsand(de-)stabilizationto the trivial empty matrix if and only if[A] =0 holds in thereducedK₁-group

Ke₁(R):=K₁(R)/{±1}=cok(K₁(Z)→K₁(R)). IfRis commutative, the determinant induces an epimorphism

det:K₁(R)→R^×,

which in general is not bijective. It is bijective, ifRis a field.

The assignmentA7→[A]∈K₁(R)can be thought of as the universal determinant forR.

2

Definition (Whitehead group)

TheWhitehead groupof a groupGis defined to be Wh(G)=K₁(ZG)/{±g|g ∈G}.

Lemma

We haveWh({1}) ={0}.

Proof.

The ringZpossesses anEuclidean algorithm.

Hence every invertible matrix overZcan be reduced via elementary row and column operations and destabilization to a (1,1)-matrix(±1).

This implies that any element inK₁(Z)is represented by±1.

LetGbe a finite group. LetF beQ,RorC.

Definer_F(G)to be the number of irreducibleF-representations of G.

The Whitehead group Wh(G)is a finitely generated abelian group of rankr_R(G)−r_Q(G).

The torsion subgroup of Wh(G)is the kernel of the map K₁(ZG)→K₁(QG).

In contrast toKe₀(ZG)the Whitehead group Wh(G)is computable.

2

Example (Non-vanishing of Wh(Z/5))

The ring homomorphismf:Z[Z/5]→C, which sends the

generator ofZ/5 to exp(2πi/5), and the mapC→R^>0taking the norm of a complex number, yield a homomorphism of abelian groups

φ: Wh(Z/5)→R^>0.

Since(1−t−t⁻¹)·(1−t²−t³) =1 inZ[Z/5], we get the unit 1−t−t⁻¹∈Z[Z/5]^×. Its class in the Whitehead group is sent to (1−2 cos(2pi/5))6=1 and hence is an element of infinite order.

Indeed, this element is a generator of the infinite cyclic group Wh(Z/5).

Conjecture

If G is torsionfree, thenWh(G)is trivial.

Torsion invariants for chain complex

LetR be an associative ring with unit, not necessarily commutative.

R-modules are by default leftR-modules.

AnR-chain complexC∗ is calledfiniteif there exists a natural numberN such thatC_n=0 for everyn∈Zwith|n|>N andC_nis finitely generated for everyn∈Z.

AnR-chain complexC∗ is calledfreeorprojectiveifCnis free or projective for alln∈Z.

A freeR-chain complexC∗is calledbased freeifC_ncomes with a (unordered) basisB_nforn∈Z.

AnR-chain complexC_∗ is calledacyclic ifH_n(C_∗)vanishes for everyn∈Z.

2

Anchain contractionγ∗ for aR-chain complexC∗ = (C∗,c∗)is a sequence ofR-mapsγn:C_n→C_n+1satisfying

c_n+1◦γn+γn−1◦cn=id_Cn for alln∈Z.

AnR-chain complexC∗ is calledcontractibleif it possesses a chain contraction.

A contractibleR-chain complex is acyclic.

The converse is not true in general, e.g.,R=ZandC∗ is concentrated in dimensions 0,1,2 and given there by the exact sequenceZ

2·id_Z

−−−→Z

−pr→Z/2.

However, a projectiveR-chain complexC_∗is acyclic if and only if it iscontractible.

LetC∗ be an acyclic finite projectiveR-chain complex.

PutC_odd=L

n∈ZC_2n+1andCev =L

n∈ZC_2n.

Letγ∗ andδ∗be two chain contractions. PutΘ∗ =γ∗+1◦δ∗. Then the composite

C_odd−−−−−→^(c+γ)odd C_ev −−−−−→^(id+Θ)ev C_ev−−−−→^(c+δ)ev C_odd is given by an upper triangular matrix whose entries on the diagonal are identity morphisms. Also(id+Θ)evis given by an upper triangular matrix whose entries on the diagonal are identity morphisms. The analogous statement holds if we interchange odd and ev, andγ∗ andδ∗.

In particular we see that(c+γ)odd:C_odd →C_ev is an isomorphism.

2

Letf:M −^∼=→Nbe an isomorphism of finitely generated based free R-modules. LetAbe the matrix describingf with respect to the given bases. Then we define

τ(f)∈Ke₁(R)

by the class ofAcoming from the identification ofK₁(R)with GL(R)/[GL(R),GL(R)], where

Ke₁(R):=cok(K₁(Z)→K₁(R)) =K₁(R)/{(±1)}.

Equivalently, choose an isomorphismb:N→M which respects the given bases. Thenτ(f) = [b◦f].

The fact that the bases are only unordered does not affect the definition ofτ(f)since we are working inKe₁(R).

If we have a commutative diagram of finitely generated based free R-modules

0 ^//M ^//

f

N ^//

g

P ^//

g

0

0 ^//M^{0 //}N^{0 //}P^{0 //}0

such that the vertical arrows are bijective and the rows are based exact, then we get inKe₁(R).

τ(g) =τ(f) +τ(h).

Iff:M →Nandg:N →Pare isomorphisms of finitely generated based freeR-modules, then we get inKe₁(R)

τ(g◦f) =τ(g) +τ(f).

2

LetC∗ be an acyclic finite based freeR-chain complex. Choose a chain contractionγ∗. Define

τ(C∗)∈Ke₁(R) byτ (c+γ)_odd:C_odd→C_ev

.

This is independent of the choice ofγ∗, since we get for any other chain contractionδ∗ from the facts above and the observation that an upper triangular matrix with identities on the diagonal

represents zero inKe₁(R)the equality inKe₁(R) τ (c+γ)odd

=−τ (c+δ)ev

.

Letf∗:C∗→D∗ be anR-chain homotopy equivalence of finite based freeR-chain complexes. Let cone(f∗)be its mapping cone

· · · →Cn⊕D_n+1





−c_n 0 fn d_n+1





−−−−−−−−−−−→Cn−1⊕Dn→ · · · This is a contractible finite based freeR-chain complex.

Definition (Whitehead torsion) Define theWhitehead torsionoff∗by

τ(f∗):=τ(cone(f∗))∈Ke₁(R).

2

If we have a commutative diagram of finitely generated based free R-chain complexes

0 ^//C∗ //

f∗

D∗ //

g∗

E∗ //

h∗

0

0 ^//C_∗^{0 //}D_∗^{0 //}E_∗^{0 //}0

such that the vertical arrows areR-chain homotopy equivalences and the rows are based exact, then we get inKe₁(R)

τ(g∗) =τ(f∗) +τ(h∗).

Iff∗:C∗ →D∗ andg∗:D∗→E∗ areR-chain homotopy

equivalences of finite based freeR-chain complexes, then we get inKe₁(R)

τ(g∗◦f∗) =τ(g∗) +τ(f∗).

Letf∗,g∗:C∗ →D∗beR-chain homotopy equivalences of finite based freeR-chain complexes. If they areR-chain homotopic, then we get inKe₁(R)

τ(f∗) =τ(g∗).

Letf∗:C∗→D∗ be an isomorphism of finite based freeR-chain complexes, not necessarily preserving the bases. Then

τ(f∗) =X

n∈Z

(−1)ⁿ·τ(fn).

2

Example (1-dimensional case)

If the acyclic finite based freeC∗is concentrated in two consecutive dimensionsnandn−1, then then-th differentialc_n:C_n→C_n−1is an isomorphism of finitely generated based freeR-modules and

τ(C∗) = (−1)ⁿ⁺¹·τ(c_n).

since we get a chain contractionγ∗ by puttingγn−1=c_n⁻¹and [cn] =−[γ_n]holds inKe₁(R).

Example (2-dimensional case)

Suppose that the acyclic based freeR-chain complexC∗ is concentrated in two dimensions. Then it is the same as a short exact sequence of finitely generated based freeR-modules

0→C₂−→^c2 C₁−→^c1 C₀→0

One easily checks that there exists aR-mapγ1:C₁→C₂with γ₁◦c₂=id_C2. Moreover, for any suchγ₁, one can find aR-map γ₀:C₀→C₁, such that we get a chain contractionγ∗.

Henceτ(C∗)is represented by the isomorphism of finitely generated based freeR-modules

C₁





c₁ γ₁





−−−−→C₀⊕C₂

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R -chain complexes

Example (R =R)

Taking the logarithm of the absolute value of the determinant of an invertible matrix induces an isomorphism

Ke₁(R)−^∼=→R, [A]7→ln(|det(A)|).

Hence the Whitehead torsion is just a real number.

Consider the finite based free 1-dimensionalR-chain complexes

C∗ andD∗ given byc₁:R²





1 2 2 4





−−−−−−→R²andd₁:R−→⁰ R.

Example (Continued)

Define a chain mapf∗:C∗ →D∗ by the commutative diagram

R²





1 2 2 4





//

1 0

R²

−8 4

R

0

//R

One easily checks thatf∗ induces an isomorphism on homology and hence is aR-chain homotopy equivalence

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Example (Continued)

Its mapping cone is the 2-dimensional finite acyclic based free R-chain complex

R²









−1 −2

−2 −4

1 0









−−−−−−−−−−→R²⊕R

−8 4 0

−−−−−−−−−→R

A retractionγ₁:R²⊕R→R²of its second differential is given by 0 0 1

−1/2 0 −1/2

Example (Continued)

Hence(c+γ)oddis given by the isomorphism of finitely generated based freeR-modules

R³









0 0 1

−1/2 0 −1/2

−8 4 0









−−−−−−−−−−−−−−−→R³

Since its determinant is−2, we get τ(f∗) =ln(2).

2

Notice that we do not need aR-basis, it suffices to have a Hilbert space structure on eachR-chain module.

Namely, then one can just choose an orthonormal basis and define the torsion using this basis. If we choose another orthonormal basis, then the change of bases matrix is an orthogonal matrix and its determinant is±1.

Hence we can define for anyR-chain homotopy equivalence f∗:C∗ →D∗of finite HilbertR-chain complexes its Whitehead torsion

τ(f∗)∈R.

Definition (Laplace operator)

LetC∗ be a finite HilbertR-chain complex. Define then-thLaplace operator

∆n=c_n^∗◦c_n+c_n+1◦c_n+1^∗ :C_n→C_n.

∆nis a positiveR-homomorphism, and we have the orthogonal decomposition

C_n=M

λ≥0

E_λ(∆_n),

whereE_λ(∆n)is the eigenspace of∆n for the eigenvalueλ >0.

Suppose that∆nis invertible, or, equivalently, that 0 is no eigenvalue. Then by functional calculus we get aR-map

ln(∆n) : Cn →Cn

This is the operator which is ln(λ)·id onE_λ(∆n).

2

Obviously we get

ln(det(∆_n)) =tr(ln(∆_n)), provided that∆_nis invertible.

Notice that both sides of the equation above are defined without choosing any basis.

Lemma

Let C∗ be a finite HilbertR-chain complex.

1 C∗ is acyclic if and only if∆nis an isomorphism for each n∈Z;

2 If C∗is acyclic, then we get τ(C∗) =−1

2 ·X

n∈Z

(−1)ⁿ·n·ln(det(∆n))∈R.

Proof.

We have forx ∈C_n

h∆_n(x),xi = hc_n^∗◦c_n(x) +c_n+1◦c_n+1^∗ (x),xi

= hc_n^∗◦c_n(x),xi+hc_n+1◦c_n+1^∗ (x),xi

= hc_n(x),cn(x)i+hc^∗_n+1(x),c^∗_n+1(x)i

= ||c_n(x)||²+||c_n+1^∗ (x)||².

This implies

ker(∆_n) =ker(c_n)∩ker(c_n+1^∗ ) =ker(c_n)∩im(c_n+1)^⊥

∼=

−→ker(cn)/im(c_n+1) =Hn(C∗).

2

Proof (Continued).

We explain the proof of the second assertion only in the special case, whereC∗ is concentrated in dimensionspand(p−1).

We have

det(c_p)²=det(c_p◦c_p^∗) =det(c_p^∗◦c_p).

Hence we get

ln(|det(cp)|) = 1

2 ·ln(det(∆p−1)) = 1

2 ·ln(det(∆p)).

Proof (Continued).

We compute

τ(C∗) = (−1)^p+1·ln(|det(c_p)|)

= (−1)^p+1·1

2·ln(det(∆p))

= −1

2· (−1)^p·p·ln(det(∆p))

+(−1)^p−1·(p−1)·ln(det(∆_p))

= −1

2· (−1)^p·p·ln(det(∆p))

+(−1)^p−1·(p−1)·ln(det(∆_p−1))

= −1 2·X

n∈Z

(−1)ⁿ·n·ln(det(∆n)).

2

Change of rings

Often we will also consider the following situation, where φ:R→Sis a fixed ring homomorphism.

LetC_∗ be a finite freeR-chain complex, not necessarily based.

Suppose thatφ∗C∗:=S⊗_RC∗is acyclic.

Next we define an element

τ(C∗)∈cok φ∗:K₁(R)→K₁(S) .

Choose anR-basisB∗forC∗. It induces anS-basisφ∗B forφ∗C∗

in the obvious way. Hence(C_∗, φ∗B)is an acyclic finite based free S-chain complex andτ(C_∗, φ∗B)∈Ke₁(S)is defined.

Letτ(C∗)∈cok φ∗:K₁(R)→K₁(S)

be the class ofτ(C∗;φ∗B).

One easily checks thatτ(C∗)does not depend on the choice ofB∗.

Example (Milnor torsion)

Letφ:R→Sbe the inclusionZ→Q.

LetC∗be a finite freeZ-chain complex such thatQ⊗_ZC∗ is acyclic. The latter condition is equivalent to the requirement that Hn(C∗)is a finite abelian group for everyn∈Z.

The cokernel ofK₁(Z)→K₁(Q)is by definitionKe₁(Q). Taking the norm of the determinant of an invertible matrix yields an

isomorphism

Ke₁(Q)−^∼=→Q^>0. Henceτ(C∗)is a positive rational number.

It is not hard to check

τ(C∗) =Y

n∈Z

|H_n(C∗)|⁽⁻¹⁾ⁿ.

2

Whitehead torsion

Letf:X →Y be a homotopy equivalence of connected finite CW-complexes.

Ifπ=π1(X) =π1(Y)is the fundamental group,f lifts to a π-homotopy equivalenceef:Xe →Ye of the universal coverings which are finite freeπ-CW-complexes.

By passing to the cellular chain complexes, we obtain aZπ-chain homotopy equivalenceC∗(ef) :C∗(Xe)→C∗(Ye).

After a choice of a lift of each open celleinX to a open celleeinXe and of an orientation onee, we obtain a preferredZπ-basis for C∗(Xe), and analogously forC∗(Ye).

Hence we obtain an elementτ(C∗(ef))∈Ke₁(Zπ).

The class[τ(C∗(ef))]in Wh(π)is independent of the choices.

Definition (Whitehead torsion)

We defineWhitehead torsionof the homotopy equivalencef:X →Y of connected finiteCW-complexes

τ(f):= [τ(C_∗(ef))]∈Wh(π).

2

The pair(Dⁿ,S₊ⁿ⁻¹)carries an obvious relativeCW-structure with one(n−1)-cell and onen-cell.

DefineY as the pushoutX∪_f Dⁿfor any mapf:S₊ⁿ⁻¹→X. The inclusionj:X →Y is a homotopy equivalence and called an elementary expansion.

There is a mapr:Y →X withr ◦j =id_X. This map is unique up to homotopy relativej(X)and is called anelementary collapse.

Definition (Simple homotopy equivalence)

Letf:X →Y be a map of finiteCW-complexes. We call it asimple homotopy equivalence, if there is a sequence of maps

X =X[0]−→^f0 X[1]−→^f1 X[2]−→ · · ·^f2 −−→^fn−1 X[n] =Y

such that eachf_i is an elementary expansion or elementary collapse andf is homotopic to the composition of the mapsf_i.

Theorem (Main properties of Whitehead torsion)

A homotopy equivalence of connected finite CW -complexes is a simple homotopy equivalence if and only ifτ(f)∈Wh(π)vanishes;

The Whitehead torsionτ(f)is a homotopy invariant;

There are sum and product formulas for it;

If X and Y are finite CW -complexes and f:X →Y is a homeomorphism, then f is a simple homotopy equivalence;

The Whitehead groupWh(π)and the Whitehead torsion of an h-cobordism W over M₀defined in a previous lecture coincide with the Whitehead group and the Whitehead torsion of the inclusion M₀→W in the sense of this lecture.

2

The notion ofReidemeister torsionforlens spaces, which led to the classification of lens spacesL(V) :=SV/(Z/m)up to

isometric diffeomorphism, or diffeomorphism, or homeomorphism, is a special case of the constructions above for the ring

homomorphismφ:Z[Z/m]→Q[Z/m]→Q(Z[Z/m])/(N), where N is the norm element.

The point is that for a lens spaceL(V)theQ(Z[Z/m])/(N)-chain complexφ∗C∗(LVf)is acyclic, which is a direct consequence of the fact thatZ/macts trivially onH∗(SV) =H∗(L(V])).

The Alexander polynomial

Definition (knot)

AknotK ⊆S³is a connected oriented 1-dimensional smooth submanifold ofS³.

We call two knotsK ⊆S³andK⁰ ⊆S³equivalentif there exists an orientation preserving diffeomorphismf:S³→S³such that f(K) =K⁰ and the induced diffeomorphismf|_K:K →K⁰ respects the orientations.

One can define knots also as smooth embeddingsS¹→S³and then equivalent means isotopy of embeddings.

2

If(N, ∂N)is a tubular neighborhood ofK, then define M_K =S³\int(N).

This is a compact 3-manifoldM_K whose boundary consists of one component which is a 2-torus.

If(N⁰, ∂N⁰)is another tubular neighborhood, then there is a diffeomorphism of compact 3-manifolds with boundary

S³\int(N)−^∼=→S³\int(N⁰).

Hence we writeM_K without taking the tubular neighborhood into account.

M_k is homotopy equivalent to theknot complementM\K.

The knotK is trivial if and only ifM_K is homeomorphic toS¹×D². IfK is non-trivial, thenM_k is an irreducible compact 3-manifold with incompressible boundary.

Consider a knotK ⊆S³.

By Alexander-Lefschetz dualityH¹(M_K;Z)∼=Z. Hence there is a preferred infinite cyclic coveringp:M_K →M_k.

Consider the inclusionφ:Z[Z]→Q[Z]→Q[Z]₍₀₎, whereQ[Z]₍₀₎is the quotient field of the integral principal ideal domainQ[Z]. Then it turns out thatφ∗C_∗(M_K)isQ[Z]₍₀₎-acyclic.

We have defined above

τ(C∗(M_K))∈cok K₁(Z[Z])→K₁(Q[Z]₍₀₎) .

The determinant induces an isomorphism K₁(Q[Z]₍₀₎)−^∼=→(Q[Z]₍₀₎)^×.

Elements in(Q[Z]₍₀₎)^×are quotientsp(t)/q(t)forp(t),q(t)∈Q[t]

withq6=0. Hence we get an identification cok K₁(Z[Z])→K₁(Q[Z]₍₀₎) ^∼=

−→(Q[Z]₍₀₎)^×/{±tⁿ}.

2

So we get a knot invariant

τ(C∗(M_K))∈(Q[Z]₍₀₎)^×/{±tⁿ}.

One can assign to a knotK itsAlexander polynomial∆K which is a symmetric finite Laurent series inZ[Z]such that its evaluation at t =1 is 1.

We have the following values of∆_K

K ∆_K

unknot t⁰

trefoil t²−t⁰+t⁻² figure eight knot −t+3t⁰−t⁻¹

Theorem (Alexander polynomial and torsion invariants,(Milnor)) If K ⊆S³is a knot, then we get in(Q[Z]₍₀₎)^×/{±tⁿ}the equality

[(t−1)·∆_K(t)] =τ(C∗(M_K)).

In particular∆K andτ(C_∗(M_K))determine one another.

Question

Can we define an interesting torsion invariant for the universal covering of M_K which gives new information about K ?

2

The definition of L

-torsion

We have defined for an acyclic finite HilbertR-chain complexC∗

its torsion to be the real number τ(C_∗) :=X

n∈Z

(−1)ⁿ·n·ln(det(∆_n))∈R.

So one can try to make sense of the same expression when we consider a finiteN(G)-chain complexC∗⁽²⁾, and declare this to be theL²-torsion ofC_∗⁽²⁾.

The condition acyclic should become the conditionweakly acyclic, i.e.,b⁽²⁾_n (C∗⁽²⁾) =0, or, equivalently,H_n⁽²⁾(C∗⁽²⁾) =0 for alln∈Z. The Laplace operator can be defined as before

∆⁽²⁾_n := (c_n⁽²⁾)^∗◦c_n⁽²⁾+c_n+1⁽²⁾ ◦(c_n+1⁽²⁾ )^∗:C_n⁽²⁾→C_n⁽²⁾.

The Laplace operator∆⁽²⁾_n is a weak isomorphism for alln∈Zif and only ifC∗⁽²⁾is acyclic.

The main problem is to make sense of ln(det(∆⁽²⁾_n )). If this has been solved, we can define

Definition (L²-torsion(Lück-Rothenberg))

LetX be a connected finiteCW-complex. Then we define the L²-torsion

ρ⁽²⁾(Xe):=−1 2 ·X

n≥0

(−1)ⁿ·n·ln(det(∆⁽²⁾_n )) ∈R,

where∆⁽²⁾_n :C_n⁽²⁾(Xe)→C_n⁽²⁾(Xe)is the Laplace operator for the finite HilbertN(π)-chain complexC∗⁽²⁾(Xe) :=L²(π)⊗_ZπC∗(Xe).

2

The definition above extends to finiteCW-complexes by taking the sum of theL²-torsion for each path component.

There is also an analytic definition in terms of heat kernels of the universal covering of a closed Riemannian manifold due to MattheyandLott. Both approaches have been identified by Burghelea-Friedlander-Kappeler-Mc Donald.

Explicit computations and the proof of some general properties are based on both approaches.

The Fuglede-Kadison determinant

Here is more information about the term ln(det(∆⁽²⁾_n )).

Consider a bijective positive operatorf:V →V of finite-dimensional Hilbert spaces with trivial kernel. Let 0< λ1< λ2< λ3< . . .be its eigenvalues andµi be the multiplicity ofλ_i. Then

ln(det(f)) =X

i≥1

µ_i·ln(λ_i) =tr(ln(f)).

Define the spectral density functionF: [0,∞)→[0,∞)to be the right-continuous step function, which has a jump at each of the eigenvalues of height its multiplicity, and which is zero forλ <0.

2

We can write

ln(det(f)) =X

i≥1

µ_i·ln(λ_i) = Z ∞

0+

ln(λ)dF wheredF is the measure onRassociated to the monotone increasing right-continuous functionF which is given by dF (a,b]

:=F(b)−F(a).

Iff:L²(G)^k →L²(G)^k is a positive boundedG-equivariant operator, we define itsspectral density function

F(f)(λ):=dim_N(G) im(E_λ^f)

=tr_N(G) E_λ^f

whereE_λ^f:L²(G)^k →L²(G)^k is its spectral projection forλ≥0.

Now the following expression makes sense:

Z ∞ 0+

ln(λ)dF ∈Rq {−∞}.

We define the logarithm of theFuglede-Kadison determinant ln(det(f)) :=

Z ∞ 0+

ln(λ)dF ∈R, provided thatR∞

0+ln(λ)dF >−∞holds.

We haveR∞

0+ln(λ)dF >−∞iff is bijective, but there are weak isomorphismsf withR∞

0+ln(λ)dF =−∞.

2

Now the observation comes into play that the Laplace operator coming from a cellular structure lives already over the integers and the following

Conjecture (Determinant Conjecture)

Let A∈M_a,b(ZG)be a matrix. It defines a bounded G-equivariant operator r_A⁽²⁾:L²(G)^m →L²(G)ⁿ. We have

ln

det (r_A⁽²⁾)^∗◦r_A⁽²⁾ :=

Z ∞ 0+

ln(λ)dF ≥0.

This conjecture is known for a very large class of groups, for instance for all sofic groups.

Therefore we will tacitly assume that this conjecture holds and ln(det(∆⁽²⁾_n ))is defined for the n-th Laplace operator∆⁽²⁾_n .

Lehmer’s problem

Before we return toL²-torsion, here is a very interesting aside concerning Fuglede-Kadison determinants and Mahler measures.

Definition (Mahler measure)

Letp(z)∈C[Z] =C[z,z⁻¹]be a non-trivial element. Write it as p(z) =c·z^k ·Qr

i=1(z−a_i)for an integerr ≥0, non-zero complex numbersc,a₁,. . .,a_r and an integerk. Define itsMahler measure

M(p)=|c| · Y

i=1,2,...,r

|a_i|>1

|a_i|.

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The following famous and open problem goes back to a question ofLehmer[6].

Problem (Lehmer’s Problem)

Does there exist a constantΛ>1such that for all non-trivial elements p(z)∈Z[Z] =Z[z,z⁻¹]with M(p)6=1we have

M(p)≥Λ.

There is even a candidate for which the minimal Mahler measure is attained, namely,Lehmer’s polynomial

L(z) :=z¹⁰+z⁹−z⁷−z⁶−z⁵−z⁴−z³+z+1.

It is actual−z⁵·∆(z)for the Alexander polynomial∆(z)of a bretzel knot given by(2,3,7).

It is conceivable that for any non-trivial elementp ∈Z[Z]with M(p)>1

M(p)≥M(L) =1.17628. . . holds.

For a survey on Lehmer’s problem we refer for instance to [1, 2, 4, 8].

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Lemma

The Mahler measure m(p)is the square root of the Fuglede-Kadison determinant of the operator L²(Z)→L²(Z)given by multiplication with p(z)·p(z).

Definition (Lehmer’s constant of a group) DefineLehmer’s constantof a groupG

Λ(G)∈[1,∞)

to be the infimum of the set of Fuglede-Kadison determinants det⁽²⁾_N(G) r_A⁽²⁾:L²(G)^r →L²(G)^r

,

whereAruns through all(r,r)-matrices with coefficients inZGfor all r ≥1, for whichr_A⁽²⁾:L²(G)^r →L²(G)^r is a weak isomorphism and the Fuglede-Kadison determinant satisfies det⁽²⁾_N(G)(r_A⁽²⁾)>1.

We can show

Λ(Zⁿ)≥M(L)

for alln≥1, provided that Lehmer’s problem has a positive answer.

We know 1≤Λ(G)≤M(L)for torsionfreeG.

Problem (Generalized Lehmer’s Problem) For which torsionfree groups G does

Λ(G) =M(L) hold?

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Basic properties of L

-torsion

Next we record the basic properties ofL²-torsion. It behaves similar to the Euler characteristic.

Theorem (Simple homotopy invariance)

Let f:X →Y be a homotopy equivalence of finite CW -complexes.

Suppose thatX and hence alsoe Y are Le ²-acyclic.

Then there is a homomorphism depending only onπ Φ_π: Wh(π)→R

sendingτ(f)toρ⁽²⁾(Ye)−ρ⁽²⁾(Xe).

If Wh(π)vanishes, theL²-torsion is a homotopy invariant.

Theorem (Sum formula)

Let X be a finite CW -complex with subcomplexes X₀, X₁and X₂ satisfying X =X₁∪X₂and X₀=X₁∩X₂. SupposeXf₀,Xf₁andXf₂are L²-acyclic and the inclusions X_i →X areπ-injective.

ThenX is Le ²-acyclic and we get

ρ⁽²⁾(Xe) =ρ⁽²⁾(Xf₁) +ρ⁽²⁾(Xf₂)−ρ⁽²⁾(Xf₀).

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Theorem (Fibration formula)

Let F →E →B be a fibration of connected finite CW -complexes such thatF is Le ²-acyclic and the inclusion F →E isπ-injective.

ThenE is Le ²-acyclic and we get

ρ⁽²⁾(Ee) =χ(B)·ρ⁽²⁾(Fe).

By Poincaré duality we haveρ⁽²⁾(M) =e 0 for every even dimensional closed manifoldM, provided thatMe isL²-acyclic.

TheL²-torsion ismultiplicative under finite coverings, i.e., if X →Y is ad-sheeted covering of connected finite

CW-complexes andXe isL²-acyclic, thenYe isL²-acyclic and ρ⁽²⁾(Xe) =d·ρ⁽²⁾(Ye).

In particularSf¹isL²-acyclic and

ρ⁽²⁾(Sf¹) =0.

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Theorem (S¹-actions on aspherical manifolds(Lück))

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

ThenM is Le ²-acyclic and

ρ⁽²⁾(M) =e 0.

Theorem (L²-torsion and asphericalCW-complexes,Wegner) Let X be an aspherical finite CW -complex. Suppose that its

fundamental groupπ₁(X)contains an elementary amenable infinite normal subgroup.

ThenX is Le ²-acyclic and

ρ⁽²⁾(Xe) =0.

Theorem (Hyperbolic manifolds,(Hess-Schick, Olbrich)) There are (computable) rational numbers rn>0such that for every hyperbolic closed manifold M of odd dimension2n+1the universal coveringM is Le ²-acyclic and

ρ⁽²⁾(M) = (−1)e ⁿ·π⁻ⁿ·r_n·vol(M).

Since for every hyperbolic manifoldM we have Wh(π₁(M)) =0, we rediscover the fact that the volume of an odd-dimensional hyperbolic closed manifold depends only onπ1(M).

We also rediscover the theorem that anyS¹-action on a closed hyperbolic manifold is trivial.

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The proof is based on the fact that the analytic version of L²-torsion is of the shape

ρ⁽²⁾(M) =e Z

F

f(x)dvol_H2n+1

whereF is a fundamental domain of theπ-action on the

hyperbolic spaceH²ⁿ⁺¹andf(x)is an expression in terms of the heat kernelk(x,x)(t).

By the symmetry ofH²ⁿ⁺¹this functionk(x,x)(t)is independent ofx and hencef(x)is independent ofx.

If we taker_n= (−1)ⁿ·πⁿ·f(x)for anyx ∈H²ⁿ⁺¹, we get Z

F

f(x)dvol_H2n+1 = (−1)ⁿ·π⁻ⁿ·r_n·vol(F) = (−1)ⁿ·π⁻ⁿ·r_n·vol(M).

We haver₁= ¹₆,r₂= ³¹₄₅,r7= ²²¹₇₀.

Theorem (Lott-Lück-,Lück-Schick)

Let M be an irreducible closed3-manifold with infinite fundamental group. Let M₁, M₂, . . . , M_mbe the hyperbolic pieces in its Jaco-Shalen decomposition.

ThenM is Le ²-acyclic and

ρ⁽²⁾(M) :=e − 1 6π ·

m

X

i=1

vol(M_i).

The proof of the result above is based on the meanwhile approved Thurston Geometrization Conjecture. It reduces the claim to Seifert manifolds with incompressible torus boundary and to hyperbolic manifolds with incompressible torus boundary using the sum formula. The Seifert pieces are treated analogously to aspherical closed manifolds withS¹-action. The hyperbolic pieces require a careful analysis of the cusps.

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Combinatorial approach in low dimensions

Here is a recipe to read of theL²-torsion for an irreducible 3-manifold with incompressible toroidal boundary from its fundamental groupπ, provided thatπis infinite.

Let

π =hs₁,s₂, . . .s_g |R₁,R₂, . . .R_ri be a presentation ofπ.

Let the(r,g)-matrix

F =









∂R₁

∂s₁ . . . ^∂R_∂s¹ .. g

. . .. ...

∂Rr

∂s₁ . . . ^∂R_∂s^r

g









be the Fox matrix of the presentation (see [3, 9B on page 123], [5], [7, page 84]).

Now there are two cases:

1 Suppose∂M is non-empty andg =r+1. DefineAto be the (g−1,g−1)-matrix with entries inZπobtained from the Fox matrix F by deleting one of the columns.

2 Suppose∂M is empty andg =r. DefineAto be the

(g−1,g−1)-matrix with entries inZπobtained from the Fox matrix F by deleting one of the columns and one of the rows.

LetK be any positive real number satisfyingK ≥ kR⁽²⁾_A k. A possible choice forK is the product of(g−1)²and the maximum over the word length of those relationsR_i whose Fox derivatives appear inA.

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Define forx =P

w∈πn_w ·w ∈Zπ

tr_Zπ(x) =λe∈Z. Then the sum of non-negative rational numbers

L

X

p=1

1

2p ·tr_Zπ (1−K⁻²·AA^∗)^p converges forL→ ∞toρ⁽²⁾(M) + (ge −1)·ln(K).

More precisely, there is a constantC >0 and a numberα >0 such that we get for allL≥1

0≤ρ⁽²⁾(M) + (ge −1)·ln(K)−

L

X

p=1

1

2p ·tr_Zπ (1−K⁻²·AA^∗)^p

≤ C L^α.

Example (Figure eight knot)

LetK ⊆S³be the figure eight knot;

Its complement is a hyperbolic 3-manifold.

It fibers overS¹and the fiber is a surface whose fundamental group is the free groupF₂in two generatorss₁ands₂. The automorphism ofF₂is given bys₁7→s₂ands₂7→s₂³s₁⁻¹. We get the presentation forπ=π₁(M_K)∼=F₂o Z

π = hs₁,s₂,t |ts₁t⁻¹s₂⁻¹=ts₂t⁻¹s₁s₂⁻³=1i.

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Example (continued)

If we delete from the Fox matrix the column belonging tos₂, we obtain the matrix

A =

t 1−s₂ s³₂s⁻¹₁ 1−s³₂s⁻¹₁

The numberK =4 is greater or equal to the operator norm of the boundedπ-equivariant operator induced byA.

Example (continued)

Define the(2,2)-matrixB= (b_i,j)overZπ by b_1,1 = 13+s₂+s⁻¹₂ ;

b_1,2 = −1+s₂+s₁s₂³−s₂s₁s₂⁻³−ts₁s₂⁻³;

b_2,1 = −1+s₂⁻¹+s³₂s⁻¹₁ −s³₂s⁻¹₁ s⁻¹₂ −s³₂s⁻¹₁ t⁻¹; b_2,2 = 13+s₂³s₁⁻¹+s₁s⁻³₂ .

SinceB=16−AA^∗, we get:

1

6π vol(M_K) =−ln(ρ(K₈)) =8·ln(2)−

∞

X

p=1

1

p·16^p ·tr_Zπ(B^p).

The volume ofM_K is about 2.02988.

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Group automorphisms

Letf:G→Gbe an automorphism of a groupGfor which there exists a finiteCW-modelX forBG.

Letbf:X →X be any selfhomotopy equivalence such thatπ₁(bf) andf agree up to inner automorphisms ofG.

Then the mapping torusT

bf is a connected finiteCW-complex, which isL²-acyclic. Hence itsL²-torsionρ⁽²⁾(Te_f)∈Ris defined.

It depends only onf and not on the choices ofX andbf since the simple homotopy type ofT

bf is independent of these choices.

Hence we get a well-defined element ρ⁽²⁾(f):=ρ⁽²⁾(T

bf)∈R.

Theorem

Suppose that all groups appearing below have finite CW -models for their classifying spaces.

Suppose that G is the amalgamated product G₁∗_G0 G₂for subgroups G_i ⊂G and the automorphism f:G→G is the amalgamated product f₁∗_f

0f₂for automorphisms f_i:G_i →G_i. Then

ρ⁽²⁾(f) =ρ⁽²⁾(f₁) +ρ⁽²⁾(f₂)−ρ⁽²⁾(f₀);

Let f:G→H and g:H→G be isomorphisms of groups. Then ρ⁽²⁾(f◦g) =ρ⁽²⁾(g◦f).

In particularρ⁽²⁾(f)is invariant under conjugation with automorphisms;

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Theorem (continued)

Suppose that the following diagram of groups 1 ^//G₁

f₁

i //G₂

f₂

p //G₃

id //1

1 ^//G₁ ^{i //}G₂ ^{p //}G₃ ^//1 commutes, has exact rows and its vertical arrows are automorphisms. Then

ρ⁽²⁾(f₂) =χ(BG₃)·ρ⁽²⁾(f₁);

Let f:G→G be a group automorphism. Then for all integers n≥1

ρ⁽²⁾(fⁿ) =n·ρ⁽²⁾(f);

Theorem (continued)

Suppose that G contains a subgroup G₀of finite index[G:G₀].

Let f:G→G be an automorphism with f(G₀) =G₀. Then ρ⁽²⁾(f) = 1

[G:G₀] ·ρ⁽²⁾(f|_G

0);

We haveρ⁽²⁾(f) =0if G satisfies one of the following conditions:

All L²-Betti numbers of the universal covering of BG vanish;

G contains an amenable infinite normal subgroup.

If h:S→S is a pseudo-Anosov selfhomeomorphism of a connected orientable surface, and f:π1(S)−^∼=→π1(S)is the induced automorphism, then its mapping torus T_his a hyperbolic 3-manifold and

ρ⁽²⁾(f) =− 1

6π ·vol(T_h).

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Approximation

The following conjecture combines and generalizes Conjectures byBergeron-Venkatesh,Hopf,Singer, andLück.

IfGis a finitely generated group, we denote byd(G)the minimal number of generators.

Achainfor a groupGis a sequence of inGnormal subgroups G=G₀⊇G₁⊇G₂⊇ · · ·

such that[G:G_i]<∞andT

i≥0G_i ={1}.

We denote byRGtherank gradientintroduced byLackenby.

RG(G,{G_i}) = lim

i→∞

d(G_i) [G:G_i].

Conjecture (Homological growth andL²-invariants for aspherical closed manifolds)

Let M be an aspherical closed manifold of dimension d and

fundamental group G=π₁(M). LetM be its universal covering. Thene For any natural number n with2n6=d we get

b⁽²⁾_n (M) =e 0.

If d =2n, we have

(−1)ⁿ·χ(M) =b_n⁽²⁾(M)e ≥0.

If d =2n and M carries a Riemannian metric of negative sectional curvature, then

(−1)ⁿ·χ(M) =b_n⁽²⁾(M)e >0;

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Conjecture (Continued)

Let(G_i)i≥0be any chain. Put M[i] =G_i\M.e

Then we get for any natural number n and any field F b⁽²⁾_n (M) =e lim

i→∞

b_n(M[i];F) [G:G_i] = lim

i→∞

d H_n(M[i];Z) [G:G_i] ; and for n=1

b₁⁽²⁾(M) =e lim

i→∞

b₁(M[i];F) [G:G_i] = lim

i→∞

d G_i/[G_i,G_i] [G:G_i]

=RG(G,(G_i)i≥0) =

(0 d 6=2;

−χ(M) d =2;

Conjecture (Continued)

If d =2n+1is odd, we have

(−1)ⁿ·ρ⁽²⁾ Me

≥0;

If d =2n+1is odd and M carries a Riemannian metric with negative sectional curvature, we have

(−1)ⁿ·ρ⁽²⁾ Me

>0;

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Conjecture (Continued)

Let(G_i)_i≥0be a chain. Put M[i] =G_i\M.e

Then we get for any natural number n with2n+16=d

i→∞lim ln

tors Hn(M[i]) [G:G_i] =0, and we get in the case d =2n+1

i→∞lim ln

tors H_n(M[i])

[G:G_i] = (−1)ⁿ·ρ⁽²⁾ Me

≥0.

The conjecture above is very optimistic, but we do not know a counterexample.

It is related to theApproximation Conjecture for the Fuglede-Kadison determinant.

The main issue here areuniform estimates about the spectrum of then-th Laplace operatorsonM[i]which are independent ofi.

Abert-Nikolovhave settled the rank gradient part ifGcontains an infinite normal amenable subgroup.

Kar-Kropholler-Nikolovhave settled the part about the growth of the torsion in the homology ifGis infinite amenable.

Abert-Gelander-Nikolovdeal with the rank gradient and the growth of the torsion in the homology for right angled lattices.

Li-Thomdeal with the vanishing ofL²-torsion for amenableG.

Bridson-Kochloukovadeal with limit groups, where the limits are not necessarily zero.

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Theorem (Lück)

Let M be an aspherical closed manifold with fundamental group G=π₁(M). Suppose that M carries a non-trivial S¹-action or suppose that G contains a non-trivial elementary amenable normal subgroup.

Then we get for all n≥0and fields F and any chain(G_i)i≥0 i→∞lim

b_n(M[i];F)

[G:G_i] = 0;

i→∞lim

d Hn(M[i];Z)

[G:G_i] = 0;

i→∞lim ln

tors H_n(M[i])

[G:G_i] = 0;

b⁽²⁾_n (M)e = 0;

ρ⁽²⁾(M)e = 0.

LetM be a closed hyperbolic 3-manifold. Then the conjecture above predicts for any chain(G_i)i≥0

i→∞lim ln

tors H₁(G_i) [G:G_i] = 1

6π ·vol(M).

Since the volume is always positive, the equation above implies that|tors H₁(G_i)

|growth exponentially in[G:G_i].

In particular this would allow to read of the volume from the profinite completion ofπ₁(M).

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