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/

Vancouver, July, 2015

Review of classical L

-invariants

LetG→X →X be aG-covering of a connected finite CW-complexX.

The cellular chain complex ofX is a finitely generated free ZG-chain complex:

· · ·−^c−−ⁿ⁻¹→M

In

ZG−→^cn M

in−1

ZG−^c−−ⁿ⁻¹→ · · · The associatedL²-chain complex

C_∗⁽²⁾(X):=L²(G)⊗_ZGC_∗(X)

has Hilbert spaces with isometric linearG-action as chain modules and boundedG-equivariant operators as differentials

· · · ^c

(2)

−−−n−1→M

In

L²(G) ^c

(2)

−−→n M

in−1

L²(G) ^c

(2)

−−−n−1→ · · ·

Definition (L²-homology andL²-Betti numbers) Define then-thL²-homologyto be the Hilbert space

H_n⁽²⁾(X):=ker(c_n⁽²⁾)/im(c⁽²⁾_n+1).

Define then-thL²-Betti number

b⁽²⁾_n (X):=dim_N(G) H_n⁽²⁾(X)

∈R^≥0.

The original notion is due toAtiyahan 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|·b_n(X).

IfG=Z, we have

b⁽²⁾_n (X) =dim_C[Z](0) C[Z]₍₀₎⊗_C[Z]Hn(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 the 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|.

Definition (L²-torsion)

Suppose thatX isL²-acyclic, i.e., allL²-Betti numbersb_n⁽²⁾(X)vanish.

Let∆⁽²⁾_n :C_n⁽²⁾(X)→C_n⁽²⁾(X)be then-Laplace operatorgiven by c⁽²⁾_n+1◦ c_n⁽²⁾∗

+ c_n−1⁽²⁾ ∗

◦c⁽²⁾_n . Define theL²-torsion

ρ⁽²⁾(X):=X

n≥0

(−1)ⁿ·n·ln det⁽²⁾(∆⁽²⁾_n )

∈R.

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 3π ·

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 induced bounded G-equivariantL²(G)^m →L²(G)^m map 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)∈K₁^w(ZG).

The universalL²-torsion is defined by the same expression as the L²-torsion, but now using the fact that the combinatorial Laplace operator can be thought of as an element inK₁^w(Z[G]), namely by

ρ⁽²⁾_u (X) :=X

n≥0

(−1)ⁿ·n·[∆^c_n] ∈K₁^w(ZG).

for∆^c_n:=c_n+1◦c_n^∗+c_n−1^∗ ◦c_n.

The universalL²-torsion is asimple homotopy invariant.

It satisfies usefulsum formulasandproduct formulas. There are also formulas for appropriatefibrationsandS¹-actions.

IfGis finite, we rediscover essentially the classicalReidemeister torsion.

Many other invariants come from the universalL²-torsion by applying a homomorphismK₁^w(ZG)→Aof abelian groups.

For instance, the Fuglede-Kadison determinant defines a homomorphism

det⁽²⁾: Wh^w(ZG)→R

which maps the universalL²-torsionρ⁽²⁾_u (X)to the (classical) L²-torsionρ⁽²⁾(X).

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, 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.

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)∈Zfor everyn≥0.

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 (with the exception of some graph manifolds) 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

χ−(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).

Polytopes

Consider a finitely generated abelian free abelian groupA. Let A_R:=R⊗_ZAbe the real vector space containingAas a spanning lattice;

ApolytopeP ⊆A_Ris a convex bounded subset which is the convex hull of a finite subsetS;

It is calledintegral, ifSis contained inA;

TheMinkowski sumof two polytopesP andQis defined by P+Q={p+q|p∈P,q ∈Q};

It iscancellative, i.e., it satisfiesP₀+Q=P₁+Q =⇒ P₀=P₁;

TheNewton polytope

N(p)⊆Rⁿ of a polynomial

p(t₁,t₂, . . . ,tn) = X

i1,...,in

a_i1,i2,...,in·t₁ⁱ¹t₂ⁱ²· · ·t_nⁱⁿ

innvariablest₁,t₂, . . . ,tnis defined to be the convex hull of the elements{(i₁,i₂, . . .i_n)∈Zⁿ |a_i1,i2,...,in 6=0};

One has

N(p·q) =N(p) +N(q).

Definition (Polytope group)

LetP(A)be the Grothendieck group of the abelian monoid of integral polytopes inA_R.

ForA=Zⁿwe obtain a well-defined homomorphism of abelian groups

Q[Zⁿ]₍₀₎×

→ P(A), p

q 7→[N(p)]−[N(q)].

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).

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(H₁(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(H₁(G)_f) factories to thepolytope homomorphism

P: Wh^w(G)→ P(H₁(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(H₁(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)^∗.

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 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 forBG, the Whitehead group Wh(G)vanishes, andGsatisfies the Atiyah Conjecture. Then we can define the universalL²-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.

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).

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

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

Summary (continued)

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

Thehigher order Alexander polynomialsdue toHarveyand Cochraneare special cases of the 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.