Introduction to the Farrell-Jones Conjecture

Introduction to the Farrell-Jones Conjecture

Wolfgang Lück Münster Germany

email lueck@math.uni-muenster.de http://www.math.uni-muenster.de/u/lueck/

Münster, August 2009

Outline

Group rings

The projective class group Wall’s finiteness obstruction The Whitehead group Whitehead torsion NegativeK-theory

Homology theories and spectra

The Isomorphism Conjectures for torsionfree groups

Consequences of the Isomorphism Conjectures for torsionfree groups

The general formulation the Isomorphism Conjectures The status of the Farrell-Jones Conjecture

Computational aspects Comments on the proof K-theory versusL-theory Concluding Remarks

Group rings

LetR be a ring, i.e., an associative ring with unit. LetGbe a group.

Definition (Group ringRG)

Thegroup ring RGis theR-algebra whose underlyingR-module is the freeR-module generated by the setGand whose multiplication comes from the multiplication inG.

Elements inRGare formal sumsP

g∈Grg·g, whererg ∈R and only finitely many elementsr_g are different from 0.

The multiplication is given by





Xr_g·g



· X s_h·h

!

:= X





X r_gs_h



·k.

LetM be aR-module. Suppose thatGacts onMbyR-linear maps. Then these data determine the structure of anRG-module onM.

The converse is also true.

AR[Z]-moduleM is the same as aR-module together with a R-automorphism ofM.

R[Z]agrees with the ringR[z,z⁻¹]of finite Laurent series.

LetX be a path connectedCW-complex with fundamental group π.

Letp:Xe →X be its universal covering. Thenπ acts freely onXe by deck transformations.

Xe inherits fromX aCW-structure. Theπ-action permutes the cells.

LetC∗(Xe)be its cellularZ-chain complex. By the inducedπ-action it becomes aZπ-chain complex which consists of free

Zπ-modules.

Many constructions of invariants forC_∗(X)can be generalized to much more refined invariants usingC_∗(Xe)with itsZπ-structure.

TakeX =S¹. ThenC∗(Xe)looks like

· · · →0→Z[Z|]−−→^t−1 Z[Z|],

Group rings are in general very complicated.

IfRis Noetherian andGis virtually poly-cyclic, thenRGis Noetherian.

There is the conjecture that the converse is true. In particularRG is in general not Noetherian.

Suppose thatg ∈Ghas finite order, let us sayn. Put N =Pn

i=1gⁱ. Then

N·N =n·N.

HenceRGcontains a zero-divisor since

(n·1−N)·N =n·N−N·N =0.

Ifnis invertible inR, thenRGcontains an idempotent, namely ^N_n.

Conjecture (Kaplansky Conjecture)

TheKaplansky Conjecturesays for a torsionfree group G that any idempotent of RG belongs to R.

Conjecture (Unit Conjecture)

Let G be a torsionfree group. Then every unit in RG is of the form r·g for some unit r ∈R and some g∈G.

It is an easy exercise to verify the conjectures above forG=Zby looking at coefficients with the lowest and highest degree.

The projective class group

Definition (ProjectiveR-module)

AnR-moduleP is calledprojectiveif it satisfies one of the following equivalent conditions:

P is a direct summand in a freeR-module;

The following lifting problem has always a solution M ^{p //}N ^//0

P

f

``@@

@@ ^f

OO

If 0→M₀→M₁→M₂→0 is an exact sequence ofR-modules, then 0→hom_R(P,M₀)→hom_R(P,M₁)→hom_R(P,M₂)→0 is

Over a field or, more generally, over a principal ideal domain, every projective module is free.

IfRis a principal ideal domain, then a finitely generatedR-module is projective (and hence free) if and only if it is torsionfree.

For instanceZ/nis forn≥2 never projective asZ-module.

LetR andS be rings andR×Sbe their product. ThenR× {0}is a finitely generated projectiveR×S-module which is not free.

Example (Representations of finite groups)

LetF be a field of characteristicp forpa prime number or 0. LetGbe a finite group.

ThenF with the trivialG-action is a projectiveFG-module if and only if p=0 orp does not divide the order ofG. It is a freeFG-module only if Gis trivial.

Definition (Projective class groupK0(R)) LetRbe an (associative) ring (with unit).

Define itsprojective class group K₀(R)

to be the abelian group whose generators are isomorphism classes[P]

of finitely generated projectiveR-modulesP and whose relations are [P₀] + [P₂] = [P₁]for every exact sequence 0→P₀→P₁→P₂→0 of finitely generated projectiveR-modules.

This is the same as theGrothendieck constructionapplied to the abelian monoid of isomorphism classes of finitely generated projectiveR-modules under direct sum.

Thereduced projective class groupKe₀(R)is the quotient ofK₀(R) by the subgroup generated by the classes of finitely generated

LetP be a finitely generated projectiveR-module. It isstably free, i.e.,P⊕R^m ∼=Rⁿfor appropriatem,n∈Z, if and only if[P] =0 in Ke₀(R).

Ke₀(R)measures thedeviationof finitely generated projective R-modules from being stably finitely generated free.

The assignmentP 7→[P]∈K₀(R)is theuniversal additive invariantordimension functionfor finitely generated projective R-modules.

Induction

Letf:R→Sbe a ring homomorphism. Given anR-moduleM, let f∗Mbe theS-moduleS⊗_RM. We obtain a homomorphism of abelian groups

f∗:K₀(R)→K₀(S), [P]7→[f∗P].

Compatibility with products

The two projections fromR×StoRandSinduce isomorphisms K₀(R×S)−→^∼= K₀(R)×K₀(S).

Morita equivalence

LetR be a ring andMn(R)be the ring of(n,n)-matrices overR.

We can considerRⁿas aM_n(R)-R-bimodule and as a R-M_n(R)-bimodule.

Tensoring with these yields mutually inverse isomorphisms K₀(R) −^∼=→ K₀(Mn(R)), [P] 7→ [_Mn(R)Rⁿ_R⊗_RP];

K₀(Mn(R)) −^∼=→ K₀(R), [Q] 7→ [_RRⁿ_Mn(R)⊗_Mn(R)Q].

Example (Principal ideal domains)

IfRis a principal ideal domain. LetF be its quotient field. Then we obtain mutually inverse isomorphisms

Z

∼=

−→ K₀(R), n 7→ [Rⁿ];

K₀(R) −→^∼= Z, [P] 7→ dim_F(F ⊗_RP).

Example (Representation ring)

LetGbe a finite group and letF be a field of characteristic zero. Then therepresentation ringR_F(G)is the same asK₀(FG). Taking the character of a representation yields an isomorphism

R_C(G)⊗_ZC=K₀(CG)⊗_ZC

∼=

−→class(G,C), whereclass(G;C)is the complex vector space ofclass functions

Example (Dedekind domains)

LetR be a Dedekind domain, for instance the ring of integers in an algebraic number field.

Call two idealsIandJ inRequivalent if there exists non-zero elementsr andsinRwithrI =sJ. Theideal class group C(R)is the abelian group of equivalence classes of ideals under

multiplication of ideals.

Then we obtain an isomorphism

C(R)−→^∼= Ke₀(R), [I]7→[I].

The structure of the finite abelian group

C(Z[exp(2πi/p)])∼=Ke₀(Z[exp(2πi/p)])∼=Ke₀(Z[Z/p])

Theorem (Swan (1960))

If G is finite, thenKe₀(ZG)is finite.

TopologicalK-theory

LetX be a compact space. LetK⁰(X)be the Grothendieck group of isomorphism classes of finite-dimensional complex vector bundles overX. This is the zero-th term of a generalized cohomology theoryK^∗(X)calledtopologicalK-theory. It is 2-periodic, i.e.,Kⁿ(X) =Kⁿ⁺²(X), and satisfiesK⁰(pt) =Zand K¹(pt) ={0}.

LetC(X)be the ring of continuous functions fromX toC. Theorem (Swan (1962))

There is an isomorphism

Wall’s finiteness obstruction

Definition (Finitely dominated)

ACW-complexX is calledfinitely dominatedif there exists a finite (=

compact)CW-complexY together with mapsi:X →Y andr:Y →X satisfyingr ◦i'id_X.

A finiteCW-complex is finitely dominated.

A closed smooth manifold is a finiteCW-complex.

Problem

Is a given finitely dominated CW -complex homotopy equivalent to a finite CW -complex?

Definition (Wall’sfiniteness obstruction)

A finitely dominatedCW-complexX defines an element o(X)∈K₀(Z[π1(X)])

called itsfiniteness obstructionas follows.

LetXe be the universal covering. The fundamental group π =π₁(X)acts freely onXe.

LetC∗(Xe)be the cellular chain complex. It is a freeZπ-chain complex.

SinceX is finitely dominated, there exists a finite projective Zπ-chain complexP∗ withP∗ '_Zπ C∗(Xe).

Define

o(X) :=X

(−1)ⁿ·[Pn]∈K₀(Zπ).

This is a kind of Euler characteristic but now counting modules themselves and not their rank.

Theorem (Wall (1965))

A finitely dominated CW -complex X is homotopy equivalent to a finite CW -complex if and only if its reduced finiteness obstruction

o(Xe )∈Ke₀(Z[π₁(X)])vanishes.

A finitely dominated simply connectedCW-complex is always homotopy equivalent to a finiteCW-complex sinceKe₀(Z) ={0}.

Given a finitely presented groupGandξ∈K₀(ZG), there exists a finitely dominatedCW-complexX withπ₁(X)∼=Gando(X) =ξ.

Theorem (Geometric characterization ofKe0(ZG) = {0})

The following statements are equivalent for a finitely presented group G:

Every finite dominated CW -complex with G∼=π₁(X)is homotopy equivalent to a finite CW -complex.

Ke₀(ZG) ={0}.

Conjecture (Vanishing ofKe0(ZG)for torsionfreeG) If G is torsionfree, then

Ke₀(ZG) ={0}.

The Whitehead group

Definition (K1-groupK1(R)) Define theK₁-group of a ring R

K₁(R)

to be the abelian group whose generators are conjugacy classes[f]of automorphismsf:P→P of 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 [f₀] + [f₂] = [f₁];

[g◦f] = [f] + [g].

This is the same asGL(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 thereduced K₁-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.

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

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

Whitehead torsion

Definition (h-cobordism)

Anh-cobordismover a closed manifoldM₀is a compact manifoldW whose boundary is the disjoint unionM₀qM₁such that both inclusions M₀→W andM₁→W are homotopy equivalences.

Theorem (s-Cobordism Theorem,Barden, Mazur, Stallings, Kirby-Siebenmann, mid 60-s)

Let M₀be a closed (smooth) manifold of dimension≥5. Let (W;M₀,M₁)be an h-cobordism over M₀.

Then W is homeomorphic (diffeomorpic) to M₀×[0,1]relative M₀if and only if itsWhitehead torsion

τ(W,M₀)∈Wh(π₁(M₀))

Conjecture (Poincaré Conjecture)

Let M be an n-dimensional topological manifold which is a homotopy sphere, i.e., homotopy equivalent to Sⁿ.

Then M is homeomorphic to Sⁿ. Theorem

For n≥5the Poincaré Conjecture is true.

Proof.

We sketch the proof forn≥6.

LetM be an-dimensional homotopy sphere.

LetW be obtained fromMby deleting the interior of two disjoint embedded disksDⁿ₁andD₂ⁿ. ThenW is a simply connected h-cobordism.

Since Wh({1})is trivial, we can find a homeomorphism f:W −→^∼= ∂Dⁿ₁×[0,1]which is the identity on∂D₁ⁿ=D₁ⁿ× {0}.

By theAlexander trickwe can extend the homeomorphism f|_Dn

1×{1}:∂Dⁿ₂−^∼=→∂D₁ⁿ× {1}to a homeomorphismg:D₁ⁿ→D₂ⁿ. The three homeomorphismsid_Dⁿ

1,f andgfit together to a homeomorphismh:M →D₁ⁿ∪_∂Dⁿ

1×{0}∂Dⁿ₁×[0,1]∪_∂Dⁿ

1×{1}D₁ⁿ. The target is obviously homeomorphic toSⁿ.

The argument above does not imply that for a smooth manifoldM we obtain a diffeomorphismg:M →Sⁿ.

The Alexander trick does not work smoothly.

Indeed, there exists so calledexotic spheres, i.e., closed smooth manifolds which are homeomorphic but not diffeomorphic toSⁿ. Thes-cobordism theorem is a key ingredient in thesurgery programfor the classification of closed manifolds due toBrowder, Novikov, SullivanandWall.

Given a finitely presented groupG, an elementξ∈Wh(G)and a closed manifoldMof dimensionn≥5 withG∼=π₁(M), there exists anh-cobordismW overMwithτ(W,M) =ξ.

Theorem (Geometric characterization of Wh(G) ={0})

The following statements are equivalent for a finitely presented group G and a fixed integer n ≥6

Every compact n-dimensional h-cobordism W with G∼=π₁(W)is trivial;

Wh(G) ={0}.

Conjecture (Vanishing of Wh(G)for torsionfreeG) If G is torsionfree, then

Wh(G) ={0}.

Negative K -theory

Definition (Bass-Nil-groups) Define forn=0,1

NKn(R):=coker(Kn(R)→Kn(R[t])).

Theorem (Bass-Heller-Swan decomposition forK1 (1964)) There is an isomorphism, natural in R,

K₀(R)⊕K₁(R)⊕NK₁(R)⊕NK₁(R)−^∼=→K₁(R[t,t⁻¹]) =K₁(R[Z]).

Definition (NegativeK-theory) Define inductively forn=−1,−2, . . .

K_n(R):=coker

K_n+1(R[t])⊕K_n+1(R[t⁻¹])→K_n+1(R[t,t⁻¹]) .

Define forn=−1,−2, . . .

NK_n(R):=coker(K_n(R)→K_n(R[t])).

Theorem (Bass-Heller-Swan decomposition for negative K-theory)

For n≤1there is an isomorphism, natural in R,

K_n−1(R)⊕K_n(R)⊕NK_n(R)⊕NK_n(R)−^∼=→K_n(R[t,t⁻¹]) =K_n(R[Z]).

Definition (Regular ring)

A ringRis calledregularif it is Noetherian and every finitely generated R-module possesses a finite projective resolution.

Principal ideal domains are regular. In particularZand any field are regular.

IfRis regular, thenR[t]andR[t,t⁻¹] =R[Z]are regular.

IfRis regular, thenRGin general is not Noetherian or regular.

Theorem (Bass-Heller-Swan decomposition for regular rings) Suppose that R is regular. Then

Kn(R) = 0 for n≤ −1;

NK_n(R) = 0 for n≤1,

and the Bass-Heller-Swan decomposition reduces for n ≤1to the natural isomorphism

K_n−1(R)⊕K_n(R)−^∼=→K_n(R[t,t⁻¹]) =K_n(R[Z]).

There are alsohigher algebraicK-groupsK_n(R)forn≥2 due to Quillen (1973). They are defined as homotopy groups of certain spaces or spectra.

Most of the well known features ofK₀(R)andK₁(R)extend to both negative and higher algebraicK-theory. For instance the Bass-Heller-Swan decomposition holds also for higher algebraic K-theory.

Notice the following formulas for a regular ringRand a generalized homology theoryH_∗, which look similar:

Kn(R[Z]) ∼= Kn(R)⊕Kn−1(R);

H_n(BZ) ∼= H_n(pt)⊕ H_n−1(pt).

IfGandK are groups, then we have the following formulas, which look similar:

Ke_n(Z[G∗K]) ∼= Ke_n(ZG)⊕Ke_n(ZK);

He_n(B(G∗K)) ∼= He_n(BG)⊕He_n(BK).

Homology theories and spectra

Definition (Spectrum) Aspectrum

E={(E(n), σ(n))|n∈Z}

is a sequence of pointed spaces{E(n)|n∈Z}together with pointed maps calledstructure maps

σ(n) :E(n)∧S¹−→E(n+1).

Amap of spectra

f:E→E⁰

is a sequence of mapsf(n) :E(n)→E⁰(n)which are compatible with the structure mapsσ(n), i.e.,f(n+1)◦σ(n) = σ⁰(n)◦(f(n)∧id_S1) holds for alln∈ .

Given two pointed spacesX = (X,x₀)andY = (Y,y₀), their one-point-unionand theirsmash productare defined to be the pointed spaces

X∨Y := {(x,y₀)|x ∈X} ∪ {(x₀,y)|y ∈Y} ⊆X×Y; X∧Y := (X×Y)/(X∨Y).

We haveSⁿ⁺¹∼=Sⁿ∧S¹.

Thesphere spectrumShas asn-th spaceSⁿand asn-th structure map the homeomorphismSⁿ∧S¹−^∼=→Sⁿ⁺¹.

LetX be a pointed space. Itssuspension spectrumΣ^∞X is given by the sequence of spaces{X ∧Sⁿ|n≥0}with the

homeomorphism(X ∧Sⁿ)∧S¹∼=X ∧Sⁿ⁺¹as structure maps.

We haveS= Σ^∞S⁰.

Definition (Ω-spectrum)

Given a spectrumE, we can consider instead of the structure map σ(n) :E(n)∧S¹→E(n+1)its adjoint

σ⁰(n) :E(n)→ΩE(n+1) =map(S¹,E(n+1)).

We callEanΩ-spectrumif each mapσ⁰(n)is a weak homotopy equivalence.

Definition (Homotopy groups of a spectrum)

Given a spectrumE, define forn∈Zitsn-th homotopy group πn(E):=colim

k→∞ π_k+n(E(k))

to be the abelian group which is given by the colimit over the directed system indexed byZwithk-th structure map

π_k+n(E(k)) ^σ

0(k)

−−−→π_k+n(ΩE(k+1)) =π_k+n+1(E(k+1)).

Homotopy groups of spectra are always abelian (in contrast to the fundamental group of a space).

Notice that a spectrum can have in contrast to a space non-trivial negative homotopy groups.

IfEis anΩ-spectrum, thenπn(E) =πn(E(0))for alln≥0.

Eilenberg-MacLane spectrum

LetAbe an abelian group. Then-thEilenberg-MacLane space EM(A,n)associated toAforn≥0 is aCW-complex with π_m(EM(A,n)) =Aform=nandπ_m(EM(A,n)) ={0}form6=n.

The associatedEilenberg-MacLane spectrumH(A)has asn-th spaceEM(A,n)and asn-th structure map a homotopy

equivalenceEM(A,n)→ΩEM(A,n+1).

AlgebraicK-theory spectrum

For a ringR there is thealgebraicK-theory spectrumK_Rwith the property

π_n(K_R) =K_n(R) forn∈Z.

AlgebraicL-theory spectrum

For a ring with involutionRthere is thealgebraicL-theory spectrumL^h−∞i_R with the property

πn(L^h−∞i_R ) =L^h−∞i_n (R) forn∈Z.

TopologicalK-theory spectrum

ByBott periodicitythere is a homotopy equivalence β:BU×Z−^'→Ω²(BU×Z).

ThetopologicalK-theory spectrumK^tophas in even degrees BU×Zand in odd degreesΩ(BU×Z).

The structure maps are given in even degrees by the mapβ and in odd degrees by the identity id: Ω(BU×Z)→Ω(BU×Z).

Definition (Homology theory)

LetΛbe a commutative ring, for instanceZorQ.

Ahomology theoryH_∗with values inΛ-modules is a covariant functor from the category ofCW-pairs to the category ofZ-gradedΛ-modules together with natural transformations

∂n(X,A) :H_n(X,A)→ H_n−1(A) forn∈Zsatisfying the following axioms:

Homotopy invariance.

Long exact sequence of a pair.

Excision.

If(X,A)is aCW-pair andf:A→Bis a cellular map , then H (X,A)−^∼=→ H (X ∪ B,B).

Definition (continued) Disjoint union axiom.

M

i∈I

H_n(X_i)−→ H^∼= _n a

i∈I

X_i

! .

Definition (Smash product)

LetEbe a spectrum andX be a pointed space. Define thesmash productX ∧Eto be the spectrum whosen-th space isX∧E(n)and whosen-th structure map is

X ∧E(n)∧S^{1 id}−−−−−−^X∧σ(n)→X ∧E(n+1).

Theorem (Homology theories and spectra)

LetEbe a spectrum. Then we obtain a homology theoryH∗(−;E)by Hn(X,A;E) :=πn((X ∪_Acone(A))∧E).

which satisfies

Hn(pt;E) =πn(E).

Example (Stable homotopy theory)

The homology theory associated to the sphere spectrumSisstable homotopyπ^s_∗(X).

The groupsπ^s_n(pt)are finite abelian groups forn6=0 by a result of Serre (1953).

Their structure is only known for smalln.

Example (Singular homology theory with coefficients)

The homology theory associated to the Eilenberg-MacLane spectrum H(A)issingular homology with coefficients inA.

Example (TopologicalK-homology)

The homology theory associated to the topologicalK-theory spectrum K^topisK-homologyK∗(X). We have

K (pt)∼

Z neven;

The Isomorphism Conjectures for torsionfree groups

Conjecture (K-theoretic Farrell-Jones Conjecture for torsionfree groups)

TheK -theoretic Farrell-Jones Conjecturewith coefficients in the regular ring R for the torsionfree group G predicts that theassembly map

Hn(BG;K_R)→Kn(RG) is bijective for all n∈Z.

Kn(RG)is the algebraicK-theory of the group ringRG;

K_Ris the (non-connective) algebraicK-theory spectrum ofR;

Hn(pt;K_R)∼=πn(K_R)∼=Kn(R)forn∈Z.

BGis theclassifying spaceof the groupG, i.e., the base space of the universalG-principalG-bundleG→EG→BG. Equivalently,

Conjecture (L-theoretic Farrell-Jones Conjecture for torsionfree groups)

TheL-theoretic Farrell-Jones Conjecturewith coefficients in the ring with involution R for the torsionfree group G predicts that theassembly map

H_n(BG;L^h−∞i_R )→L^h−∞i_n (RG) is bijective for all n∈Z.

L^h−∞i_n (RG)is the algebraicL-theory ofRGwith decorationh−∞i;

L^h−∞i_R is the algebraicL-theory spectrum ofRwith decoration h−∞i;

Hn(pt;L^h−∞i_R )∼=πn(L^h−∞i_R )∼=L^h−∞i_n (R)forn∈Z.

Conjecture (Baum-Connes Conjecture for torsionfree groups) TheBaum-Connes Conjecturefor the torsionfree group predicts that theassembly map

K_n(BG)→K_n(C_r^∗(G)) is bijective for all n∈Z.

K_n(BG)is the topologicalK-homology ofBG, where

K∗(−) =H∗(−;K^top)forK^topis the topologicalK-theory spectrum.

K_n(C_r^∗(G))is the topologicalK-theory of the reduced complex groupC^∗-algebraC_r^∗(G)ofGwhich is the closure in the norm topology ofCGconsidered as subalgebra ofB(l²(G)).

There is also areal versionof the Baum-Connes Conjecture KO_n(BG)→K_n(C_r^∗(G;R)).

Consequences of the Isomorphism Conjectures for torsionfree groups

In order to illustrate the depth of the Farrell-Jones Conjecture and the Baum-Connes Conjecture, we present some conclusions which are interesting in their own right.

LetFJ_K(R)andFJ_L(R)respectively be the class of groups which satisfy theK-theoretic andL-theoretic respectively Farrell-Jones Conjecture for the coefficient ringR.

LetBC be the class of groups which satisfy the Baum-Connes Conjecture.

Lemma

Let R be a regular ring. Suppose that G is torsionfree and G∈ FJ_K(R). Then:

K_n(RG) =0for n≤ −1;

The change of rings map K₀(R)→K₀(RG)is bijective. In particularKe₀(RG)is trivial if and only ifKe₀(R)is trivial.

Lemma

Suppose that G is torsionfree and G ∈ FJ_K(Z). Then the Whitehead groupWh(G)is trivial.

Proof.

The idea of the proof is to study theAtiyah-Hirzebruch spectral sequenceconverging toH_n(BG;K_R)whoseE²-term is given by

2

Proof (continued).

SinceRis regular by assumption, we getK_q(R) =0 forq ≤ −1.

Hence the edge homomorphism yields an isomorphism K₀(R) =H₀(pt,K₀(R))−→^∼= H₀(BG;K_R)∼=K₀(RG).

We haveK₀(Z) =ZandK₁(Z) ={±1}.

We get an exact sequence

0→H₀(BG;K₁(Z)) ={±1} →H₁(BG;K_Z)∼=K₁(ZG)

→H₁(BG;K₀(Z)) =G/[G,G]→0.

This implies Wh(G) :=K₁(ZG)/{±g|g ∈G} ∼=0.

In particular we get for a torsionfree groupG∈ FJ_K(Z):

Kn(ZG) =0 forn≤ −1;

Ke₀(ZG) =0;

Wh(G) =0;

Every finitely dominatedCW-complexX withG=π₁(X)is homotopy equivalent to a finiteCW-complex;

Every compacth-cobordismW of dimension≥6 withπ₁(W)∼=G is trivial;

Conjecture (Kaplansky Conjecture)

TheKaplansky Conjecturesays for a torsionfree group G and an integral domain R that0and1are the only idempotents in RG.

Theorem (The Farrell-Jones Conjecture and the Kaplansky Conjecture,Bartels-Lück-Reich(2008))

Let F be a skew-field and let G be a group with G ∈ FJ_K(F). Suppose that one of the following conditions is satisfied:

F is commutative and has characteristic zero and G is torsionfree;

G is torsionfree and sofic, e.g., residually amenable;

The characteristic of F is p, all finite subgroups of G are p-groups and G is sofic.

Then0and1are the only idempotents in FG.

Proof.

Letpbe an idempotent inFG. We want to showp∈ {0,1}.

Denote by:FG→F the augmentation homomorphism sending P

g∈Grg·g toP

g∈Grg. Obviously(p)∈F is 0 or 1. Hence it suffices to showp=0 under the assumption that(p) =0.

Let(p)⊆FGbe the ideal generated bypwhich is a finitely generated projectiveFG-module.

SinceG∈ FJ_K(F), we can conclude that

i∗:K₀(F)⊗_ZQ→K₀(FG)⊗_ZQ is surjective.

Hence we can find a finitely generated projectiveF-modulePand integersk,m,n≥0 satisfying

(p)^k⊕FG^m∼=_FGi_∗(P)⊕FGⁿ.

Proof (continued).

If we now applyi∗◦∗and use◦i =id,i∗◦∗(FG^l)∼=FG^l and (p) =0 we obtain

FG^m ∼=i_∗(P)⊕FGⁿ. Inserting this in the first equation yields

(p)^k ⊕i∗(P)⊕FGⁿ∼=i∗(P)⊕FGⁿ.

Our assumptions onF andGimply thatFGisstably finite, i.e., ifA andBare square matrices overFGwithAB=I, thenBA=I.

This implies(p)^k =0 and hencep=0.

Theorem (The Baum-Connes Conjecture and the Kaplansky Conjecture)

Let G be a torsionfree group with G∈ BC. Then0and1are the only idempotents inCG.

Proof.

We can prove the claim even forp∈C_r^∗(G).

There is a trace map

tr:C_r^∗(G)→C

which sendsf ∈C_r^∗(G)⊆ B(l²(G))tohf(e),ei_l2(G).

TheL²-index theoremdue toAtiyah (1976)shows that the composite

K₀(BG)→K₀(C_r^∗(G))−→^tr C coincides with

Proof (continued).

HenceG∈ BCimplies tr(p)∈Z.

Since tr(1) =1, tr(0) =0, 0≤p≤1 andp²=p, we get tr(p)∈R and 0≤tr(p)≤1.

We conclude tr(0) =tr(p)or tr(1) =tr(p).

This implies alreadyp=0 orp=1.

Conjecture (Borel Conjecture)

TheBorel Conjecture for Gpredicts for two closed aspherical manifolds M and N withπ₁(M)∼=π₁(N)∼=G that any homotopy equivalence M →N is homotopic to a homeomorphism and in particular that M and N are homeomorphic.

The Borel Conjecture can be viewed as the topological version of Mostow rigidity. A special case of Mostow rigidity says that any homotopy equivalence between closed hyperbolic manifolds of dimension≥3 is homotopic to an isometric diffeomorphism.

The Borel Conjecture is not true in the smooth category by results ofWall(1969)andFarrell-Jones(1989).

There are also non-aspherical manifolds which are topologically rigid in the sense of the Borel Conjecture (seeKreck-Lück (2009)).

Theorem (The Farrell-Jones Conjecture and the Borel Conjecture)

If the K - and L-theoretic Farrell-Jones Conjecture hold for G in the case R =Z, then the Borel Conjecture is true in dimension≥5and in dimension4if G is good in the sense of Freedman.

The Borel Conjecture in dimension 1 and 2 is obviously true.

Thurston’s Geometrization Conjectureimplies the Borel Conjecture in dimension 3.

Definition (Structure set)

Thestructure setS^top(M)of a manifoldM consists of equivalence classes of orientation preserving homotopy equivalencesN→M with a manifoldN as source.

Two such homotopy equivalencesf₀:N₀→Mandf₁:N₁→Mare equivalent if there exists a homeomorphismg:N₀→N₁with f₁◦g'f₀.

Theorem

The Borel Conjecture holds for a closed manifold M if and only if S^top(M)consists of one element.

Theorem (Ranicki (1992))

There is an exact sequence of abelian groups, calledalgebraic surgery exact sequence, for an n-dimensional closed manifold M

. . .−^σ−−ⁿ⁺¹→H_n+1(M;Lh1i)−−−→^An+1 L_n+1(Zπ₁(M))−^∂−−ⁿ⁺¹→

S^top(M)−→^σn H_n(M;Lh1i)−→^An L_n(Zπ1(M))−→^∂n . . . It can be identified with the classical geometric surgery sequence due toSullivan and Wallin high dimensions.

S^top(M)consist of one element if and only ifA_n+1is surjective and Anis injective.

H_k(M;Lh1i)→H_k(M;L)is bijective fork ≥n+1 and injective for k =n.

The general formulation the Isomorphism Conjectures

The versions of the Farrell-Jones Conjecture and the Baum- Connes Conjecture above become false for finite groups unless the group is trivial.

For instance the version of the Baum-Connes Conjecture above would predict for a finite groupG

K₀(BG)∼=K₀(C_r^∗(G))∼=R_C(G).

However,K₀(BG)⊗_ZQ∼=_Q K₀(pt)⊗_ZQ∼=_Q Qand R_C(G)⊗_ZQ∼=_QQholds if and only ifGis trivial.

IfGis torsionfree, then the version of theK-theoretic Farrell-Jones Conjecture predicts

H_n(BZ;K_R) =H_n(S¹;K_R) =H_n(pt;K_R)⊕H_n−1(pt;K_R)

=K_n(R)⊕K_n−1(R)∼=K_n(RZ).

In view of the Bass-Heller-Swan decomposition this is only possible if NK_n(R)vanishes which is true for regular ringsRbut not for general ringsR.

However, there are more technical versions of the Farrell-Jones Conjecture which make sense for all groups and more general coefficients, where one allows twisted group rings or orientation homomorphisms.

We present their formulation without giving details.

Definition (Additive category)

Anadditive categoryAis a small categoryAsuch that for two objects AandBthe morphism set morA(A,B)has the structure of an abelian group and the direct sumA⊕Bof two objectsAandBexists and the obvious compatibility conditions hold.

Example

Examples of additive categories are the category ofR-modules and of finitely generated projectiveR-modules. Further examples are the category ofR-chain complexes and the homotopy category ofR-chain complexes.

Definition (TheK-theoretic Farrell-Jones Conjecture with additive categories as coefficients)

TheK-theoretic Farrell-Jones Conjecture forGwith additive categories as coefficientssays that the projectionE G→G/Ginduces for all n∈Zand all additive categoriesAwith rightG-action an isomorphism

H_n^G(E G;KA)−→^∼= H_n^G(G/G;KA) =Kn

Z

G

A

.

IFAis the additive category of finitely generated projective R-modules equipped with the trivialG-action, then the assembly map above can be identified with the classical one

H_n^G(E G;K_R)−^∼=→H_n^G(G/G;K_R) =Kn(RG)

Roughly speaking, the Farrell-Jones Conjecture predicts how one can computeKn(RG)if one knowsKm(RV)for all virtually cyclic subgroupsV ofGand allm≤ntaking their relation coming from inclusions and conjugation into account.

The advantage of the approach via additive categories is that it includes the case of twisted group rings and more generally of crossed product rings and that many inheritance properties such as the inheritance to subgroups is built in.

In some sense the coefficient ringRbecomes a kind of dummy variable.

There is also a version forL-theory and a version for topological K-theory of groupC^∗-algebras which is called the Baum-Connes Conjecture.

Conjecture (Novikov Conjecture)

TheNovikov Conjecture for Gpredicts for a closed oriented manifold M together with a map f:M→BG that for any x ∈H^∗(BG)thehigher signature

sign_x(M,f):=hL(M)∪f^∗x,[M]i

is an oriented homotopy invariant of(M,f), i.e., for every orientation preserving homotopy equivalence of closed oriented manifolds g:M₀→M₁and homotopy equivalence f_i:M_i →BG with f₁◦g 'f₂ we have

sign_x(M₀,f₀) =sign_x(M₁,f₁).

Both the Farrell-Jones Conjecture forL-theory and the Baum-Connes Conjecture imply the Novikov Conjecture.

The status of the Farrell-Jones Conjecture

Theorem (Main Theorem Bartels-Farrell-Lück-Reich (2008/2009))

LetFJ be the class of groups for which both the K -theoretic and the L-theoretic Farrell-Jones Conjectures hold with coefficients in any additive G-category (with involution). It has the following properties:

Hyperbolic group and virtually nilpotent groups belongs toFJ; If G₁and G₂belong toFJ, then G₁×G₂belongs toFJ; Let{G_i |i ∈I}be a directed system of groups (with not

necessarily injective structure maps) such that G_i ∈ FJ for i ∈I.

Thencolim_i∈IG_i belongs toFJ;

If H is a subgroup of G and G∈ FJ, then H ∈ FJ;

Theorem (continued)

If we demand for the K -theory version only that the assembly map is1-connected and keep the full L-theory version, then the

properties above remain valid and the classFJ contains also all CAT(0)-groups;

The last statement is also true all cocompact lattices in almost connected Lie groups.

For all applications presented in these talks the version, where we demand for theK-theory version only that the assembly map is 1-connected and keep the fullL-theory version, is sufficient.

Limit groupsin the sense ofZelaare CAT(0)-groups (Alibegovic-Bestvina (2005)).

There are manyconstructions of groups with exotic properties which arise as colimits of hyperbolic groups.

One example is the construction ofgroups with expandersdue to Gromov. These yieldcounterexamplesto theBaum-Connes Conjecture with coefficients(seeHigson-Lafforgue-Skandalis (2002)).

However, our results show that these groups do satisfy the Farrell-Jones Conjecture in its most general form and hence also the other conjectures mentioned above.

Mike Davis (1983)has constructed exotic closed aspherical manifolds usinghyperbolization techniques. For instance there are examples which donot admit a triangulationor whose universal covering is not homeomorphic to Euclidean space.

However, in all cases the universal coverings are CAT(0)-spaces and hence the fundamental groups are CAT(0)-groups.

Hence by our main theorem they satisfy the Farrell-Jones Conjecture and hence the Borel Conjecture in dimension≥5.

There are still many interesting groups for which the Farrell-Jones Conjecture in its most general form is open. Examples are:

Amenable groups;

Sln(Z)forn≥3;

Mapping class groups;

Out(Fn);

Thompson groups.

If one looks for a counterexample, there seems to be no good candidates which do not fall under our main theorems and have some exotic properties which may cause the failure of the Farrell-Jones Conjecture.

One needs a property which can be used to detect a non-trivial element which is not in the image of the assembly map or is in its kernel.

Computational aspects

Theorem (The algebraic K-theory of torsionfree hyperbolic groups)

Let G be a torsionfree hyperbolic group and let R be a ring (with involution). Then we get an isomorphisms

H_n(BG;K_R)⊕

M

(C),C⊆G,C6=1 C maximal cyclic

NK_n(R) _∼

−→= K_n(RG);

and

Hn(BG;L^h−∞i_R ) −^∼=→ L^h−∞i_n (RG);

Theorem (Lück (2002))

Let G be a group. Let T be the set of conjugacy classes(g)of elements g ∈G of finite order. There is a commutative diagram

L

p+q=n

L

(g)∈TH_p(BC_Ghgi;C)⊗_ZK_q(C) //

Kn(CG)⊗_ZC

L

p+q=n

L

(g)∈THp(BC_Ghgi;C)⊗_ZK_q^top(C) ^//K_n^top(C_r^∗(G))⊗_ZC The vertical arrows come from the obvious change of rings and of K-theory maps.

The horizontal arrows can be identified with the assembly maps occurring in the Farrell-Jones Conjecture and the Baum-Connes Conjecture by the equivariant Chern character.

Splitting principle.

Comments on the proof

Here are the basic steps of the proof of the main Theorem.

Step 1: Interpret the assembly map as aforget control map. Then the task is to give a way ofgaining control.

Step 2: Show for a finitely generated groupGthatG∈ FJ holds if one can construct the followinggeometric data:

AG-spaceX, such that the underlying spaceX is the realization of an abstract simplicial complex;

AG-spaceX, which containsX as an openG-subspace. The underlying space ofX should becompact,metrizableand contractible,

such that the following assumptions are satisfied:

Z-set-condition

There exists a homotopyH:X×[0,1]→X, such thatH₀=id_X andHt(X)⊂X for everyt >0;

Long thin coverings

There exists anN ∈Nthat only depends on theG-spaceX, such that for everyβ ≥1 there exists aVCyc-coveringU(β)ofG×X with the following two properties:

For everyg ∈Gandx ∈X there exists aU∈ U(β)such that with respect to the word metric;

The dimension of the coveringU(β)is smaller than or equal toN.

Step 3: Prove the existence of the geometric data above.

This is often done by constructing a certainflow spaceand use the flow to let a given not yet perfect covering flow into a good one. The construction of the flow space for CAT(0)-space is one of the main ingredients.

K -theory versus L-theory

So far theK-theory case has been easier to handle.

The reason is that at some point atransfer argumentcomes in.

After applying the transfers the element gets controlled on the total space level and then is pushed down to the base space.

The transferp^!for a fiber bundleF:E →B has inK-theory the property thatp^!◦p∗ is multiplication with theEuler characteristic.

In most situationsF is contractible and hence obviouslyp^!◦p∗ is the identity what is needed for the proof.

In theL-theory casep^!◦p∗is multiplication with thesignature. If the fiber is a sphere, thenp^!◦p∗ is zero.

One needs a construction which makes out of a finite

CW-complex with Euler characteristic 1 a finite Poincare complex

Such a construction is given by themultiplicative hyperbolic form.

Given a finitely projectiveR-moduleP over the commutative ring R, define a symmetric bilinearR-formH⊗(P)by

P⊗P^∗

× P⊗P^∗

→R, (p⊗α,q⊗β)7→α(q)·β(p).

If one replaces⊗by⊕and·by+, this becomes the standard hyperbolic form.

The multiplicative hyperbolic form induces aring homomorphism K₀(R)→L⁰(R), [P]7→[H⊗(P)].

It is anisomorphism forR=Z.

Concluding Remarks