Inheritance of Isomorphism Conjectures under colimits

Inheritance of Isomorphism Conjectures under colimits

Wolfgang Lück

Münster

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

May 2007

Outline

We define the notion of anequivariant homology theory.

We explain the notion of aclassifyingG-space of a family of subgroups.

We explain what anIsomorphism Conjectureis.

We give someapplicationsof the Farrell-Jones Conjecture.

We prove inheritance properties undercolimits.

We explainconsequencesof these inheritance properties.

Convention: group will always meandiscrete group.

Equivariant homology theories

Definition (G-homology theory)

AG-homology theoryH∗ is a covariant functor from the category of G-CW-pairs to the category ofZ-graded abelian groups together with natural transformations

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

G-homotopy invariance;

Long exact sequence of a pair;

Excision;

Disjoint union axiom.

Definition (Equivariant homology theory)

Anequivariant homology theoryH^?_∗ assigns to every groupGa G-homology theoryH^G_∗. These are linked together with the following so calledinduction structure: given a group homomorphismα:H→G and aH-CW-pair(X,A)there are for alln∈Znatural homomorphisms

ind_α:H^H_n(X,A) → H^G_n(ind_α(X,A)) satisfying:

Bijectivity

If ker(α)acts freely onX, then ind_α is a bijection;

Compatibility with the boundary homomorphisms;

Functoriality inα;

Compatibility with conjugation.

Example (Equivariant homology theories)

Given aK∗ non-equivariant homology theory, put H^G_∗(X) := K_∗(X/G);

H^G_∗(X) := K_∗(EG×_GX) Borel homology.

Equivariant bordismΩ^?_∗(X);

Equivariant topologicalK-homologyK_∗^?(X)in the sense of Kasparov.

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∈Z.

Given a spectrumE, a classical construction in algebraic topology assigns to it a homology theoryH∗(−,E)with the property

Hn(pt;E) =πn(E).

Put

Hn(X;E) :=πn(X₊∧E).

The basic example of a spectrum is thesphere spectrumS.

Itsn-th space isSⁿand itsn-th structure map is the standard homeomorphismSⁿ∧S¹−→^∼= Sⁿ⁺¹.

Its associated homology theory isstable homotopy π_∗^s(−) =H∗(−;S).

This construction can be extended to the equivariant setting as follows.

Theorem (L.-Reich (2005))

Given a functorE:Groupoids→Spectrasending equivalences to weak equivalences, there exists an equivariant homology theory H^?_∗(−;E)satisfying

H^H_n(pt)∼=H^G_n(G/H)∼=πn(E(H)).

Theorem (Equivariant homology theories associated toK and L-theory,Davis-L. (1998))

Let R be a ring (with involution). There exist covariant functors K_R,L^h−∞i_R ,K^top

l¹ :Groupoids → Spectra;

K^top:Groupoids^inj → Spectra, with the following properties:

They send equivalences to weak equivalences;

For every group G and all n∈Zwe have:

πn(K_R(G)) ∼= K_n(RG);

π_n(L^h−∞i_R (G)) ∼= L^h−∞i_n (RG);

πn(K^top(G)) ∼= K_n(C_r^∗(G));

πn(K^top

l¹ (G)) ∼= Kn(l¹(G)).

Example (Equivariant homology theories associated toK and L-theory)

We get equivariant homology theories:

H_∗^?(−;K_R);

H_∗^?(−;L^h−∞i_R );

H_∗^?(−;K^top);

H_∗^?(−;K^top_l1 ),

satisfying forH ⊆G:

H_n^G(G/H;K_R) ∼= H_n^H(pt;K_R) ∼= Kn(RH);

H_n^G(G/H;L^h−∞i_R ) ∼= H_n^H(pt;L^h−∞i_R ) ∼= L^h−∞i_n (RH);

H_n^G(G/H;K^top) ∼= H_n^H(pt;K^top) ∼= K_n(C_r^∗(H));

H_n^G(G/H;K^top_l1 ) ∼= H_n^H(pt;K^top_l1 ) ∼= Kn(l¹(H)).

Classifying spaces for families of subgroups

Definition (G-CW-complex)

AG-CW -complexX is aG-space together with aG-invariant filtration

∅=X₋₁⊆X₀⊆. . .⊆X_n⊆. . .⊆ [

n≥0

X_n=X

such thatX carries thecolimit topologywith respect to this filtration, andX_nis obtained fromX_n−1for eachn≥0 byattaching equivariant n-dimensional cells, i.e., there exists aG-pushout

`

i∈InG/H_i×Sⁿ⁻¹

‘

i∈Inq_iⁿ

//

X_n−1

`

i∈I G/H_i×Dⁿ

‘

i∈InQⁿ_i

//Xn

Example (Simplicial actions)

LetX be a simplicial complex. Suppose thatGacts simplicially onX. ThenGacts simplicially also on thebarycentric subdivisionX⁰, and the G-spaceX⁰ inherits the structure of aG-CW-complex.

Example (Smooth actions)

IfGacts properly and smoothly on a smooth manifoldM, thenM inherits the structure ofG-CW-complex.

Definition (Family of subgroups)

AfamilyF of subgroupsofGis a set of subgroups ofGwhich is closed under conjugation and taking subgroups.

Examples forF are:

T R = {trivial subgroup};

F IN = {finite subgroups};

VCYC = {virtually cyclic subgroups};

ALL = {all subgroups}.

Definition (ClassifyingG-space for a family of subgroups,tom Dieck(1974))

LetF be a family of subgroups ofG. A model for theclassifying G-space for the familyF is aG-CW-complexEF(G)which has the following properties:

All isotropy groups ofEF(G)belong toF;

For anyG-CW-complexY, whose isotropy groups belong toF, there is up toG-homotopy precisely oneG-mapY →X.

We abbreviateE G:=EF IN(G)and call it theuniversal G-space for proper G-actions.

We also writeEG=ET R(G).

IfF ⊆ Gare families of subgroups ofG, there is up toG-homotopy precisely oneG-mapEF(G)→EG(G).

Theorem (Homotopy characterization ofEF(G)) LetF be a family of subgroups.

There exists a model for EF(G)for any familyF;

Two models for EF(G)are G-homotopy equivalent;

A G-CW -complex X is a model for EF(G)if and only if all its isotropy groups belong toF and for each H∈ F the H-fixed point set X^H is contractible.

IfF ⊆ Gare families of subgroups ofG, thenEF(G)×EG(G)is a model forEF(G).

The spacesE G are interesting in their own right and have often very nice geometric models which are rather small. For instance

Rips complexfor word hyperbolic groups;

Teichmüller spacefor mapping class groups;

Outer spacefor the group of outer automorphisms of free groups;

L/K for an almost connected Lie groupL, a maximal compact subgroupK ⊆LandG⊆La discrete subgroup;

CAT(0)-spaceswith proper isometricG-actions, e.g., simply connected Riemannian manifolds with non-positive sectional curvature or trees.

Isomorphism Conjectures

Conjecture (Isomorphism Conjecture)

LetH_∗^?be an equivariant homology theory. It satisfies theIsomorphism Conjecturefor the group G and the familyF if the projection

E_F(G)→pt induces for all n∈Za bijection H^G_n(EF(G))→ H_n^G(pt).

The point is to find an as small as possible familyF.

The Isomorphism Conjecture is always true forF=ALLsince it becomes a trivial statement because ofEALL(G) =pt.

Thephilosophyis to be able to compute the functor of interest for Gby knowing it on the values of elements inF.

Example (Farrell-Jones Conjecture)

The Farrell-Jones ConjectureforK-theory orL-theory respectively with coefficients inRis the Isomorphism Conjecture forH^?_∗ =H∗(−;K_R)or H^?_∗ =H∗(−;L^h−∞i_R )respectively andF =VCYC.

In other words, it predicts that the assembly map

H_n^G(EVCYC(G),K_R)→H_n^G(pt,K_R) =Kn(RG) or

H_n^G(EVCYC(G),L^h−∞i_R )→H_n^G(pt,L^h−∞i_R ) =L^h−∞i_n (RG) respectively is bijective for alln∈Z.

Example (Baum-Connes Conjecture)

The Baum-Connes Conjectureis the Isomorphism Conjecture for H^?_∗ =K_∗^? =H_∗^?(−;K^top)andF =F IN.

In other words it predicts that the assembly map

K_n^G(E G) =H_n^G(EF IN(G),K^top)→H_n^G(pt,K^top) =Kn(C_r^∗(G)) is bijective for alln∈Z.

Example (Bost Conjecture)

The Bost Conjectureis the Isomorphisms Conjecture for H^?_∗ =K_∗^? =H_∗^?(−;K^top_l1 )andF =F IN.

In other words it predicts that the assembly map

K_n^G(E G) =H_n^G(EF IN(G),K^top_l1 )→H_n^G(pt,K^top_l1 ) =Kn(l¹(G))

Applications

Definition (Whitehead group)

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

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)

Let M₀be a closed (smooth) manifold of dimension n≥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)) vanishes.

Thes-cobordism theorem is a key ingredient in thesurgery programfor the classification of closed manifolds due toBrowder, Novikov, SullivanandWall.

If Wh(G)vanishes, everyh-cobordism(W;M₀,M₁)of dimension

≥6 withG∼=π₁(W)is trivial and in particularM₀∼=M₁.

TheK-theoretic Farrell-Jones Conjecture implies for a torsionfree groupGthat Wh(G)is trivial.

ThePoincaré Conjecturein dimension≥5 is a consequence of thes-cobordism theorem since Wh({1})vanishes.

Conjecture (Kaplansky Conjecture)

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

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

If the torsionfree group G satisfies the Baum-Connes Conjecture, then the Kaplansky Conjecture is true for C_r^∗(G)and hence forCG.

Theorem (The Farrell-Jones Conjecture and the Kaplansky Conjecture,Bartels-L.-Reich (2007))

] Let F be a skew-field and let G be a group satisfying the K -theoretic Farrell-Jones Conjecture with coefficients in 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.

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 is homotopic to an isometric diffeomorphism.

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

There are also non-aspherical manifolds which are topological rigid in the sense of the Borel Conjecture (seeKreck-L. (2005)).

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.

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

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

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:M₀→M₁with f₁◦g 'f₂ we have

sign_x(M₀,f₀) =sign_x(M₁,f₁).

Theorem (The Farrell-Jones, the Baum-Connes and the Novikov Conjecture)

Suppose that one of the following assembly maps

H_n^G(EVCYC(G),L^−∞_R ) → H_n^G(pt,L^−∞_R ) =L^−∞_n (RG);

K_n^G(E G) =H_n^G(E_{F IN}(G),K^top) → H_n^G(pt,K^top) =K_n(C_r^∗(G)), is rationally injective.

Then the Novikov Conjecture holds for the group G.

Inheritance properties

Fix an equivariant homology theoryH_∗^?. Theorem (Transitivity Principle)

SupposeF ⊆ G are two families of subgroups of G. Assume that for every element H ∈ G the group H satisfies the Isomorphism

Conjecture forF |_H ={K ⊆H |K ∈ F }.

Then the map

H_n^G(EF(G))→ H^G_n(EG(G)) is bijective for all n∈Z.

Moreover,(G,G)satisfies the Isomorphism Conjecture if and only if (G,F)satisfies the Isomorphism Conjecture.

Sketch of proof.

For aG-CW-complexX with isotropy group inGconsider the natural map induced by the projection

s^G_∗(X) :H^G_∗(X×E_F(G))→ H^G_∗(X).

This a natural transformation ofG-homology theories defined for G-CW-complexes with isotropy groups inG.

In order to show that it is a natural equivalence it suffices to show thats_n^G(G/H)is an isomorphism for allH ∈ G andn∈Z.

Sketch of proof (continued).

TheG-spaceG/H×EF(G)isG-homeomorphic to G×_Hres^H_GEF(G)and res^H_GEF(G)is a model forE_{F |H}(H).

Hence by the induction structures_n^G(G/H)can be identified with the assembly map

H^H_∗(E_{F |H}(H))→ H^H_∗(pt), which is bijective by assumption.

Now apply this toX =EG(G)and observe thatEG(G)×EF(G)is a model forEF(G).

Example (Baum-Connes Conjecture andVCYC) Consider the Baum-Connes setting, i.e., takeH_∗^?=K_∗^?. Consider the familiesF IN ⊆ VCYC.

For every virtually cyclic groupV the Baum-Connes Conjecture is true, i.e.,

K_n^V(EF IN(V))→Kn(C_r^∗(V)) is bijective forn∈Z.

Hence by the Transitivity principle the following map is bijective for all groupsGand alln∈Z

K_n^G(E G) =K_n^G(EF IN(G))→K_n^G(EVCYC(G)).

This explains why in the Baum-Connes setting it is enough to deal withF IN instead ofVCYC.

This is not true in the Farrell-Jones setting and causes many extra difficulties there (NILandUNIL-phenomena).

This difference is illustrated by the following isomorphisms due to Pimsner-VoiculescuandBass-Heller-Swan:

K_n(C_r^∗(Z)) ∼= K_n(C)⊕K_n−1(C);

Kn(R[Z]) ∼= Kn(R)⊕Kn−1(R)⊕NKn(R)⊕NKn(R).

Consider a directed system of groups{G_i |i ∈I}with structure mapsψ_i:G_i →Gfori ∈I. PutG=colim_i∈IG_i.

LetX be aG-CW-complex.

We have the canonicalG-map

ad: (ψi)∗ψ^∗_iX =G×_Gi X →X, (g,x)7→gx.

Define a homomorphism t_n^G(X): colim

i∈I H^G_nⁱ(ψ_i^∗X) −^∼=→ H^G_n(X) by the colimit of the system of maps indexed byi∈I

H^G_nⁱ(ψ^∗_iX)−−−→ H^{indψi G}_n ((ψi)∗ψ_i^∗X) ^H

Gn(ad)

−−−−−→ H^G_n(X).

Definition (Strongly continuous equivariant homology theory) An equivariant homology theoryH^?_∗ is calledstrongly continuousif for every groupGand every directed system of groups{G_i |i ∈I}with G=colim_i∈IG_i the map

t_n^G(pt) : colim

i∈I H^G_nⁱ(pt)→ H_n^G(pt) is an isomorphism for everyn∈Z.

Lemma

Consider a directed system of groups{G_i |i∈I}with G=colim_i∈IG_i. Let X be a G-CW -complex. Suppose thatH_∗^?is strongly continuous.

Then the homomorphism t_n^G(X) : colim

i∈I H^G_nⁱ(ψ^∗_iX) −→ H^∼= _n^G(X) is bijective for every n∈Z.

Idea of proof.

Show thatt_∗^G is a transformation ofG-homology theories.

Prove that the strong continuity implies thatt_n^G(G/H)is bijective for alln∈ZandH⊆G.

Then a general comparison theorem gives the result.

Letφ:K →Gbe a group homomorphism and letF be a family of subgroups ofG.

Define the familyφ^∗F of subgroups ofK by φ^∗F :={L⊆K |φ(L)∈ F }.

Basic property: φ^∗EF(G) =Eφ∗F(K).

Lemma

LetF be a family of subgroups of G. Let{G_i |i ∈I}be a directed system of groups with G =colimi∈IG_i and structure mapsψ_i:G_i →G.

Suppose thatH^?_∗ is strongly continuous and for every i∈I the Isomorphism Conjecture holds for G_i andψ^∗_iF.

Then the Isomorphism Conjecture holds for G andF.

Proof.

This follows from the following commutative square, whose horizontal arrows are bijective because of the last lemma, and the identification ψ_i^∗EF(G) =E_ψ^∗

iF(G_i)

colim_i∈IH_n^Gi(E_ψ^∗

iF(G_i)) ^t

nG(EF(G))

∼= //

H^G_n(EF(G))

colim_i∈IH^G_nⁱ(pt) ^t

nG(pt)

∼= //H^G_n(pt)

Fix a class of groupsCclosed under isomorphisms, taking subgroups and taking quotients, e.g., the class of finite groups or the class of virtually cyclic groups.

For a groupGletC(G)be the family of subgroups ofGwhich belong toC.

Theorem (Inheritance under colimits for Isomorphism Conjectures)

Let{G_i |i∈I}be a directed system of groups with G=colimi∈IG_i. Suppose thatH^?_∗ is strongly continuous and that the Isomorphism Conjecture is true for(H,C(H))for every i∈I and every subgroup H ⊆G_i.

Then for every subgroup K ⊆G the Isomorphism Conjecture is true for K andC(K).

Proof.

IfGis the colimit of the directed system{G_i |i∈I}, then the subgroupK ⊆Gis the colimit of the directed system

{ψ_i⁻¹(K)|i ∈I}. Hence we can assumeG=K without loss of generality.

SinceCis closed under quotients by assumption, we have C(G_i)⊆ψ_i^∗C(G)for everyi ∈I. Hence we can consider for any i ∈Ithe composition

H_n^Gi(E_C(Gi)(G_i))→H_n^Gi(E_ψ^∗

iC(G)(G_i))→H_n^Gi(pt).

By the last lemma it suffices to show that the second map is bijective.

By assumption the composition of the two maps is bijective.

Hence it remains to show that the first map is bijective.

Proof (continued).

By the Transitivity Principle this follows from the assumption that the Isomorphism Conjecture holds for every subgroupH ⊆G_i and in particular for anyH∈ψ_i^∗C(G)forC(G_i)|_H =C(H).

Notice that it is very convenient for the proof to allow arbitrary families of subgroups and to have the definition ofH^G_∗(X)at hand for arbitrary (not necessarily proper)G-CW-complexesX.

Lemma

The homology theories

H_∗^?(−;K_R);

H_∗^?(−;L^h−∞i_R );

H_∗^?(−;K^top

l¹ ), are strongly continuous.

For instance one has to show that the canonical map induced by the various structure mapsG_i →Ginduces an isomorphism

colim

i∈I Kn l¹(G_i) ^∼₌

−→Kn l¹ colim

i∈I G_i .

This statement does not make sense for the reduced group C^∗-algebra since it is not functorial under arbitrary group homomorphisms.

For instance,C_r^∗(Z∗Z)is a simpleC^∗-algebra and hence no epimorphismC_r^∗(Z∗Z)→C_r^∗({1})exists.

Let{G_i |i ∈I}be a directed system of groups with (not necessarily injective) structure mapsφi,j:G_i →G_j. Let G=colimi∈IG_i be its colimit.

Next we pass totwisted coefficients: LetRbe a ring (with involution) and letAbe aC^∗-algebra, both with structure preservingG-action.

Giveni ∈Iand a subgroupH ⊆G_i, we letH act onRandAby restriction with the group homomorphism(ψi)|_H:H →G.

The following result follows for untwisted coefficients from the previous result. In the twisted case one has to modify the setting by considering everything over a fixed reference group.

Theorem (Inheritance under colimits for the Farrell-Jones and the Bost Conjecture,Bartels-Echterhoff-Lück (2007))

In the situation above we get:

Suppose that the assembly map

H_n^H(EVCYC(H);K_R)→H_n^H(pt;K_R) =K_n(RoH) is bijective for all n∈Z, all i ∈I and all subgroups H ⊆G_i. Then for every subgroup K of G the assembly map

H_n^K(EVCYC(K);K_R)→H_n^K(pt;K_R) =Kn(RoK) is bijective for all n∈Z.

The corresponding version is true for the assembly maps in the L-theory setting and for the Bost Conjecture.

Theorem (Bartels-L.-Reich (2007))

Let G be a subgroup of a finite product of hyperbolic groups. Let R be a ring with structure preserving G-action.

Then the K -theoretic Farrell-Jones Conjecture holds for G and R, i.e., the assembly map

H_n^G(E_VCYC(K);K_R)→H_n^G(pt;K_R) =K_n(RoG) is bijective for all n∈Z.

Theorem (Lafforgue (2002))

Let G be a subgroup of a hyperbolic group. Let A be a C^∗-algebra with structure preserving G-action.

Then the Bost Conjecture holds for G and A, i.e., the assembly map H_n^G(E G;K^top_A,l1)→H_n^G(pt;K^top_A,l1) =K_n(Ao_l¹G)

is bijective for all n∈Z.

Theorem (The Farrell-Jones and the Bost Conjecture with coefficients for colimits of hyperbolic groups,

Bartels-Echterhoff-Lück (2007))

Both the K -theoretic Farrell-Jones Conjecture and the Bost Conjecture with twisted coefficients hold for a group G if G is a subgroup of a colimit of directed system of hyperbolic groups (with not necessarily injective structure maps).

The theorem above is not true for the Baum-Connes Conjecture because of the lack of functoriality of the reduced group

C^∗-algebra.

One needs for the Baum-Connes setting that all structure maps have amenable kernels.

The groups above are certainly wild inBridson’suniverse of groups.

Many recent constructions of groups with exotic properties are given by colimits of directed systems of hyperbolic groups.

Examples are.

groups with expanders;

Lacunary hyperbolic groupsin the sense ofOlshanskii-Osin-Sapir;

Tarski monsters, i.e., groups which are not virtually cyclic and whose proper subgroups are all cyclic;

certain infinite torsion groups.

Miscellaneous

The Baum-Connes Conjecture and the Farrell-Jones Conjecture do not seem to be known forSLn(Z)forn≥3,mapping class groupsandOut(Fn);

Certaingroups with expandersyield counterexamples to the Baum-Connes Conjecture with coefficients by a construction due toHigson-Lafforgue-Skandalis (2002).

TheK-theoretic Farrell-Jones conjecture and the Bost Conjecture are true for these groups as shown above.

So the counterexample ofHigson-Lafforgue-Skandalis (2002) shows that the mapKn(Aol¹G)→Kn(Aor G)is not bijective in general.

It is not known whether there are counterexamples to the Farrell-Jones Conjecture or the Baum-Connes Conjecture.

There seems to be no promising candidate of a groupGwhich is a potential counterexample to theK- orL-theoretic Farrell-Jones Conjecture or the Bost Conjecture.

The Baum-Connes Conjecture is the one for which it is most likely that there may exist a counterexample.

One reason is the existence of counterexamples to the version with coefficients and thatK_n(C^∗_r(G))has certainfailures concerning functorialitywhich do not exists forK_n^G(E G).

These failures are not present forKn(RG),L^h−∞i(RG)and K_n(l¹(G)).

Bartels and L.have a program to prove theL-theoretic

Farrell-Jones Conjecture for all coefficient rings and the same class of groups for which theK-theoretic versions have been proved.

Bartels and L.have a program to prove the Farrell-Jones

Conjecture forGand all twisted coefficients ifGacts properly and cocompactly on a simply connected CAT(0)-space. This would yield the same result for all subgroups of cocompact lattices in almost connected Lie groups.

H_n^G(EF IN(G);L^p

Z)[1/2] −−−−→^∼= L^p_n(ZG)[1/2]



 y

∼=



 y

∼=

H_n^G(EF IN(G);L^p

R)[1/2] −−−−→^∼= L^p_n(RG)[1/2]



 y

∼=



 y

∼=

H_n^G(EF IN(G);L^p_C∗

r(?;R))[1/2] −−−−→^∼= L^p_n(C_r^∗(G;R))[1/2]



 y

∼=



 y

∼=

H_n^G(EF IN(G);K^top

R )[1/2] −−−−→^∼= K_n(C_r^∗(G;R))[1/2]



 y



 y

H_n^G(EF IN(G);K^top)[1/2] −−−−→^∼= Kn(C_r^∗(G))[1/2]