On the Conjectures of Bost and of Baum-Connes and the generalized Trace Conjecture

On the Conjectures of Bost and of Baum-Connes and the generalized Trace Conjecture

Wolfgang Lück

Münster

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

May 2007

Outline

Equivariant homology theory.

ClassifyingG-space for proper actions.

Conjecture due toBostand toBaum-Connes.

Inheritance properties underdirected colimits.

Equivariant Chern characters.

(Generalized) Trace Conjectured.

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.

Recall forH ⊆Gfinite

K_n^G(G/H)∼=K_n^H({•})∼=

(R_C(H) neven;

{0} nodd.

Classifying spaces for proper actions

Definition (ClassifyingG-space for properG-actions,tom Dieck(1974))

A model for theclassifying G-space for proper G-actionsis a proper G-CW-complexE Gsuch that for any properG-CW-complexY there is up toG-homotopy precisely oneG-mapY →E G.

Theorem (Homotopy characterization ofEF(G)) There exists a model for E G;

Two models for E G are G-homotopy equivalent;

A proper G-CW -complex X is a model for E G if and only if for each H ∈ F the H-fixed point set X^H is contractible.

We haveEG=E Gif and only ifGis torsionfree.

We haveE G={•}if and only ifGis finite.

A model forE D∞is the real line with the obvious D∞=Z o Z/2=Z/2∗Z/2-action.

Every model forED_∞is infinite dimensional, e.g., the universal covering ofRP^∞∨RP^∞.

The spacesE Gare interesting in their own right and have often very nice geometric modelswhich are rather small.

On the other hand anyCW-complex is homotopy equivalent to G\E Gfor some groupG(seeLeary-Nucinkis (2001)).

Conjectures due to Bost and Baum-Connes

Conjecture (Baum-ConnesConjecture)

The Baum-Connes Conjecturepredicts that the assembly map K_n^G(E G)→Kn(C_r^∗(G))

is bijective for all n∈Z.

Conjecture (BostConjecture)

The Bost Conjecturepredicts that the assembly map K_n^G(E G)→K_n(l¹(G)) is bijective for all n∈Z.

These conjecture have versions, where one allowscoefficients in aG-C^∗algebraA

K_n^G(E G;A) → K_n(AoC_r^∗G);

K_n^G(E G;A) → K_n(Aol¹G).

There is a natural map

ι:Kn(Aol¹G)→Kn(AoC^∗_r G) map.

The composite of the assembly map appearing in the Bost Conjecture withιis the assembly map appearing in the Baum-Connes Conjecture.

We will see that the Bost Conjecture has a better chance to be true than the Baum-Connes Conjecture.

On the other hand the Baum-Connes Conjecture has a higher potential for applications since it is related to index theory and thus has interesting consequences for instance to the Conjectures due toBass,Gromov-Lawson-Rosenberg,Novikov,Kadison, Kaplansky.

These conjecture have been proved for interesting classes of groups. Prominent papers have been written for instance by Connes,Gromov,Higson,Kasparov,Lafforgue,Mineyev, Skandalis,Yu,Weinbergerand others.

Inheritance properties under colimits

Letψ:H →Gbe a (not necessarily injective) group homomorphism.

GivenG-CW-complexY, letψ^∗Y be theH-CW-complex obtained fromY byrestrictingtheG-action to aH-action viaψ.

GivenH-CW-complexX, letψ∗X be theG-CW-complex obtained fromY byinductionwithψ, i.e.,ψ∗X =G×_ψX.

Consider a directed system of groups{G_i |i ∈I}with (not necessarily injective) structure mapsψ_i:G_i →Gfori∈I. Put G=colim_i∈IG_i.

LetX be aG-CW-complex.

We have the canonicalG-map

ad: (ψ_i)∗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 ((ψ_i)∗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({•}) : colim

i∈I H^G_nⁱ({•})→ H^G_n({•}) is an isomorphism for everyn∈Z.

Theorem (Bartels-Echterhoff-Lück (2007))

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.

Theorem (Bartels-Echterhoff-Lück (2007))

Let{G_i |i ∈I}be a directed system of groups with G=colimi∈IG_i and (not necessarily injective) structure mapsψ_i:G_i →G. Suppose that H^?_∗ is strongly continuous and for every i∈I and subgroup H ⊆G_i the assembly map

H_n^H(E H)→H_n^H({•}) is bijective.

Then for every subgroup K ⊆G (and in particular for K =G) also the assembly map

H_n^K(E K)→H_n^K({•}) is bijective.

Lemma (Davis-Lück(1998))

There are equivariant homology theoriesH^?_∗(−;C_r^∗)andH^?_∗(−;l¹) defined for all equivariant CW -complexes with the following properties:

If H ⊆G is a (not necessarily finite) subgroup, then

H^G_n(G/H;C_r^∗) ∼= H_n^H({•};C_r^∗) ∼= K_n(C_r^∗(H));

H^G_n(G/H;l¹) ∼= H^H_n({•};l¹) ∼= Kn(l¹(H));

H^?_∗(−,l¹)is strongly continuous;

BothH^?_∗(−;C_r^∗)andH^?_∗(−;l¹)agree for proper equivariant CW -complexes with equivariant topological K -theory K_∗^?in the sense of Kasparov.

One ingredient in the proof of the strong continuity ofH^?_∗(−;l¹)is to show

colim

i∈I Kn(l¹(G_i))∼=Kn(l¹(G)).

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.

HenceH^?_∗(−;C_r^∗)isnotstrongly continuous.

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

Let{G_i |i ∈I}be a directed system of groups with G=colimi∈IG_i and (not necessarily injective) structure mapsψ_i:G_i →G. Suppose that the Bost Conjecture with C^∗-coefficients holds for all groups G_i. Then the Bost Conjecture with C^∗-coefficients holds for G.

Theorem (Lafforgue (2002))

The Bost Conjecture holds with C^∗-coefficients holds for all hyperbolic groups.

Corollary

Let{G_i |i∈I}be a directed system of hyperbolic groups with (not necessarily injective structure maps).

Then the Bost Conjecture holds with C^∗-coefficients holds for colim_i∈IG_i.

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.

Certaingroups with expandersyield counterexamples to the surjectivity of the assembly map appearing Baum-Connes Conjecture with coefficients by a construction due

toHigson-Lafforgue-Skandalis (2002).

These implies that the mapKn(Aol¹G)→Kn(Aor G)is not surjective in general.

The main critical point concerning the Baum-Connes Conjecture is that the reduced groupC^∗-algebra of a group lacks certain functorial properties which are present on the left side of the assembly map. This is not true if one deals withl¹(G)or groups ringsRG.

The counterexamples above raised the hope that one may find counterexamples to the conjectures due toBaum-Connes,Borel, Bost,Farrell-Jones,Novikov.

The results above due toBartels-Echterhoff-Lück (2007)and unpublished work byBartels-Lück (2007)prove all these conjectures (with coefficients) except the Baum-Connes Conjecture for colimits of hyperbolic groups.

There is no counterexample to the Baum-Connes Conjecture (without coefficients) in the literature.

Equivariant Chern character

Theorem (Artin’s Theorem) Let G be finite. Then the map

M

C⊂G

ind^G_C: M

C⊂G

R_C(C)→R_C(G)

is surjective after inverting|G|, where C⊂G runs through the cyclic subgroups of G.

LetC be a finite cyclic group.

TheArtin defectis the cokernel of the map M

D⊂C,D6=C

ind^C_D: M

D⊂C,D6=C

R_C(D)→R_C(C).

For an appropriate idempotentθ_C ∈R_Q(C)⊗_ZZ h 1

|C|

i

the Artin defect is after inverting the order of|C|canonically isomorphic to

θ_C·R_C(C)⊗_ZZ 1

|C|

.

Theorem (Lück(2002))

Let X be a proper G-CW -complex. LetZ⊆Λ^G ⊂Qbe the subring of Qobtained by inverting the orders of all the finite subgroups of G.

Then there is a natural isomorphism ch^G: M

(C)

K_n(C_GC\X^C)⊗_Z[WGC]θ_C·R_C(C)⊗_ZΛ^G

∼=

−→K_n^G(X)⊗_ZΛ^G, where(C)runs through the conjugacy classes of finite cyclic

subgroups and W_GC=N_GC/C·C_GC.

Example (Improvement of Artin’s Theorem)

Consider the special case whereGis finite andX ={•}Then we get an improvement of Artin’s theorem, namely,

M

(C)

Z⊗_Z[WGC]θ_C·R_C(C)⊗_ZZ 1

|G|

∼=

−→R_C(G)⊗_ZZ 1

|G|

Example (X =E G)

In the special caseX =E Gwe get an isomorphism M

(C)

K_n(BC_GC)⊗_Z[W

GC]θ_C·R_C(C)⊗_ZΛ^G −→^∼= K_n^G(E G)⊗_ZΛ^G,

Conjecture (Trace Conjecture forG) The image of the trace map

K₀(C_r^∗(G))−→^tr R

is the additive subgroup ofRgenerated by{_|H|¹ |H ⊂G,|H|<∞}.

Lemma

Let G be torsionfree. Then the Baum-Connes Conjecture for G implies the Trace Conjecture for G.

Proof.

The following diagram commutes because of theL²-index theoremdue toAtiyah(1974).

K₀^G(EG) ^//

∼=

K₀(C_r^∗(G))^{tr //}R

K₀(BG) ^//K₀({•}) ^{∼= //}Z

OO

Theorem (Roy(1999))

The Trace Conjecture is false in general.

Conjecture (Modified Trace Conjecture)

LetΛ^G⊂Qbe the subring ofQobtained fromZby inverting the orders of finite subgroups of G. Then the image of the trace map

K₀(C_r^∗(G))−−−→^trN(G) R is contained inΛ^G.

Theorem (Image of the traceLueck(2002)) The image of the composite

K₀^G(E G)−−−→^asmb K₀(C_r^∗(G))−−−→^trN(G) R is contained inΛ^G.

In particular the Baum-Connes Conjecture implies the Modified Trace Conjecture.

Problem: What is the image of the trace map in terms ofG?

TakeX =E G. Elements inK∗(E G)are given by elliptic G-operatorsP over cocompact properG-manifolds with Riemannian metrics.

Problem: What is the concrete preimage of its class under ch^G_∗? One term could be the index ofP^C onM^Cgiving an element in K₀(C_GC\E^C)which isK₀(BC_GC)after tensoring withΛ^G.

Another term could come from the normal data ofM^C inMwhich yields an element inθ_C·R_C(C).

The failure of the Trace Conjecture shows that this is more complicated than one anticipates. The answer to the question above would lead to a kind oforbifoldL²-index theoremwhose possible denominators, however, are not of the expected shape

n forH ⊆Gfinite.