A Survey on the

Baum-Connes Conjecture

Wolfgang Lück^{∗}

Fachbereich Mathematik und Informatik

Westfälische Wilhelms-Universität

Münster Einsteinstr. 62 48149 Münster

Germany

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

June 2004

1. The statement of the Baum-Connes Conjecture

G will always denote a discrete group.

Conjecture 1 (Baum-Connes Conjecture for G) The assembly map

asmb : K_{p}^{G}(EG) → K_{p}(C_{r}^{∗}(G))

which sends [M, P^{∗}] to index _{C_{r}^{∗}(G)}(P^{∗}) is bijective for p ∈ ℤ.

Here are some explanations:

• C_{r}^{∗}(G) is the reduced C^{∗}-algebra of G, i.e. the completion of the complex group ring with respect to the operator norm coming from the embedding

ℂG → B(l^{2}(G), l^{2}(G)), g ↦ r_{g},

where r_{g} is given by right multiplication with g ∈ G. For example

C_{r}^{∗}(ℤ^{n}) = C(T^{n}).

• K_{p}(C_{r}^{∗}(G)) is the topological K-theory of C_{r}^{∗}(G).

This is for p = 0 the same as the algebraic K-group. So elements in K_{0}(C_{r}^{∗}(G)) are represented by finitely generated modules over the ring C_{r}^{∗}(G).

These groups are two-periodic, i.e.

K_{p}(C_{r}^{∗}(G)) = K_{p+2} (C_{r}^{∗}(G))

for p ∈ ℤ.

In odd dimensions the topology plays a role, namely

K_{n}(C_{r}^{∗}(G)) = π_{n−1}(GL(C_{r}^{∗}(G)))

for n ≥ 1.

• EG is the classifying space for proper G-actions. It is characterized uniquely up to G-homotopy by the property that it is a G-CW -complex whose isotropy groups are all finite and whose H-fixed point sets for H ⊂ G are contractible.

If G is torsionfree, this coincides with EG.

These spaces are interesting in their own right and have often very nice geometric models. For instance

– Rips complex for word hyperbolic groups;

– Teichmüller space for mapping class groups;

– Outer space for the group of outer automorphisms of free groups;

– L/K for a connected Lie group L, a maximal compact subgroup K ⊆ L and G ⊆ L a discrete subgroup;

– Riemannian manifolds with non-positive sectional curvature and proper isometric G-action;

– Trees with proper G-action.

• K_{p}^{G}(X) for a proper G-CW -complex X is the equivariant K-homology of X as defined for instance by Kasparov.

If G acts freely on X, there is a canonical isomorphism

K_{0}^{G}(X) ⟶^{≅} K_{0}(G\X)

to the K-homology of G\X.

For H ⊂ G finite, K_{0}^{G}(G/H) is Rep _{ℂ}(H).

An element in K_{p}^{G}(EG) is given by a pair (M, P^{∗}) which consists of a smooth manifold with proper cocompact G-action and an elliptic G-complex P^{∗} of differential operators of order 1;

• index _{C_{r}^{∗}(G)} is the C_{r}^{∗}(G)-valued index due to Mishchenko and Fomenko; Remark 2 The Baum-Connes Conjecture makes also sense for topological groups and is in particular for Lie groups and for p-adic groups closely related to their representation theory.

Remark 3 The Farrell-Jones Isomorphisms Conjecture for the algebraic K-theory K_{n}(RG) and the algebraic L-theory L_{n}(RG) of group rings is analogous to the the Baum-Connes Conjecture. Both can be viewed as a special case of an assembly principle and their setup and computational questions can be treated simultaneously. However, the proofs of these in special cases have so far been rather different.

The Farrell-Jones Conjecture plays an important role in the classification of manifolds, in particular in connection with the Borel Conjecture that two aspherical closed topological manifolds are homeomorphic if and only if their fundamental groups are isomorphic.

2. Some applications of the Baum-Connes Conjecture

Next we explain the relevance of the Baum-Connes Conjecture.

• Computations Since K_{p}^{G}(−) is an equivariant homology theory for proper G-CW -complexes, it is much easier to compute K_{p}^{G}(EG) than to compute K_{p}(C_{r}^{∗}(G)). Techniques like Mayer-Vietoris sequences apply.

• Novikov-Conjecture for G The Hirzebruch signature formula says

sign (M) = ⟨L(M), [M]⟩.

Given a map f : M → BG and x ∈ H^{∗}(BG), define the higher signature by

sign _{x}(M, f) = ⟨f^{∗}(x) ∪ L(M), [M]⟩.

The Novikov Conjecture says that these are homotopy invariants, i.e. for f :

M → BG, g : N → BG and a homotopy equivalence u : M → N with g ∘ u ≃ f we have

sign _{x}(M, f) = sign _{x}(N, g).

The Baum-Connes Conjecture for G implies the Novikov Conjecture for G.

• Stable Gromov-Lawson-Rosenberg Conjecture for G Let M be a closed Spin-manifold with fundamental group G. Let B be the Bott manifold. The Stable Gromov-Lawson-Rosenberg Conjecture says that M ×B^{k} carries a Riemannian metric of positive scalar curvature for some k ≥ 0 if and only if

index _{C_{r}^{∗}(G)}(M̃, D̃) = 0.

Here D is the Dirac operator and D̃ its lift to M̃.

Stolz has shown that the Baum-Connes Conjecture for G implies the stable Gromov-Lawson-Rosenberg Conjecture for G. The unstable version of the Gromov-Lawson-Rosenberg Conjecture , i.e. k = 0, is false in general by a construction of Schick.

• The Kadison Conjecture The Baum-Connes Conjecture implies

Conjecture 4 (Kadison Conjecture) Let G be torsionfree. Let p ∈ C_{r}^{∗}(G) be an idempotent, i.e. p^{2} = p. Then p = 0, 1.

3. Status of the Baum-Connes Conjecture

There is a stronger version of the Baum-Connes conjecture, where certain coefficients are allowed:

Conjecture 5 (Baum-Connes Conjecture with Coefficients)

For every separable C^{∗}-algebra A with an action of a countable group G and every n ∈ ℤ the assembly map

K_{n}(EG; A) → K_{n}(A ⋊ G)

is an isomorphism.

Definition 6 (a-T-menable group) A group G is a-T-menable, or, equivalently, has the Haagerup property if G admits a metrically proper isometric action on some affine Hilbert space.

Metrically proper means that for any bounded subset B the set {g ∈ G | gB ∩ B ≠ ∅} is finite. Amenable groups and free groups are a-T-menable.

In the following table we list prominent classes of groups and state whether they are known to satisfy the Baum-Connes Conjecture (with coefficients). The proofs are due to Connes, Guentner, Kasparov, Higson, Lafforgue, Weinberger, Yu and others.

type of group

Status

a-T-menable groups true with coef-

ficients

discrete subgroups of Lie groups with finitely many path components injectivity true type of group

Status

linear groups injectivity is true arithmetic groups injectivity is true Groups with finite BG and finite asymptotic dimension

injectivity is true

G acts properly and isometrically on a complete Riemannian manifold M with non-positive sectional curvature rational injectivity is true

π_{1}(M) for a closed Riemannian manifold M with negative sectional curvature true for all subgroups

word hyperbolic groups true for all subgroups (true with coefficients by an unpublished result of Lafforgue) type of group

Status

one-relator groups true with coeffi-

cients

Haken 3-manifold groups (in particular knot groups) true with coefficients

SL(n, ℤ), n ≥ 3

injectivity is true

Artin’s braid group B_{n} true with coeffi-

cients

Thompson’s group F true with coefficients

Remark 7 A group which both is a-T-menable and has Kazhdan’s property T is finite. Meanwhile Lafforgue has shown that there are groups which have Kazhdan’s property T and do satisfy the Baum-Connes Conjecture.

Remark 8 There are counterexamples to the Baum-Connes Conjecture with coefficients. No counterexamples to the Baum-Connes Conjecture are known. Groups with expanders are the only candidates which may yield counterexamples.

4. The Trace Conjecture

Conjecture 9 (Trace Conjecture for G) The image of the composite

K_{0}(C_{r}^{∗}(G)) → K_{0}(N (G)) ⟶^{tr_{N (G)}} ℝ

is the additive subgroup of ℝ generated by \{\frac{1}{|H|} | H ⊂ G, |H| < ∞\}. Here N (G) is the group von Neumann algebra and tr_{N (G)} the von Neumann trace.

Notice that ℂ[G] ⊂ C_{r}^{∗}(G) ⊂ N (G) and equality holds if and only if G is finite.

Lemma 10 The Trace Conjecture for G implies the Kadison Conjecture for torsionfree G.

Proof:

0 ≤ p ≤ 1 ⇒ 0 = tr(0) ≤ tr(p) ≤ tr(1) = 1

⇒ tr(p) ∈ ℤ ∩ [0, 1] ⇒ tr(p) = 0, 1 ⇒ tr(p) = tr(0), tr(1) ⇒ p = 0, 1.

Lemma 11 Let G be torsionfree. Then the Baum-Connes Conjecture for G implies the Trace Conjecture for G.

Proof: The following diagram commutes

[figure: K_{0}^{G}(EG); ≅; K_{0}(C_{∗}^{r}(G)); K_{0}(N (G)); ℝ; K_{0}(BG); K_{0}(∗); ≅; ℤ]

This follows from the Atiyah index theorem. Namely, the upper horizontal composite sends [M, P^{∗}] ∈ K_{0}^{G}(EG) to the L^{2}- index in the sense of Atiyah

L^{2} − index (M, P^{∗}) ∈ ℝ,

the right vertical arrow sends [M, P^{∗}] to [G\M, G\P^{∗}] and the lower horizontal composite sends [G\M, G\P^{∗}] to the ordinary index

index (G\M, G\P^{∗}) ∈ ℤ.

The L^{2}-index theorem of Atiyah says

L^{2} − index (M, P^{∗}) = index (G\M, G\P^{∗}).

5. Roy’s counterexample to the trace conjecture

Theorem 12 (Roy 99) The Trace Conjecture is false in general.

Proof: Define an algebraic smooth variety

M = {[z_{0}, z_{1}, z_{2}, z_{3}] ∈ ℂℙ^{3} | z_{0}^{15} + z_{1}^{15} + z_{2}^{15} + z_{3}^{15} = 0}.

The group G = ℤ/3 × ℤ/3 acts on it by

[z_{0}, z_{1}, z_{2}, z_{3}] ↦ [exp (2πi/3) · z_{0}, z_{1}, z_{2}, z_{3}]
[z_{0}, z_{1}, z_{2}, z_{3}] ↦ [z_{0}, z_{3}, z_{1}, z_{2}]

One obtains

M^{G} = ∅;
sign (M) = −1105;
π_{1}(M) = {1}.

An equivariant version of a construction due to Davis and Januszkiewicz yields

• A closed oriented aspherical manifold N with G-action; • A G-map f : N → M of degree one;

• An isomorphism f^{∗}T M ≅ T N.

There is an extension of groups

1 → π = π_{1}(N) → Γ ⟶^{≅} G → 1

and a Γ-action on Ñ extending the π-action on Ñ and covering the G-action on N. We compute using the Hirzebruch signature formula

sign (N) = ⟨L(N), [N]⟩ = ⟨f^{∗}L(M), [N]⟩ = ⟨L(M), f_{∗}([N])⟩ = ⟨L(M), [M])⟩ = sign (M).

Next we prove that any finite subgroup H ⊂ Γ satisfies

|H| ∈ {1, 3}.

Since Ñ turns out to be a CAT(0)-space, any finite subgroup H ⊂ Γ has a fixed point by a result of Bruhat and Tits. This implies

Ñ^{H} ≠ ∅ ⇒ N^{p(H)} ≠ ∅ ⇒ M^{p(H)} ≠ ∅ ⇒ p(H) ≠ G.

Since π_{1}(N) is torsionfree, p|_{H} : H → p(H) is bijective.

On Ñ we have the signature operator S̃. We claim that the composite

K_{0}^{Γ}(EΓ) ⟶^{asmb} K_{0}(C_{r}^{∗}(Γ)) → K_{0}(N (Γ)) ⟶^{tr_{N (Γ)}} ℝ

sends [Ñ, S̃] to

\frac{1}{[Γ : π]} · sign (N) = \frac{−1105}{9} .

The Trace Conjecture for Γ says

\frac{−1105}{9} ∈ \{r ∈ ℝ | 3 · r ∈ ℤ\}.

This is not true (by some very deep number theoretic considerations).

6. The Modified Trace Conjecture

Conjecture 13 Modified Trace Conjecture) Let Λ^{G} ⊂ ℚ be the subring of ℚ obtained from ℤ by inverting the orders of finite subgroups of G. Then the image of composite

K_{0}(C_{r}^{∗}(G)) → K_{0}(N (G)) ⟶^{tr_{N (G)}} ℝ

is contained in Λ^{G}.

Theorem 14 (The Baum-Connes Conjecture implies the Modified Trace Conjecture (L.)) The image of the composite

K_{0}^{G}(EG) ⟶^{asmb} K_{0}(C_{r}^{∗}(G))

→ K_{0}(N (G)) ⟶^{tr_{N (G)}} ℝ

is contained in Λ^{G}.

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

7. The equivariant Chern character

Theorem 15 (Artin’s Theorem) Let G be finite. Then the map

_{C⊂G}^{⨁} ind _{C}^{G} : _{C⊂G}^{⨁} Rep _{ℂ}(C) → Rep _{ℂ}(G)

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

Let C be a finite cyclic group. The Artin defect is the cokernel of the map

_{D⊂C,D≠C}^{⨁} ind _{D}^{C} : _{D⊂C,D≠C}^{⨁} Rep _{ℂ}(D) → Rep _{ℂ}(C).

For an appropriate idempotent

θ_{C} ∈ Rep _{ℚ}(C) ⊗_{ℤ} ℤ [\frac{1}{|C|}]

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

θ_{c} · Rep _{ℂ}(C) ⊗_{ℤ} ℤ [\frac{1}{|C|}] .

Theorem 16 (Equivariant Chern character (L.)) Let X be a proper G-CW - complex. For a finite cyclic subgroup C ⊂ G let (C) be its conjugacy class, N_{G}C ist normalizer, C_{G}C its centralizer and W_{G}C = N_{G}C/C_{G}C. Then there is a natural isomorphism called equivariant Chern character ⨁_{(C)} K_{p}(C_{G}C\X^{C}) ⊗_{ℤ[W_{G}C]} θ_{c} · Rep _{ℂ}(C) ⊗_{ℤ} Λ^{G}

ch^{G}↓≅
K_{p}^{G}(X) ⊗_{ℤ} Λ^{G}

Example 17 Suppose that G is torsionfree. Then the trivial subgroup {1} is the only finite cyclic subgroup of C. We have C_{G}{1} = N_{G}{1} = G and W_{G}{1} = {1}. We get an isomorphism

⨁_{(C)} K_{p}(C_{G}C\X^{C}) ⊗_{ℤ[W_{G}C]} θ_{c} · Rep _{ℂ}(C) ⊗_{ℤ} Λ^{G}

↓≅
K_{0}(G\X) ⊗_{ℤ} ℤ
↓≅
K_{p}(G\X)

Under this identification the inverse of ch^{G} becomes the canonical isomorphism

K_{p}^{G}(X) ⟶^{≅} K_{p}(G\X).

Example 18 Let G be finite and X = {∗}. Then we get an improvement of Artin’s theorem, namely, the equivariant Chern character induces an isomorphism

⨁_{(C)} ℤ ⊗_{ℤ[W_{G}C]} θ_{c} · Rep _{ℂ}(C) ⊗_{ℤ} ℤ [\frac{1}{|C|}]

ch^{G}↓≅
Rep _{ℂ}(G) ⊗_{ℤ} ℤ [\frac{1}{|C|}]

Example 19 Take G to be any (discrete) group and X = EG. There is a natural isomorphism

K_{p}(BC_{G}C)⊗_{ℤ}Λ^{G} ⟶^{≅} K_{p}(C_{G}C\(EG)^{C})⊗_{ℤ}Λ^{G}.

The equivariant Chern character induces an isomorphism

⨁_{(C)} K_{p}(BC_{G}C) ⊗_{ℤ[W_{G}C]} θ_{c} · Rep _{ℂ}(C) ⊗_{ℤ} Λ^{G}

ch^{G}↓≅
K_{0}^{G}(EG) ⊗_{ℤ} Λ^{G}

Corollary 20 The ordinary Chern character induces for a CW -complex Y an isomorphism

⊕_{k}H_{2k+p} (Y) ⊗_{ℤ} ℚ ⟶^{≅} K_{p}(Y) ⊗_{ℤ} ℚ

If the Baum-Connes Conjecture holds for G, then we obtain an isomorphism ⨁_{(C)} ⨁_{k} H_{p+2k} (BC_{G}C) ⊗_{ℤ[W_{G}C]} θ_{c} · Rep _{ℂ}(C) ⊗_{ℤ} ℚ

ch^{G}↓≅
K_{p}^{G}(C_{r}^{∗}(G)) ⊗_{ℤ} ℚ

8. A generalization of Atiyah’s L^{2}-index theorem

Theorem 21 (Generalized L^{2}-index theorem (L 01)) The following diagram commutes

\xymatrix{K_{0}^{G}(EG) \ar[r] \ar[d]^{≅} & K_{0}^{G}(EG) \ar[r] & K_{0}(C_{∗}^{r}(G)) \ar[r] & K_{0}(N (G) \\ K_{0}(BG) \ar[r] & K_{0}(∗) \ar[rr]^{≅} &  & K_{0}(N (1)) \ar[u]}

or, equivalently, we get for a free cocompact G-manifold M with elliptic G- complex P^{∗} of differential operators of order 1 in K_{0}(N (G))

index _{N (G)}(M, P^{∗}) = index (G\M, G\P^{∗})·[N (G)].

Example 22 Let M be a closed oriented 4k-dimensional manifold. Suppose that the finite group G acts on M freely and orientation preserving. Define the equivariant signature

sign ^{G}(M) ∈ Rep _{ℂ}(G)

by

sign ^{G}(M) = [H_{2k}(M; ℂ)^{+} ] −[H_{2k}(M; ℂ)^{−}] .

Then the theorem above implies the wellknown statement that for a free G-action we get

sign ^{G}(M) = sign (G\M) · [ℂG];
sign (M) = |G| · sign (G\M).

9. Proof of Theorem 14

Let X be a proper G-CW -complex. Define two homomorphisms

⨁_{(C)} K_{0}(C_{G}C\X^{C}) ⊗_{ℤ[W_{G}C]} θ_{c} · Rep _{ℂ}(C) ⊗_{ℤ} Λ^{G}

ξ_{i}↓
K_{0}(N (G)) ⊗_{ℤ} Λ^{G}

as follows. The first one is the composition of the equivariant Chern character with the assembly map

asmb ^{G} ⊗ id : K_{0}^{G}(X)⊗_{ℤ}Λ^{G} → K_{0}(C_{r}^{∗}(G))⊗_{ℤ}Λ^{G}

and the change of rings homomorphism

K_{0}(C_{r}^{∗}(G)) ⊗_{ℤ} Λ^{G} → K_{0}(N (G)) ⊗_{ℤ} Λ^{G}.

This is the homomorphism which we want to understand. In particular we are interested in its image. We want to identify it with the easier to compute homomorphism ξ_{2}.

The homomorphism ξ_{2} is induced by the composition

⨁_{(C)} K_{0}(C_{G}C\X^{C}) ⊗_{ℤ} θ_{c} · Rep _{ℂ}(C) ⊗_{ℤ} Λ^{G}

⨁_{(C)} K_{0}(pr)⊗_{ℤ}incl ↓
⨁_{(C)} K_{0}(∗) ⊗_{ℤ} Rep _{ℂ}(C) ⊗_{ℤ} Λ^{G}
≅↓
⨁_{(C)} ℤ ⊗_{ℤ} Rep _{ℂ}(C) ⊗_{ℤ} Λ^{G}
≅↓
⨁_{(C)} Rep _{ℂ}(C) ⊗_{ℤ} Λ^{G}
⨁_{(C)} ind _{C}^{G}↓
K_{0}(N (G)) ⊗_{ℤ} Λ^{G}

The proof of the next result uses the generalized L^{2}-Atiyah index theorem.

Theorem 23 Let X be a proper G-CW - complex. Then the maps ξ_{1} and ξ_{2} agree.

Theorem 24 The image of the composite

K_{0}(EG) ⊗_{ℤ} Λ^{G} → K_{0}(C_{r}^{∗}(G)) ⊗_{ℤ} Λ^{G} → K_{0}(N (G)) ⊗_{ℤ} Λ^{G}

is contained in the image of

_{(C)}^{⨁} ind _{C}^{G} : _{(C)}^{⨁} Rep _{ℂ}(C)⊗_{ℤ}Λ^{G} → K_{0}(N (G)⊗_{ℤ}Λ^{G}.

Remark 25 If we compose the second map above with

tr_{N (G)} : K_{0}(N (G)) ⊗_{ℤ} Λ^{G} → ℝ

it is easy to see that its image is contained in Λ^{G}. Hence the following composition has Λ^{G} as image

K_{0}^{G}(EG) ⟶^{asmb} K_{0}(C_{r}^{∗}(G))

→ K_{0}(N (G)) ⟶^{tr_{N (G)}} ℝ.