The relation between the

Baum-Connes Conjecture

and the Trace Conjecture

Wolfgang Lück^{∗}

Fachbereich Mathematik und Informatik

Westfälische Wilhelms-Universität

Münster Einsteinstr. 62 48149 Münster

Germany

June 7, 2003 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.

• C_{r}^{∗}(G) is the reduced C^{∗}-algebra of G;

• 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);

• 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; • 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;

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

• 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));

• 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 of dimension ≥ 5. Let B be the Bott manifold. Then 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; Conjecture 2 (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.

Conjecture 3 (Kadison Conjecture for G) Let G be torsionfree. Let p ∈ C_{r}^{∗}(G) be an idempotent, i.e. p^{2} = p. Then p = 0, 1.

Lemma 4 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 5 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^{∗}).

Theorem 6 (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 ⟶^{p} 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).

Conjecture 7 (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 8 (L 01) 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.

Theorem 9 (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 10 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).

Theorem 11 (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 |G|, 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 12 (L 01) Let X be a proper G-CW -complex. For a finite cyclic subgroup C ⊂ G let (C) be its conjugacy class, N_{G}C its 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 13 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_{p}(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 14 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 15 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_{p}^{G}(EG) ⊗_{ℤ} Λ^{G}

Corollary 16 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

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

Theorem 18 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 19 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.