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

October 17, 2001

Abstract

We prove a version of the L^{2}-index Theorem of Atiyah, which uses the universal center-valued trace instead of the standard trace. We construct for G-equivariant K-homology an equivariant Chern character, which is an isomorphism and lives over the ring ℤ ⊂ Λ^{G} ⊂ ℚ obtained from the integers by inverting the orders of all finite subgroups of G. We use these two results to show that the Baum-Connes Conjecture implies the modified Trace Conjecture, which says that the image of the standard trace K_{0}(C_{r}^{∗}(G)) → ℝ takes values in Λ^{G}. The original Trace Conjecture predicted that its image lies in the additive subgroup of ℝ generated by the inverses of all the orders of the finite subgroups of G, and has been disproved by Roy [15]. Key words: Baum-Connes Conjecture, Trace Conjecture, equivariant Chern character, L^{2}-index theorem. Mathematics subject classification 2000: 19L47, 19K56, 55N91

0. Introduction and statements of results

Throughout this paper let G be a discrete group. The Baum-Connes Conjecture for G says that the assembly map

asmb^{G} : K_{0}^{G}(\underline{E}G) → K_{0}(C_{r}^{∗}(G))

from the equivariant K-homology of the classifying space for proper G-actions \underline{E}G to the topological K-theory of the reduced C^{∗}-algebra C_{r}^{∗}(G) is bijective [3, page 8], [5, Conjecture 3.1]. In connection with this conjecture Baum and Connes [3, page 21] also made the sometimes so called Trace Conjecture. It says that the image of the composition

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

∗email: lueck@math.uni-muenster.de, www: http://www.math.uni-muenster.de/u/lueck, FAX: 49 251 8338370

is the additive subgroup of ℚ generated by all numbers \frac{1}{|H|}, where H ⊂ G runs though all finite subgroups of G. Here N (G) is the group von Neumann algebra, i the change of rings homomorphism associated to the canonical inclusion C_{r}^{∗}(G) → N (G) and tr_{N (G)} is the map induced by the standard von Neumann trace tr_{N (G)} : N (G) → ℂ. Roy has construced a counterexample to the Trace Conjecture in this form in [15] based on her article [16]. She constructs a group Γ, whose finite subgroups are all of order 1 or 3, together with an element in K_{0}^{G}(\underline{E}G), whose image under tr_{N (Γ)} ∘i ∘ asmb is −\frac{1105}{9}. The point is that 3 · \frac{1105}{9} is not an integer. Notice that Roy’s example does not imply that the Baum-Connes Conjecture does not hold for Γ. Since the group Γ contains a torsionfree subgroup of index 9 and the Trace Conjecture for torsionfree groups does follow from the Baum-Connes Conjecture, the Baum-Connes Conjecture predicts that the image of tr_{N (Γ)} ∘i : K_{0}(C_{r}^{∗}(Γ)) → ℝ is contained in {r ∈ ℝ | 9 · r ∈ ℤ}. So one could hope that the following version of the Trace Conjecture is still true. Denote by

Λ^{G} := ℤ [\frac{1}{|Fin(G)|}]  (0.1)

the ring ℤ ⊂ Λ^{G} ⊂ ℚ obtained from ℤ by inverting all the orders |H| of finite subgroups of G. For Roy’s group Γ this is \{m · 3^{−n} | m, n ∈ ℤ, n ≥ 0\} and obviously contains −\frac{1105}{9}.

Conjecture 0.2 (Modified Trace Conjecture for a group G) The image of the composition

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

is contained in Λ^{G}.

The motivation for this paper is to prove

Theorem 0.3 The image of the composition

Λ^{G} ⊗_{ℤ} K_{0}^{G}(\underline{E}G) ⟶^{id ⊗ asmb^{G}} Λ^{G} ⊗_{ℤ} K_{0}(C_{r}^{∗}(G)) ⟶^{i} Λ^{G} ⊗_{ℤ} K_{0}(N (G)) ⟶^{tr_{N (G)}} ℝ

is Λ^{G}.

In particular the modified Trace Conjecture 0.2 holds for G, if the assembly map asmb^{G} : K_{0}^{G}(\underline{E}G) → K_{0}(C_{r}^{∗}(G)) appearing in the Baum-Connes Conjecture is surjective.

In order to prove Theorem 0.3 (actually a generalization of it in Theorem 0.8), we will prove a slight generalization of Atiyah’s L^{2}-Index Theorem and construct an equivariant Chern character for equivariant K-homology of proper G-CW -complexes, which is bijective and defined after applying Λ^{G} ⊗_{ℤ} −.

Let M be a closed Riemannian manifold and D^{∗} = (D^{∗}, d^{∗}) be an elliptic complex of differential operators of order 1 on M. Denote by index(D^{∗}) ∈ ℤ its index. Let \overline{M} → M be a G-covering. Then one can lift D^{∗} to an elliptic G-equivariant complex \overline{D}^{∗}. Using the trace tr_{N (G)} : N (G) → ℂ Atiyah [1] defines its L^{2}-index index_{N (G)}(\overline{D}^{∗}) ∈ ℝ and shows

index(D^{∗}) = index_{N (G)}(\overline{D}^{∗}).

Let EG → BG be the universal G-covering. The L^{2}-index theorem of Atiyah implies that the composition

K_{0}^{G}(EG) ⟶^{asmb^{G}} K_{0}(C_{r}^{∗}(G)) ⟶^{i} K_{0}(N (G)) ⟶^{tr_{N (G)}} ℝ

agrees with the composition

K_{0}^{G}(EG) ⟶^{ind_{G→{1}}} K_{0}(BG) ⟶^{K_{0}(pr)} K_{0}(∗) ⟶^{asmb^{{1}}} K_{0}(C_{r}^{∗}({1})) ⟶^{dim_{ℂ}} ℤ ↪ ℝ.

Since for a torsionfree group G the spaces EG and \underline{E}G agree, the Baum-Connes Conjecture for

a torsionfree group G does imply that the image of K_{0}(C_{r}^{∗}(G)) ⟶^{i} K_{0}(N (G)) ⟶^{tr_{N (G)}} ℝ is ℤ [3, Corollary 1 on page 21]. Instead of using the the standard von Neumann trace tr_{N (G)} : N (G) → ℂ, one can use the universal center-valued trace tr_{N (G)}^{u} : N (G) → Z(N (G)) to define an index

index_{N (G)}^{u}(\overline{D}^{∗}) ∈ Z(N (G)),

which takes values in the center Z(N (G)) of the group von Neumann algebra N (G). Thus we get additional information, namely, for any element g ∈ G, whose conjugacy class (g) is finite, we get a complex number. However, it turns out that the value at classes (g) with g ≠ 1 is zero and that the value at (1) is the index of D^{∗}. Namely, we will show in Section 1

Theorem 0.4 Under the conditions above we get in Z(N (G))

index_{N (G)}^{u}(\overline{D}^{∗}) = index(D^{∗}) · 1_{N (G)}.

As an illustration we discuss the special case, where G is finite, M is an oriented closed 4kdimensional manifold with free orientation preserving G-action and D^{∗} is the signature operator. Then Theorem 0.4 reduces to the well-known statement that the equivariant signature

sign^{G}(M) := [H_{2k}(M)^{+}] − [H_{2k}(M)^{−}] ∈ Rep_{ℂ}(G)

is equal to sign(G\M) · [ℂG] for sign(G\M) ∈ ℤ the (ordinary) signature of G\M. We mention that this implies sign(M) = |G| · sign(G\M). Theorem 0.4 is a special case of Theorem 5.4 but we will need it in the proof of Theorem 5.4 and therefore will have to prove it first.

The second ingredient is a variation of the equivariant Chern character of [13] for equivariant Khomology of proper G-CW -complexes. Recall that proper means that all isotropy groups are finite. The construction in [13] works for equivariant homology theories with a Mackey structure on the coefficient system in general, but requires to invert all primes. The construction we will give here works after applying Λ^{G}⊗_{ℤ}? and has a different source.

Denote for a proper G-CW -complex X by F(X) the set of all subgroups H ⊂ G, for which X^{H} ≠ ∅, and by

Λ^{G}(X) := ℤ [\frac{1}{F(X)}]  (0.5)

the ring ℤ ⊂ Λ^{G}(X) ⊂ Λ^{G} obtained from ℤ by inverting the orders of all subgroups H ∈ F(X). Denote by

J^{G} resp. J^{G}(X)  (0.6)

the set of conjugacy classes (C) of finite cyclic subgroups C ⊂ G resp. the subset J^{G}(X) ⊂ J^{G} of conjugacy classes (C) of finite cyclic subgroups C ⊂ G, for which X^{C} is non-empty. Obviously Λ^{G} = Λ^{G}(\underline{E}G) and J^{G} = J^{G}(\underline{E}G) since \underline{E}G is characterized up to G-homotopy by the property that \underline{E}G^{H} is contractible (and hence non-empty) for finite H ⊂ G and empty for infinite H ⊂ G. Let C ⊂ G be a finite cyclic subgroup. Let C_{G}C be the centralizer and N_{G}C be the normalizer of C ⊂ G.

Let W_{G}C be the quotient N_{G}C/C_{G}C. We will construct an idempotent θ_{C} ∈ Λ^{C} ⊗_{ℤ} Rep_{ℚ}(C) which acts on Λ^{C} ⊗_{ℤ} Rep_{ℂ}(C). We will see in Lemma 3.4 (b) that the cokernel of

⊕_{D⊂C,D≠C} ind_{D}^{C} : ⊕_{D⊂C,D≠C} ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℂ}(D) → ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℂ}(C)

is isomorphic to the image of the idempotent endomorphism

θ_{C} : ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℂ}(C) → ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℂ}(C).

After introducing and proving some preliminary results about modules over a category and representation theory of finite groups in Sections 2 and 3, we will prove in Section 4

Theorem 0.7 Let X be a proper G-CW -complex. Put Λ = Λ^{G}(X) and J = J^{G}(X). Then there is for p = 0, 1 a natural isomorphism called equivariant Chern character

ch_{p}^{G}(X) : ⊕_{(C)∈J} Λ ⊗_{ℤ} K_{p}(C_{G}C\X^{C}) ⊗_{Λ[W_{G}C]} im (θ_{C} : Λ ⊗_{ℤ} Rep_{ℂ}(C) → Λ ⊗_{ℤ} Rep_{ℂ}(C))

⟶^{≅} Λ ⊗_{ℤ} K_{p}^{G}(X).

Notice that the equivariant Chern character of Theorem 0.7 reduces to the obvious isomorphism K_{0}(G\X) ⊗_{ℤ} Rep_{ℂ}({1}) ⟶^{≅} K_{0}^{G}(X), if G acts freely on X. In the special case, where G is finite, X is the one-point-space {∗} and p = 0, the equivariant Chern character reduces to an isomorphism

⊕_{(C)∈J^{G}} ℤ [\frac{1}{|G|}] ⊗_{ℤ[\frac{1}{|G|}][W_{G}C]} im (θ_{C} : ℤ [\frac{1}{|G|}] ⊗_{ℤ} Rep_{ℂ}(C) → ℤ [\frac{1}{|G|}] ⊗_{ℤ} Rep_{ℂ}(C))

⟶^{≅} ℤ [\frac{1}{|G|}] ⊗_{ℤ} Rep_{ℂ}(G).

This is a strong version of the well-known theorem of Artin that the map induced by induction

⊕_{(C)∈J^{G}} ℚ ⊗_{ℤ} Rep_{ℂ}(C) → ℚ ⊗_{ℤ} Rep_{ℂ}(G)

is surjective for any finite group G. Artin’s theorem is the reason why it does suffice rationally to consider all finite cyclic subgroups of G instead of all finite subgroups in Theorem 0.7. One might expect (and has to do integrally) in view of the Baum-Connes Conjecture and the fact that \underline{E}G involves all finite subgroups that one has to take all finite subgroups into account.

Theorem 0.7 gives a computation of Λ^{G} ⊗ K_{0}^{G}(\underline{E}G), namely

⊕_{(C)∈J^{G}} Λ^{G} ⊗_{ℤ} K_{p}(B(C_{G}C)) ⊗_{Λ^{G}[W_{G}C]} im (θ_{C} : Λ^{G} ⊗_{ℤ} Rep_{ℂ}(C) → Λ^{G} ⊗_{ℤ} Rep_{ℂ}(C))

⟶^{≅} Λ^{G} ⊗_{ℤ} K_{p}^{G}(\underline{E}G).

Another construction of an equivariant Chern character using completely different methods can be found in [4]. However, it works only after applying ℂ ⊗_{ℤ} − and therefore cannot be used for our purposes here.

In Theorem 5.4 we will identify the composition of the Chern character of Theorem 0.7 with the map

Λ^{G} ⊗_{ℤ} K_{0}^{G}(\underline{E}G) ⟶^{id ⊗ asmb^{G}} Λ^{G} ⊗_{ℤ} K_{0}(C_{r}^{∗}(G)) ⟶^{id ⊗i} Λ^{G} ⊗_{ℤ} K_{0}(N (G))

with an easier to understand and to calculate homomorphism, whose image is obvious from its definition. This will immediately imply Theorem 0.8 Let Λ^{G} resp. J^{G} be the ring resp. set introduced in (0.1) resp. (0.6). Then the image of the composition

Λ^{G} ⊗_{ℤ} K_{0}^{G}(\underline{E}G) ⟶^{id ⊗_{ℤ} asmb^{G}} Λ^{G} ⊗_{ℤ} K_{0}(C_{r}^{∗}(G)) ⟶^{id ⊗_{ℤ}i} Λ^{G} ⊗_{ℤ} K_{0}(N (G))

is the image of the map given by induction

⊕_{(C)∈J^{G}} id ⊗ ind_{C}^{G} : ⊕_{(C)∈J^{G}} Λ^{G} ⊗_{ℤ} Rep_{ℂ}(C) → Λ^{G} ⊗_{ℤ} K_{0}(N (G)).

Now Theorem 0.3 follows from Theorem 0.8.

The change of rings and K-theory map l : K_{0}(ℂG) → K_{0}(C_{r}^{∗}(G)) from the algebraic K_{0}-group of the complex group ring ℂG to the topological K_{0}-group of C_{r}^{∗}(G) is in general far from being surjective. There is some evidence that it is injective after applying Λ⊗_{ℤ}? (see [13, Theorem 0.1]). Theorem 0.8 gives some evidence for the conjecture that the image of Λ^{G} ⊗_{ℤ} K_{0}(C_{r}^{∗}(G)) ⟶^{id ⊗_{ℤ}i} Λ^{G} ⊗_{ℤ} K_{0}(N (G)) agrees with the image of the composition Λ^{G}⊗_{ℤ}K_{0}(ℂG) ⟶^{l} Λ^{G}⊗_{ℤ}K_{0}(C_{r}^{∗}(G)) ⟶^{id ⊗_{ℤ}i} Λ^{G}⊗_{ℤ}K_{0}(N (G)).

Alain Valette pointed out to the author that the Modified Trace Conjecture 0.2 implies the following conjecture of Farkas [8, page 593]

Conjecture 0.9 (Farkas) If the rational number m/n is in the image of the composition

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

and the prime p divides n but not m, then G has an element of order p.

Notice that the Modified Trace Conjecture 0.2 implies that the image of the composition

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

is contained in ℚ which is not known to be true in general. Some evidence for this claim comes from the theorem of Zalesskii that the image of the composition

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

is contained in ℚ. For its proof see [6, Section 3], [19].

The paper is organized as follows

1. The L^{2}-index theorem
2. Modules over a category
3. Some representation theory for finite groups
4. The construction of the Chern character
5. The Baum-Connes Conjecture and the Trace Conjecture

References

The author wants to thank the Max-Planck-Institute for Mathematics in Bonn for the hospitality during his stay in January and February 2001, when parts of the paper were written. Moreover, he wants to thank the referee for its very detailed and helpful report.

1. The L^{2}-index theorem

In this section we prove a slight generalization of the L^{2}-index theorem of Atiyah [1]. Let \overline{M} be a Riemannian manifold (without boundary) together with a cocompact free proper action of G by isometries. In other words, M = G\\overline{M} is a closed Riemannian manifold, the projection p : \overline{M} → M is a G-covering and \overline{M} is equipped with the Riemannian metric induced by the one of M. Let D^{∗} = (D^{∗}, d^{∗}) be an elliptic complex of differential operators d^{p} : D^{p} → D^{p+1} of order 1 acting on the space of sections D^{p} = C^{∞}(E^{p}) of vector bundles E^{p} → M. Define \overline{E}^{p} by p^{∗}E^{p} and \overline{D}^{p} by L^{2}C^{∞}(\overline{E}_{p}). Then G acts on \overline{E}^{p} and \overline{D}^{p}. Since differential operators are local operators, there is a unique lift of each operator d^{p} to a G-equivariant differential operator d̂^{p} : C^{∞}(\overline{E}^{p}) → C^{∞}(\overline{E}^{p+1}). We obtain an elliptic G-complex (C^{∞}(\overline{E}^{∗}), d̂^{∗}). Let \overline{d}^{p} : \overline{D}^{p} → \overline{D}^{p+1} be the minimal closure of d̂^{p} which is the same as its maximal closure [1, Proposition 3.1].

Since D^{∗} is elliptic, each cohomology module H^{p}(D^{∗}) := ker(d^{p})/ im(d^{p−1}) is a finitely generated ℂ-module. Hence we can define the index of the elliptic complex D^{∗} by

index(D^{∗}) := ∑_{p≥0}(−1)^{p} · dim_{ℂ}(H^{p}(D^{∗})) ∈ ℤ.  (1.1)

Next we want to define an analogous invariant for the lifted complex \overline{D}^{∗}. The group von Neumann algebra N (G) of G is the ∗-algebra B(l^{2}(G))^{G} of all bounded G-equivariant operators l^{2}(G) → l^{2}(G), where we equip l^{2}(G) with the obvious left G-action. Let

tr_{N (G)} : N (G) → ℂ  (1.2)

be the standard von Neumann trace, which sends f ∈ N (G) = B(l^{2}(G))^{G} to ⟨f(e), e⟩_{l^{2}(G)}, where e denotes the element in l^{2}(G) given by the unit element in G ⊂ l^{2}(G). Denote by Z(N (G)) the center of N (G). There is the universal center-valued trace [9, Theorem 7.1.12 on page 462, Proposition 7.4.5 on page 483, Theorem 8.2.8 on page 517, Proposition 8.3.10 on page 525, Theorem 8.4.3 on page 532]

tr_{N (G)}^{u} : N (G) → Z(N (G))  (1.3)

which is uniquely determined by the following two properties:

(a) tr^{u} is a trace with values in the center, i.e. tr^{u} is ℂ-linear, for a ∈ N (G) with a ≥ 0 we have tr^{u}(a) ≥ 0 and tr^{u}(ab) = tr^{u}(ba) for all a, b ∈ N (G);

(b) tr^{u}(a) = a for all a ∈ Z(N (G)).

The map tr^{u} has the following further properties:

(c) tr^{u} is faithful;

(d) tr^{u} is normal. Equivalently, tr^{u} is continuous with respect to the ultraweak topology on N (G);

(e) || tr^{u}(a)|| ≤ ||a|| for a ∈ N (G);

(f) tr^{u}(ab) = a tr^{u}(b) for all a ∈ Z(N (G)) and b ∈ N (G);

(g) Let p and q be projections in N (G). Then p and q are equivalent, i.e. p = vv^{∗} and q = v^{∗}v, if and only if tr^{u}(p) = tr^{u}(q); (h) Any linear functional f : N (G) → ℂ, which is continuous with respect to the norm topology on

N (G) and which is central, i.e. f(ab) = f(ba) for all a, b ∈ N (G), factorizes as

N (G) ⟶^{tr^{u}} Z(N (G)) ⟶^{f|_{Z(N (G))}} ℂ.

In particular tr_{N (G)} ∘ tr_{N (G)}^{u} = tr_{N (G)}.

A Hilbert N (G)-module V is a Hilbert space V together with a G-action by isometries such that there exists a Hilbert space H and a G-equivariant projection p : H ⊗ l^{2}(G) → H ⊗ l^{2}(G) with the property that V and im(p) are isometrically G-linearly isomorphic. Here H ⊗ l^{2}(G) is the tensor product of Hilbert spaces and G acts trivially on H and on l^{2}(G) by the obvious left multiplication. Notice that p is not part of the structure, only its existence is required. We call V finitely generated if H can be chosen to be finite-dimensional.

Our main examples of Hilbert N (G)-modules are the Hilbert spaces \overline{D}^{p} which are isometrically G-isomorphic to L^{2}(C^{∞}(E^{p})) ⊗ l^{2}(G). This can be seen using a fundamental domain F for the Gaction on \overline{M} which is from a measure theory point of view the same as M. A morphism f : V → W of Hilbert N (G)-modules is a densely defined closed G-equivariant operator. The differentials \overline{d}^{p} are morphisms of Hilbert N (G)-modules.

Let f : V → V be a morphism of Hilbert N (G)-modules which is positive. Choose a G-projection p : H ⊗ l^{2}(G) → H ⊗ l^{2}(G) and an isometric invertible G-equivariant operator u : im(p) → V. Let \{b_{i} | i ∈ I\} be a Hilbert basis for H. Let \overline{f} be the composition

H ⊗ l^{2}(G) ⟶^{p} im(p) ⟶^{u} V ⟶^{f} V ⟶^{u^{−1}} im(p) ↪ H ⊗ l^{2}(G).

Define the von Neumann trace of f : V → V by

tr_{N (G)}(f) := ∑_{i∈I}⟨\overline{f}(b_{i} ⊗ e), b_{i} ⊗ e⟩_{H⊗l^{2}(G)}  ∈ [0, ∞].  (1.4)

This is indeed independent of the choice of p, u and the Hilbert basis {b_{i} | i ∈ I}. If V is finitely generated and f is bounded, then tr_{N (G)}(f) < ∞ is always true. Define the von Neumann dimension of a Hilbert N (G)-module V by

dim_{N (G)}(V) := tr_{N (G)}(id : V → V)  ∈ [0, ∞].  (1.5)

If V is a finitely generated Hilbert N (G)-module, we define the universal center-valued von Neumann dimension

dim_{N (G)}^{u}(V) := tr_{N (G)}^{u}(id : V → V) ∈ Z(N (G))  (1.6)

analogously to dim_{N (G)}(V) replacing tr_{N (G)} by tr_{N (G)}^{u}. Given a finitely generated Hilbert N (G)- module V, we have tr_{N (G)}(dim_{N (G)}^{u}(V)) = dim_{N (G)}(V).

Define the L^{2}-cohomology H_{(2)}^{p}(\overline{D}^{∗}) to be ker(\overline{d}^{p})/ clos(im(\overline{d}^{p−1})), where clos(im(\overline{d}^{p−1})) is the

closure of the image of \overline{d}^{p−1}. Define the p-th Laplacian by \overline{Δ}_{p} = (\overline{d}^{p})^{∗}\overline{d}^{p} + \overline{d}^{p−1}(\overline{d}^{p−1})^{∗}. By the L^{2}- Hodge-deRham Theorem we get a G-equivariant isometric isomorphism ker(\overline{Δ}_{p}) ⟶^{≅} H_{(2)}^{p}(\overline{D}^{∗}). Thus

H_{(2)}^{p}(\overline{D}^{∗}) inherits the structure of a Hilbert N (G)-module. Moreover, it turns out to be a finitely generated Hilbert N (G)-module. This can be deduced from the results of [14], where an index already over C_{r}^{∗}(G) is defined and the problem of getting finitely generated modules over C_{r}^{∗}(G) is treated. Namely, one can deduce from [14] after passing to the group von Neumann algebra, that there are finitely generated Hilbert N (G)-modules U_{1}, U_{2}, V_{1} and V_{2} and Hilbert N (G)-modules W_{1} and W_{2} together with a morphism v : V_{1} → V_{2} and isomorphisms of Hilbert N (G)-modules w : W_{1} ⟶^{≅} W_{2},

u_{1} : \overline{D}^{p} ⊕ U_{1} ⟶^{≅} V_{1} ⊕ W_{1} and u_{2} : \overline{D}^{p} ⊕ U_{2} ⟶^{≅} V_{2} ⊕ W_{2} such that u_{2} ∘ (Δ_{p} ⊕ 0) = (v ⊕ w) ∘ u_{1}.

Obviously the kernel of v and hence the kernel of \overline{Δ}_{p} are finitely generated Hilbert N (G)-modules.

Define the center-valued L^{2}-index and the L^{2}-index

index_{N (G)}^{u}(\overline{D}^{∗}) := ∑_{p≥0}(−1)^{p} · dim_{N (G)}^{u}(H_{(2)}^{p}(\overline{D}^{∗})) ∈ Z(N (G));  (1.7)
index_{N (G)}(\overline{D}^{∗}) := ∑_{p≥0}(−1)^{p} · dim_{N (G)}(H_{(2)}^{p}(\overline{D}^{∗})) ∈ ℝ.  (1.8)

The rest of this section is devoted to the proof of Theorem 0.4

Notation 1.9 Denote by con(G)_{cf} the set of conjugacy classes (g) of elements g ∈ G such that the set (g) is finite, or, equivalently, the centralizer C_{G}(g) = {g′ ∈ G | g′g = gg′} has finite index in G. For c ∈ con(G)_{cf} let N_{c} be the element ∑_{g∈c} g ∈ ℂG. In the sequel L_{c} resp. L_{g} denotes left multiplication with N_{c} resp. g for c ∈ con(G)_{cf} resp. g ∈ G.

Notice for the sequel that N_{c} ∈ Z(N (G)) and L_{c} is G-equivariant and commutes with all Goperators.

Lemma 1.10 Consider a ∈ Z(N (G)). Then we have a = 0 if and only if tr_{N (G)}(N_{c}a) = 0 holds for any c ∈ con_{cf}(G).

\underline{\underline{Proof}} : Consider a ∈ N (G) = B(l^{2}(G))^{G} which belongs to Z(N (G)). Write a(e) = ∑_{g∈G} λ_{g} · g ∈ l^{2}(G). Since aR_{g} = R_{g}a holds for g ∈ G and R_{g} : l^{2}(G) → l^{2}(G) given by right multiplication with g ∈ G, we get λ_{g} = λ_{hgh^{−}1} for g, h ∈ G. This implies that λ_{g} = 0 if the conjugacy class (g) is infinite. On easily checks for an element g with finite (g)

|(g)| · λ_{g} = tr_{N (G)}(N_{(g^{−1})}a).

Lemma 1.11 We get under the conditions above.

tr_{N (G)} (index_{N (G)}^{u}(\overline{D}^{∗})) = index(D^{∗}).

\underline{\underline{Proof}} : The L^{2}-index theorem of Atiyah [1, (1.1)] says

index_{N (G)}(\overline{D}^{∗}) = index(D^{∗}).

We have

tr_{N (G)} (index_{N (G)}^{u}(\overline{D}^{∗})) = tr_{N (G)} (∑_{p≥0}(−1)^{p} dim_{N (G)}^{u}(H_{(2)}^{p}(\overline{D}^{∗})))

= ∑_{p≥0}(−1)^{p} tr_{N (G)} (dim_{N (G)}^{u}(H_{(2)}^{p}(\overline{D}^{∗})))
= ∑_{p≥0}(−1)^{p} dim_{N (G)} (H_{(2)}^{p}(\overline{D}^{∗}))
= index_{N (G)}(\overline{D}^{∗}).

Next we want to prove

Lemma 1.12 Consider an element c ∈ con(G)_{cf} with c ≠ (1). Then

tr_{N (G)} (N_{c} · index_{N (G)}^{u}(\overline{D}^{∗})) = 0.

\underline{\underline{Proof}} : In the sequel we denote by \overline{pr}_{p} : \overline{D}^{p} → \overline{D}^{p} the projection onto the kernel of the p-th Laplacian

\overline{Δ}_{p} = (\overline{d}^{p})^{∗}\overline{d}^{p} + \overline{d}^{p−1}(\overline{d}^{p−1})^{∗}. By the L^{2}-Hodge-deRham Theorem we get a G-equivariant isometric isomorphism im(\overline{pr}_{p}) ⟶^{≅} H_{(2)}^{p}(\overline{D}^{∗}). This implies

tr_{N (G)} (N_{c} · index_{N (G)}^{u}(\overline{D}^{∗}))
= ∑_{p≥0}(−1)^{p} · tr_{N (G)} (N_{c} · tr_{N (G)}^{u} (id : H_{(2)}^{p}(\overline{D}^{∗}) → H_{(2)}^{p}(\overline{D}^{∗})))
= ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} : H_{(2)}^{p}(\overline{D}^{∗}) → H_{(2)}^{p}(\overline{D}^{∗}))
= ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ \overline{pr}_{p} : \overline{D}^{p} → \overline{D}^{p}) .  (1.13)

The operator e^{−t\overline{Δ}_{p}} : \overline{D}^{p} → \overline{D}^{p} is a bounded G-equivariant operator and has a smooth kernel e^{−t\overline{Δ}_{p}}(\overline{x}, \overline{y}) : \overline{E}p ; \overline{x} → \overline{E}p ; \overline{y} for \overline{x}, \overline{y} ∈ \overline{M}. Thus e^{−t\overline{Δ}_{p}}(ω) applied to a section ω is given at \overline{y} ∈ \overline{M} by ∫\overline{M} e^{−t\overline{Δ}_{p}}(\overline{x}, \overline{y})(ω(\overline{x}))dvol\overline{x}. The operator L_{c} ∘e^{−t\overline{Δ}_{p}} is also a bounded G-equivariant operator and has

a smooth kernel (L_{c} ∘ e^{−t\overline{Δ}_{p}}) (\overline{x}, \overline{y}) satisfying

(L_{c} ∘ e^{−t\overline{Δ}_{p}}) (\overline{x}, \overline{y}) = ∑_{g∈c} L_{g} ∘ e^{−t\overline{Δ}_{p}}(\overline{x}, g^{−1}\overline{y}).

If F is a fundamental domain for the G-action, then [1, Proposition 4.6].

tr_{N (G)}(L_{c} ∘ e^{−t\overline{Δ}_{p}}) = ∫_{F} tr_{ℂ} ((L_{c} ∘ e^{−t\overline{Δ}_{p}}) (\overline{x}, \overline{x})) dvol\overline{x};
= ∑_{g∈c} ∫_{F} tr_{ℂ} (L_{g} ∘ e^{−t\overline{Δ}_{p}}(\overline{x}, g^{−1}\overline{x})) dvol\overline{x} .  (1.14)

where tr_{ℂ} is the trace of an endomorphism of a finite-dimensional complex vector space. We have

lim_{t→0} sup \{||e^{−t\overline{Δ}_{p}}(\overline{x}, g^{−1}\overline{x})|| | \overline{x} ∈ F\} = 0,  (1.15)

where ||e^{−t\overline{Δ}_{p}}(\overline{x}, g^{−1}\overline{x})|| is the operator norm of the linear map e^{−t\overline{Δ}_{p}}(\overline{x}, g^{−1}\overline{x}) of finite-dimensional Hilbert spaces. This follows from the finite propagation speed method of [7]. There only the standard Laplacian on 0-forms is treated, but the proof presented there carries over to the Laplacian \overline{Δ}_{p} associated to the lift \overline{D}^{∗} to the G-covering \overline{M} of an elliptic complex D^{∗} of differential operators of order 1 on a closed Riemannian manifold M in any dimension p. The point is that \overline{M} has bounded

geometry, \overline{Δ}_{p} is essentially selfadjoint and positive so that \sqrt{\overline{Δ}_{p}} makes sense, and \frac{∂^{2}}{∂t^{2}} + \overline{Δ}_{p} is strictly hyperbolic. Now one applies the results of [7, Section 1] and uses the estimate in [11, page 475], where the special case of D^{∗} being the deRham complex is treated.

Since

|tr_{ℂ} (L_{g} ∘ e^{−t\overline{Δ}_{p}}(\overline{x}, g^{−1}\overline{x}))| ≤ dim_{ℂ}(E^{p}) · ||e^{−t\overline{Δ}_{p}}(\overline{x}, g^{−1}\overline{x})||

and F is relative compact, we conclude from (1.14) and (1.15)

lim_{t→0} tr_{N (G)}(L_{c} ∘ e^{−t\overline{Δ}_{p}}) = 0.  (1.16)

Since the trace tr_{N (G)} is ultraweakly continuous and lim_{t→∞} e^{−t\overline{Δ}_{p}} = \overline{pr}_{p} in the weak topology, we get

lim_{t→∞} tr_{N (G)}(L_{c} ∘ e^{−t\overline{Δ}_{p}}) = tr_{N (G)}(L_{c} ∘ \overline{pr}_{p}).  (1.17)

We conclude from (1.13) and (1.17)

tr_{N (G)} (N_{c} · index_{N (G)}^{u}(\overline{D}^{∗})) = lim_{t→∞} ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ e^{−t\overline{Δ}_{p}}) .  (1.18)

We have

\frac{d}{dt} ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ e^{−t\overline{Δ}_{p}} : \overline{D}^{p} → \overline{D}^{p})

= ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ \frac{d}{dt}e^{−t\overline{Δ}_{p}})
= ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ (−\overline{Δ}_{p}) ∘ e^{−t\overline{Δ}_{p}})
= − ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ \overline{d}^{p−1} ∘ (\overline{d}^{p−1})^{∗} ∘ e^{−t\overline{Δ}_{p}})

− ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ (\overline{d}^{p})^{∗} ∘ \overline{d}^{p} ∘ e^{−t\overline{Δ}_{p}})

= − ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ \overline{d}^{p−1} ∘ (\overline{d}^{p−1})^{∗} ∘ e^{− \frac{t}{2} \overline{Δ}_{p}} ∘ e^{− \frac{t}{2} \overline{Δ}_{p}})

− ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ (\overline{d}^{p})^{∗} ∘ e^{−t\overline{Δ}_{p+1}} ∘ \overline{d}^{p})

= − ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ e^{− \frac{t}{2} \overline{Δ}_{p}} ∘ \overline{d}^{p−1} ∘ (\overline{d}^{p−1})^{∗} ∘ e^{− \frac{t}{2} \overline{Δ}_{p}})

+ ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ (\overline{d}^{p−1})^{∗} ∘ e^{−t\overline{Δ}_{p}} ∘ \overline{d}^{p−1})

= − ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ (\overline{d}^{p−1})^{∗} ∘ e^{− \frac{t}{2} \overline{Δ}_{p}} ∘ e^{− \frac{t}{2} \overline{Δ}_{p}} ∘ \overline{d}^{p−1})

+ ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ (\overline{d}^{p−1})^{∗} ∘ e^{−t\overline{Δ}_{p}} ∘ \overline{d}^{p−1})

= − ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ (\overline{d}^{p−1})^{∗} ∘ e^{−t\overline{Δ}_{p}} ∘ \overline{d}^{p−1})

+ ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ (\overline{d}^{p−1})^{∗} ∘ e^{−t\overline{Δ}_{p}} ∘ \overline{d}^{p−1})

= 0.  (1.19)

Here are some justifications for the calculation above. Recall that L_{c} is a bounded G-operator and commutes with any G-equivariant operator. We can commute tr_{N (G)} and \frac{d}{dt} since tr_{N (G)} is ultraweakly continuous. We conclude e^{−t\overline{Δ}_{p+1}} ∘ \overline{d}^{p} = \overline{d}^{p} ∘ e^{−t\overline{Δ}_{p}} from the fact that \overline{Δ}_{p+1} ∘ \overline{d}^{p} = \overline{d}^{p} ∘ \overline{Δ}_{p} holds on C^{∞}(\overline{E}^{p−1}). We have used at several places the typical trace relation tr_{N (G)}(AB) = tr_{N (G)}(BA) which is in each case justified by [1, section 4]. In order to be able to apply this trace relation we have splitted e^{−t\overline{Δ}_{p}} into e^{− \frac{t}{2} \overline{Δ}_{p}} ∘ e^{− \frac{t}{2} \overline{Δ}_{p}} in the calculation above.

Hence ∑_{p≥0}(−1)^{p}·tr_{N (G)} (L_{c} ∘ e^{−t\overline{Δ}_{p}} : \overline{D}^{p} → \overline{D}^{p}) is independent of t and we conclude from (1.18)

tr_{N (G)} (N_{c} · index_{N (G)}^{u}(\overline{D}^{∗})) = lim_{t→0} ∑_{p≥0}(−1)^{p} · tr_{N (G)} (L_{c} ∘ e^{−t\overline{Δ}_{p}} : \overline{D}^{p} → \overline{D}^{p}) . (1.20)

Now Lemma 1.12 follows from (1.16) (1.20).

Finally Theorem 0.4 follows from Lemma 1.10, Lemma 1.11 and Lemma 1.12.

2. Modules over a category

In this section we recall some facts about modules over the category Sub = Sub(G; F(X)) for a proper G-CW -complex X as far as needed here. For more information about modules over a category we refer to [12].

Let Sub := Sub(G; F(X)) be the following category. Objects are the elements of the set F(X) of subgroups H ⊂ G, for which X^{H} ≠ ∅. For two finite subgroups H and K in F(X) denote by conhom_{G}(H, K) the set of group homomorphisms f : H → K, for which there exists an element g ∈ G with gHg^{−1} ⊂ K such that f is given by conjugation with g, i.e. f = c(g) : H → K, h ↦ ghg^{−1}. Notice that c(g) = c(g′) holds for two elements g, g′ ∈ G with gHg^{−1} ⊂ K and g′H(g′)^{−1} ⊂ K if and only if g^{−1}g′ lies in the centralizer C_{G}H = {g ∈ G | gh = hg for all h ∈ H} of H in G. The group of inner automorphisms of K acts on conhom_{G}(H, K) from the left by composition. Define the set of morphisms mor_{Sub} (H, K) by Inn(K)\ conhom_{G}(H, K). Let N_{G}H be the normalizer {g ∈ G | gHg^{−1} = H} of H. Define H · C_{G}H = {h · g | h ∈ H, g ∈ C_{G}H}. This is a normal subgroup of N_{G}H and we define W_{G}H := N_{G}H/(H ·C_{G}H). One easily checks that W_{G}H is a finite group and that there is an isomorphism from W_{G}H to aut_{Sub} (H) which sends g(H · C_{G}H) ∈ W_{G}H to the automorphism of H represented by c(g) : H → H. Notice that there is a morphism from H to K if and only if H is subconjugated to K. There is an isomorphism from H to K if and only if H and K are conjugated. The category Sub is a so called EI-category, i.e. any endomorphism in Sub is an isomorphism.

Many constructions in equivariant topology of proper G-spaces are carried out over the orbit category Or(G; Fin). It has as objects homogeneous spaces G/H for finite subgroups H ⊂ G. Morphisms are G-maps. Notice that Sub(G; Fin) is a quotient category of Or(G; Fin). The decisive difference between Or(G; Fin) and Sub(G; Fin) is that the automorphism group of G/H in Or(G; Fin) is N_{G}H/H which is not finite in general, whereas the automorphism group of H in Sub(G; Fin) is W_{G}H := N_{G}H/(H · C_{G}H) which always is finite. We can work with Sub instead of the orbit category since we have induction homomorphisms for equivariant K-homology.

Let R be a commutative associative ring with unit. A covariant resp. contravariant RSub-module M is a covariant resp. contravariant functor from Sub to the category of R-modules. Morphisms are natural transformations. The structure of an abelian category on the category of R-modules carries over to the category of RSub-modules. In particular the notion of a projective RSub-module is defined. Given a contravariant RSub-module M and a covariant RSub-module N, one can define a R-module, their tensor product over Sub

M ⊗_{RSub} N = ⊕_{H∈F(X)} M(H) ⊗_{R} N(H)/ ∼,

where ∼ is the typical tensor relation mf ⊗ n = m ⊗ fn, i.e. for each morphism f : H → K in Sub, m ∈ M(K) and n ∈ N(H) we introduce the relation M(f)(m) ⊗ n − m ⊗ N(f)(n) = 0.

Given a left R[W_{G}H]-module N for H ∈ F(X), define a covariant RSub-module E_{H}M by

(E_{H}M)(K) := R mor_{Sub} (H, K) ⊗_{R[W_{G}H]} N  for K ⊂ G, |K| < ∞,  (2.1)

where R mor_{Sub} (H, K) is the free R-module generated by the set mor_{Sub} (H, K). Given a covariant RSub-module M and H ∈ F(X), define M(H)_{s} to be the left R-submodule of M(H), which is spanned by the images of all R-maps M(f) : M(K) → M(H), where f runs through all morphisms f : K → H in Sub, which have H as target and are not isomorphisms. Obviously M(H)_{s} is an R[W_{G}H]-submodule of M(H). Define a left R[W_{G}H]-module S_{H}M by

S_{H}M := M(H)/M(H)_{s}.  (2.2)

Both functors E_{H} and S_{H} respect direct sums and the property finitely generated and the property projective. Given a left R[W_{G}H]-module M, S_{K} ∘ E_{H}M is M, if H = K and is 0, if H and K are not conjugated in G.

Let M be a covariant RSub-module. We want to check whether it is projective or not. A necessary (but not sufficient) condition is that S_{H}M is a projective R[W_{G}H]-module. Assume that S_{H}M is R[W_{G}H]-projective for all objects H in Sub. We can choose a R[W_{G}H]-splitting σ_{H} : S_{H}M → M(H) of the canonical projection M(H) → S_{H}M = M(H)/M(H)_{s}. For a finite subgroup H ⊂ G define the morphism of covariant RSub-modules

i_{H}M : E_{H}(M(H)) → M

by (i_{H}M)(K)((f : H → K) ⊗_{R[W_{G}H]} m) = M(f)(m). We obtain after a choice of representatives H ∈ (H) for any conjugacy class (H) of subgroups H ∈ F(X) a morphism of covariant RSub-modules

T : ⊕_{(H),H∈F(X)}E_{H}S_{H}M ⟶^{⊕_{(H),H∈F(X)}E_{H}(σ_{H})} ⊕_{(H),H∈F(X)}E_{H}(M(H)) ⟶^{⊕_{(H),H∈F(X)}i_{H}M} M. (2.3)

We get as a special case of [13, Theorem 2.11]

Theorem 2.4 The morphism T is always surjective. It is bijective if and only if M is a projective RSub-module.

3. Some representation theory for finite groups

Denote for a finite group H by Rep_{ℚ}(H) resp. Rep_{ℂ}(H) the ring of finite dimensional Hrepresentations over the field ℚ resp. ℂ. Recall for the sequel that these are finitely generated free abelian groups. Given an inclusion of finite groups H ⊂ G, we denote by ind_{H}^{G} : Rep_{ℚ}(H) → Rep_{ℚ}(G) and res_{G}^{H} : Rep_{ℚ}(G) → Rep_{ℚ}(H) the induction and restriction homomorphism and similar for R ⊗_{ℤ} Rep_{ℚ}, Rep_{ℂ} and R ⊗_{ℤ} Rep_{ℂ} for a commutative ring R with ℤ ⊂ R. Let con_{ℚ}(H) be the set of ℚ-conjugacy classes of elements in H, where h and h′ are called ℚ-conjugated if the cyclic subgroups ⟨h⟩ and ⟨h′⟩ are conjugated in G. Let con(G) be the set of conjugacy classes of elements in G. Denote by class_{ℚ}(H) resp. class_{ℂ}(H) the rational resp. complex vector space of functions con_{ℚ}(H) → ℚ resp. con(G) → ℂ. Character theory yields isomorphisms [17, page 68 and Theorem 29 on page 102]

χ_{ℚ} : ℚ ⊗_{ℤ} Rep_{ℚ}(H) ⟶^{≅} class_{ℚ}(H);
χ_{ℂ} : ℂ ⊗_{ℤ} Rep_{ℂ}(H) ⟶^{≅} class_{ℂ}(H).

For a finite cyclic group C denote by θ_{C} ∈ ℚ ⊗_{ℤ} Rep_{ℚ}(C) the element whose character χ_{ℚ}(θ_{C}) sends c ∈ C to 1, if c generates C, and to 0 otherwise.

Let C ⊂ H be a cyclic subgroup of the finite group H. Then we get for h ∈ H

\frac{1}{[H : C]} · χ_{ℚ} (ind_{C}^{H} θ_{C}) (h) = \frac{1}{[H : C]} · \frac{1}{|C|} · ∑_{l∈H,l^{−1}hl∈C} χ_{ℚ} (θ_{C}) (l^{−1}hl) = \frac{1}{|H|} · ∑_{l∈H,⟨l^{−1}hl⟩=C} 1.

Denote by [ℚ] ∈ Rep_{ℚ}(H) the class of the trivial H-representation ℚ. Notice that χ_{ℚ}([ℚ]) is the constant function with values 1. We get in ℚ ⊗_{ℤ} Rep_{ℚ}(H)

1 ⊗_{ℤ} [ℚ] = ∑_{C⊂H,C cyclic} \frac{1}{[H : C]} · ind_{C}^{H} θ_{C},  (3.1)

since for any l ∈ H and h ∈ H there is precisely one cyclic subgroup C ⊂ H with C = ⟨l^{−1}hl⟩ and χ_{ℚ} is bijective. In particular we get for a finite cyclic group C in ℚ ⊗_{ℤ} Rep_{ℚ}(C)

θ_{C} = 1 ⊗_{ℤ} [ℚ] − ∑_{D⊂C,D≠C} \frac{1}{[C : D]} · ind_{D}^{C} θ_{D}.  (3.2)

Now one easily checks by induction over the order of the finite cyclic subgroup C that the element θ_{C} satisfies

θ_{C} ∈ ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℚ}(C).  (3.3)

Obviously θ_{C} is an idempotent in ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℚ}(C). By the obvious change of rings homomor-

phism, ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℂ}(C) becomes a ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℚ}(C)-module. Hence multiplication with θ_{C} defines an idempotent endomorphism

θ_{C} : ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℂ}(C) → ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℂ}(C).

It is natural with respect to group automorphisms of C, since θ_{C} is invariant under group automorphisms of C.

Lemma 3.4 (a) For a finite group H the map

⊕_{C⊂H,C cyclic} ind_{C}^{H} : ⊕_{C⊂H,C cyclic} ℤ [\frac{1}{|H|}] ⊗_{ℤ} Rep_{ℂ}(C) → ℤ [\frac{1}{|H|}] ⊗_{ℤ} Rep_{ℂ}(H)

is surjective;

(b) Let C be a finite cyclic group. Then the image resp. cokernel of

⊕_{D⊂C,D≠C} ind_{D}^{C} : ⊕_{D⊂C,D≠C} ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℂ}(D) → ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℂ}(C)

is equal resp. isomorphic to the kernel resp. image of the idempotent endomorphism

θ_{C} : ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℂ}(C) → ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℂ}(C);

(c) Let C be a finite cyclic group. The image of the idempotent endomorphism

θ_{C} : ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℂ}(C) → ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℂ}(C);

is a projective ℤ [\frac{1}{|C|}] [aut(C)]-module, where the aut(C)-operation comes from the obvious aut(C)-operation on C.

\underline{\underline{Proof}} : (a) follows from the following calculation for x ∈ ℤ [\frac{1}{|H|}] ⊗_{ℤ} Rep_{ℂ}(H) based on (3.1)

x = (1 ⊗_{ℤ} [ℚ]) · x = ( ∑_{C⊂H,C cyclic} \frac{1}{[H : C]} · ind_{C}^{H} θ_{C} ) · x = ∑_{C⊂H,C cyclic} \frac{1}{[H : C]} · ind_{C}^{H}(θ_{C} · res_{H}^{C} x).

(b) follows from the following two calculations based on (3.2) for x ∈ ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℂ}(H)

x − θ_{C} · x = (1 ⊗ [ℚ] − θ_{C}) · x
= ( ∑_{D⊂C,D≠C} \frac{1}{[C : D]} · ind_{D}^{C} θ_{D}) · x
= ∑_{D⊂C,D≠C} \frac{1}{[C : D]} · ind_{D}^{C}(θ_{D} · res_{C}^{D} x)

and for D ⊂ C, D ≠ C and y ∈ ℤ [\frac{1}{|C|}] ⊗_{ℤ} Rep_{ℂ}(D)

θ_{C} · ind_{D}^{C} y = ind_{D}^{C}(res_{C}^{D} θ_{C} · y) = ind_{D}^{C}(0 · y) = 0.

(c) Put Λ = ℤ [\frac{1}{|C|}]. Let C_{p} be the p-Sylow subgroup of C for a prime p. There are canonical isomorphisms

C ≅ ∏_{p} C_{p};
aut(C) ≅ ∏_{p} aut(C_{p});
P : ⊗_{p} Rep_{ℂ}(C_{p}) ≅ Rep_{ℂ}(C),

where p runs through the prime numbers diving |C|. The isomorphism P assigns to ⊗_{p}[V_{p}] for C_{p}- representations V_{p} the class of the C-representation ⊗_{p}V_{p} with the factorwise action of aut(C) = ∏_{p} aut(C_{p}). The following diagram commutes

⊗_{p}Λ ⊗_{ℤ} Rep_{ℂ}(C_{p}) ⟶^{P} Rep_{ℂ}(C)
⊗_{p}θ_{Cp} ↓ ↓^{θ_{C}}
⊗_{p}Λ ⊗_{ℤ} Rep_{ℂ}(C_{p}) ⟶_{P} Rep_{ℂ}(C)

Thus we obtain an isomorphism of Λ[aut(C)]-modules

⊗_{p} im(θ_{C_{p}}) ⟶^{≅} im(θ_{C}),

where aut(C) = ∏_{p} aut(C_{p}) acts factorwise on the source. Hence the claim for C follows if we know it for C_{p} for all primes p. Therefore it remains to treat the case C = ℤ/p^{n} for some prime number p and positive integer n. Notice that then Λ = ℤ [\frac{1}{p}].

In the sequel we abbreviate A(n) = aut(ℤ/p^{n}). This is isomorphic to the multiplicative group of units ℤ/p^{n}^{×} in ℤ/p^{n} and hence an abelian group of order p^{n−1} · (p − 1). Denote by A(n)_{p} the p-Sylow subgroup and by A(n)′_{p} the subgroup {a ∈ A(n) | a^{p−1} = 1} which is cyclic of order (p − 1). We get a canonical isomorphism

A(n) ≅ A(n)_{p} × A(n)′_{p}

Notice that ℤ/p^{n} has precisely one subgroup of order p^{m} for 0 ≤ m ≤ n which will be denoted by ℤ/p^{m}. These subgroups are characteristic and hence restriction to these subgroups yields homomorphisms A(n) → A(n − 1) → … → A(1). They induce epimorphisms A(m)_{p} → A(m − 1)_{p} and isomorphisms A(m)′_{p} ⟶^{≅} A(m − 1)′_{p}. Using these isomorphisms we will identify

A(n)′_{p} = A(n − 1)′_{p} = … = A(1)′_{p} = ℤ/p^{×}.

Thus we get canonical decompositions

A(n) = A(n)_{p} × ℤ/p^{×}.

Let M be a Λ[A(n)]-module. Let res M be the Λ[ℤ/p^{×}]-module obtained by restriction. The following maps are Λ[A(n)]-homomorphisms

q : Λ[A(n)_{p}] ⊗_{Λ} res M → M,  a ⊗ m ↦ am;
s : M → Λ[A(n)_{p}] ⊗_{Λ} res M,  m ↦ \frac{1}{|A(n)_{p}|} · ∑_{a∈A(n)_{p}} a ⊗ a^{−1}m,

where A(n) = A(n)_{p} × ℤ/p^{×} acts factorwise on Λ[A(n)_{p}] ⊗_{Λ} res M. They satisfy q ∘ s = id. Obviously Λ[A(n)_{p}]⊗_{Λ} res M is Λ[A(n)]-projective if res M is Λ[ℤ/p^{×}]-projective. This shows that M is Λ[A(n)]- projective if its restriction res M to a Λ[ℤ/p^{×}]-module is projective. Therefore it suffices to show that im(θ_{C}) is Λ[ℤ/p^{×}]-projective.

The composition of the induction homomorphism Rep_{ℂ}(ℤ/p^{n−1}) → Rep_{ℂ}(ℤ/p^{n}) with the restriction homomorphism Rep_{ℂ}(ℤ/p^{n}) → Rep_{ℂ}(ℤ/p^{n−1}) is p · id : Rep_{ℂ}(ℤ/p^{n−1}) → Rep_{ℂ}(ℤ/p^{n−1}). We conclude from Lemma 3.4 (b) applied with C = ℤ/p^{n} that the Λ[ℤ/p^{×}]-module im(θ_{C}) is isomorphic to the kernel of the surjective restriction homomorphism res : Λ⊗_{ℤ}Rep_{ℂ}(ℤ/p^{n}) → Λ⊗_{ℤ}Rep_{ℂ}(ℤ/p^{n−1}). Hence there is an exact sequence of Λ[ℤ/p^{×}]-modules

0 → im(θ_{C}) → Λ ⊗_{ℤ} Rep_{ℂ}(ℤ/p^{n}) → Λ ⊗_{ℤ} Rep_{ℂ}(ℤ/p^{n−1}) → 0.

It induces an exact sequence of Λ[ℤ/p^{×}]-modules

0 → im(θ_{C}) → ker (Λ ⊗_{ℤ} Rep_{ℂ}(ℤ/p^{n}) → Λ ⊗_{ℤ} Rep_{ℂ}({1}))
→ ker (Λ ⊗_{ℤ} Rep_{ℂ}(ℤ/p^{n−1}) → Λ ⊗_{ℤ} Rep_{ℂ}({1})) → 0

whose central and final terms are augmentation ideals. Hence it suffices to show that the Λ[ℤ/p^{×}]- module ker (Λ ⊗_{ℤ} Rep_{ℂ}(ℤ/p^{m}) → Λ ⊗_{ℤ} Rep_{ℂ}({1})) is projective for m = 1, 2 … n.

Recall that ℤ/p^{×} is a subgroup of A(m) = aut(ℤ/p^{m}) and thus acts on ℤ/p^{m} − \{\overline{0}\} in the obvious way. Denote for k ∈ ℤ by ℂ_{k} the one-dimensional ℤ/p^{m}-representation for which \overline{b} ∈ ℤ/p^{m} acts by multiplication with exp(\frac{2πikb}{p^{m}}). We obtain a Λ[ℤ/p^{×}]-homomorphism

Q : Λ[ℤ/p^{m} − \{\overline{0}\}] → ker (Λ ⊗_{ℤ} Rep_{ℂ}(ℤ/p^{m}) → Λ ⊗_{ℤ} Rep_{ℂ}(\{1\}))

by sending \overline{k} to [ℂ_{k}] − \frac{1}{p^{m}} · [ℂ[ℤ/p^{m}]]. This is the composition of the inclusion Λ[ℤ/p^{m} − \{\overline{0}\}] → Λ[ℤ/p^{m}], the isomorphism Λ[ℤ/p^{m}] → Λ ⊗_{ℤ} Rep(ℤ/p^{m}) sending \overline{k} to [ℂ_{k}] and the split epimorphism Λ⊗_{ℤ}Rep(ℤ/p^{m}) → ker (Λ ⊗_{ℤ} Rep_{ℂ}(ℤ/p^{m}) → Λ ⊗_{ℤ} Rep_{ℂ}(\{1\})) sending [V] to [V]− \frac{dim(V)}{p^{m}} ·[ℂ[ℤ/p^{m}]]. One easily checks that Q is an isomorphism of Λ[ℤ/p^{×}]-modules. Hence it remains to show that ℤ/p^{×}- acts freely on ℤ/p^{m} − \{\overline{0}\} because then Λ[ℤ/p^{m} − \{\overline{0}\}] is a free Λ[ℤ/p^{×}]-module.

Consider x ∈ ℤ/p^{m} with x ≠ \overline{0}. We have to show for a ∈ ℤ/p^{×} = A(m)′_{p} ⊂ A(m) that a(x) = x implies a = id. Since x is non-zero, x generates a cyclic subgroup ℤ/p^{l} for some l ∈ {1, 2, … m}. Then a ∈ A(m) restricted to A(l) is an automorphism ℤ/p^{l} → ℤ/p^{l} which sends a generator to itself. Hence this automorphism of ℤ/p^{l} is the identity. This implies that a is the identity in A(l)′_{p} = ℤ/p^{×}. This finishes the proof of Lemma 3.4.

The next result is analogous to [13, Lemma 7.4] but we have to go through its proof again because here we want to invert only the orders of finite subgroups of G, whereas in [13] we have considered everything over ℚ.

Theorem 3.5 Let G be a group and Λ = Λ^{G}(X) as defined in (0.5). Consider the covariant ΛSubmodule Λ ⊗_{ℤ} Rep_{ℂ}(?) which sends a finite subgroup group H ⊂ G to Λ ⊗_{ℤ} Rep_{ℂ}(H). Then

(a) S_{H}Λ ⊗_{ℤ} Rep_{ℂ}(?) is trivial if the finite subgroup H ⊂ G is not cyclic.

For a finite cyclic subgroup C ⊂ G, the Λ[W_{G}C]-module S_{C}Λ ⊗_{ℤ} Rep_{ℂ}(?) is isomorphic to the image of the idempotent Λ[W_{G}C]-homomorphism

θ_{C} : Λ ⊗_{ℤ} Rep_{ℂ}(C) → Λ ⊗_{ℤ} Rep_{ℂ}(C).

The isomorphism is given by the composition of the obvious inclusion im(θ_{C}) → Rep_{ℂ}(C) with the obvious projection Rep_{ℂ}(C) → S_{C}Λ ⊗_{ℤ} Rep_{ℂ}(?);

(b) Λ ⊗_{ℤ} Rep_{ℂ}(?) is a projective ΛSub-module;

(c) Let M be a contravariant ΛSub-module. There is a natural isomorphism of Λ-modules

⊕_{(C),C cyclic,C∈F(X)} M(C) ⊗_{Λ[W_{G}C]} im (θ_{C} : Λ ⊗_{ℤ} Rep_{ℂ}(C) → Λ ⊗_{ℤ} Rep_{ℂ}(C))
≅ M ⊗_{ΛSub} Λ ⊗_{ℤ} Rep_{ℂ}(?);

(d) Λ ⊗_{ℤ} Rep_{ℂ}(H) is a flat ΛSub-module, i.e. for an exact sequence 0 → M_{0} → M_{1} → M_{2} → 0 of contravariant ΛSub-modules the induced sequence of R-modules 0 → M_{0} ⊗_{ΛSub} Λ⊗_{ℤ} Rep_{ℂ}(?) → M_{1} ⊗_{ΛSub} Λ ⊗_{ℤ} Rep_{ℂ}(?) → M_{2} ⊗_{ΛSub} Λ ⊗_{ℤ} Rep_{ℂ}(?) → 0 is exact.

\underline{\underline{Proof}} : (a) We conclude from Lemma 3.4 (a) that S_{H}Λ ⊗_{ℤ} Rep_{ℂ}(?) is trivial if H is not cyclic. If H = C for a finite cyclic subgroup C ⊂ G, the assertion follows from Lemma 3.4 (b).

(b) Notice that N_{G}H/C_{G}H is a subgroup of aut(H) and all W_{G}H-operations are induced by the obvious aut(H)-operations. We conclude from Lemma 3.4 (c) and assertion (a) that S_{H}Λ ⊗_{ℤ} Rep_{ℂ}(?) is a projective Λ[W_{G}H]-module for all H ∈ F(X). Because of Theorem 2.4 it suffices to show for the morphism T for Λ⊗_{ℤ} Rep_{ℂ}(?) defined in (2.3) that T (K) is injective for any given element K ∈ F(X).

Consider an element u in the kernel of T (K). Put J(H) = mor_{Sub} (H, K)/(W_{G}H) for H ∈ F(X) and put I = {(H) | H ∈ F(X)}. Choose for any (H) ∈ I a representative H ∈ (H). Then fix for any element \overline{f} ∈ J(H) a representative f : H → K in mor_{Sub} (H, K). For the remainder of the proof of assertion (b) we abbreviate L(?) := Λ ⊗_{ℤ} Rep_{ℂ}(?). We can find elements x_{H,f} ∈ S_{H}L for (H) ∈ I and \overline{f} ∈ J(H) such that only finitely many of the x_{H,f}-s are different from zero and u can be written as

u = ∑_{(H)∈I} ∑_{\overline{f}∈J(H)}(f : H → K) ⊗_{Λ[W_{G}H]} x_{H,f}.

We want to show that all elements x_{H,f} are zero. Suppose that this is not the case. Let (H_{0}) be maximal among those elements (H) ∈ I for which there is \overline{f} ∈ J(H) with x_{H,f} ≠ 0, i.e. if for (H) ∈ I the element x_{H,f} is different from zero for some morphism f : H → K in Sub and there is a morphism H_{0} → H in Sub, then (H_{0}) = (H). In the sequel we choose for any of the morphisms f : H → K in Sub a group homomorphism denoted in the same way f : H → K representing it. Recall that f : H → K is given by conjugation with an appropriate element g ∈ G. Fix f_{0} : H_{0} → K with x_{H_{0}.f_{0}} ≠ 0. We claim that the composition

A : ⊕_{(H)∈I}E_{H} ∘ S_{H}(L(K)) ⟶^{T (K)} L(K) ⟶^{res_{K}^{im(f_{0})}} L(im(f_{0})) ⟶^{ind_{f{−1 \atop 0}:im(f_{0})→H_{0}}} L(H_{0}) ⟶^{pr_{H_{0}}} S_{H_{0}}L

maps u to m·x_{H_{0},f_{0}} for some integer m > 0 which is invertible in Λ. This would lead to a contradiction because of T (K)(u) = 0 and x_{H_{0},f_{0}} ≠ 0.

Consider (H) ∈ I and \overline{f} ∈ J(H). It suffices to show that A ((f : H → K) ⊗_{Λ[W_{G}H]} x_{H,f}) is [K ∩ N_{G} im(f_{0}) : im(f_{0})] · x_{H,f} if (H) = (H_{0}) and \overline{f} = \overline{f_{0}}, and is zero otherwise. One easily checks that A((f : H → K) ⊗_{Λ[W_{G}H]} x_{H,f}) is the image of x_{H,f} under the composition

a(H, f) : S_{H}L ⟶^{σ_{H}} L(H) ⟶^{ind_{f:H→im(f)}} L(im(f)) ⟶^{ind_{im(f)}^{K}} L(K) ⟶^{res_{K}^{im(f_{0})}} L(im(f_{0}))

⟶^{ind_{f{−1 \atop 0}:im(f_{0})→H_{0}}} L(H_{0}) ⟶^{pr_{H_{0}}} S_{H_{0}}L,

where σ_{H} is a Λ[W_{G}H]-splitting of the canonical projection L(H) → S_{H}L. It exists because S_{H}L is a projective Λ[W_{G}H]-module by assertion (a).

The Double Coset formula implies

res_{K}^{im(f_{0})} ∘ ind_{im(f)}^{K} = ∑_{k∈im(f_{0})\K/ im(f)} ind_{c(k):im(f)∩k^{−1} im(f_{0})k→im(f_{0})} ∘ res_{im(f)}^{im(f)∩k^{−1} im(f_{0})k} .

The composition pr_{H_{0}} ∘ ind_{f_{0}^{−1}:im(f_{0})→H_{0}} ∘ ind_{c(k):im(f)∩k^{−1} im(f_{0})k→im(f_{0})} is trivial, if c(k) : im(f) ∩ k^{−1} im(f_{0})k → im(f_{0}) is not an isomorphism. This follows from the definition of S_{H_{0}}L (see (2.2)). Suppose that c(k) : im(f) ∩ k^{−1} im(f_{0})k → im(f_{0}) is an isomorphism. Then k^{−1} im(f_{0})k ⊂ im(f). Since H_{0} has been chosen maximal among the H for which x_{H,f} ≠ 0 for some morphism f : H → K, this implies either that x_{H,f} = 0 or that k^{−1} im(f_{0})k = im(f). Suppose k^{−1} im(f_{0})k = im(f). Then (H) = (H_{0}) which implies H = H_{0}. Moreover, the homomorphisms in Sub represented by f_{0} and f agree. Hence the group homomorphisms f_{0} and f agree themselves and we get k ∈ N_{G} im(f_{0}) ∩ K. This implies that a(H, f) = [K ∩N_{G} im(f_{0}) : im(f_{0})]·id if (H) = (H_{0}) and \overline{f} = \overline{f_{0}}, and that otherwise a(H, f) = 0 or x_{H,f} = 0 holds. Hence the map T is injective.

(c) follows from assertion (a) and the bijectivity of the isomorphism T for Λ ⊗_{ℤ} Rep_{ℂ}(?) defined in (2.3) because there is a natural isomorphism

M ⊗_{ΛSub} E_{H}S_{H}Λ ⊗_{ℤ} Rep_{ℂ}(?) ⟶^{≅} M(H) ⊗_{Λ[W_{G}H]} S_{H}Λ ⊗_{ℤ} Rep_{ℂ}(?).  (3.6)

Now (d) follows from (c) and the fact that the Λ[W_{G}H]-module S_{H}Λ⊗_{ℤ}Rep_{ℂ}(?) ≅ im(θ_{C}) is projective. This finishes the proof of Theorem 3.5.

4. The construction of the Chern character

In this section we want to prove Theorem 0.7. There are similarities with the construction in [13]. The main difference is that here we want to give a construction, where we only have to invert the orders of elements in F(X), whereas in [13] we have worked over the rationals. In [13] we have used the Hurewicz homomorphism from stable homotopy to singular homology, which is only an isomorphism after inverting all primes. We will use the multiplicative structure of K_{∗}^{G} instead and work with a different source for the equivariant Chern character, which allows us to invert only the orders of finite subgroups of G.

In the sequel we denote by K_{p}^{G}(X) the equivariant K-homology of a proper G-CW -complex X. It is defined by colim_{Y ⊂X} KK_{G}^{p}(C_{0}(Y), ℂ), where Y runs over all cocompact G-subcomplexes of X and KK_{G}^{p}(C_{0}(Y), ℂ) denotes equivariant KK-theory of the G-C^{∗}-algebra C_{0}(X) of continuous functions X → ℂ, which vanish at infinity, and the C^{∗}-algebra ℂ with the trivial G-action. Given a homomorphism ϕ : H → G of groups and a proper H-CW -complex, then ind_{ϕ} X := G ×_{ϕ} X is a proper G-CW -complex and there is an induction homomorphism

ind_{ϕ} : K_{0}^{H}(X) → K_{0}^{G}(ind_{ϕ} X).

If the kernel of ϕ acts freely on X, then ind_{ϕ} is bijective. In particular we get for a proper G-CW - complex X a homomorphism

K_{p}^{G}(X) ⟶^{ind_{G→{1}}} K_{p}(G\X),

which is bijective if G acts freely on X. There is an external product

μ : K_{p}^{G}(X) × K_{q}^{G′}(X′) → K_{p+q}^{G×G′}(X × X′)

for groups G and G′, a proper G-CW -complex X and a proper G′-CW -complex X′. External products and induction are compatible. For more information about equivariant K-homology and KK-theory we refer to [10] and in particular for the induction homomorphisms to [18].

Let X be a proper G-CW -complex. We have introduced the ring Λ = Λ^{G}(X) in (0.5). We want to construct for H ∈ F(X) and p = 0, 1 a Λ-homomorphism

\underline{ch}_{p}^{G}(X)(H) : Λ ⊗_{ℤ} K_{p}(C_{G}H\X^{H}) ⊗_{ℤ} Rep_{ℂ}(H) → Λ ⊗_{ℤ} K_{p}^{G}(X),  (4.1)

where K_{p}(C_{G}H\X^{H}) is the (non-equivariant) K-homology of the CW -complex C_{G}H\X^{H}. The map will be defined by the following composition

Λ ⊗_{ℤ} K_{p}(C_{G}H\X^{H}) ⊗_{ℤ} Rep_{ℂ}(H)
id ⊗_{ℤ}K_{p}(pr_{1})⊗_{ℤ}id↑≅
Λ ⊗_{ℤ} K_{p}(EG ×_{C_{G}H} X^{H}) ⊗_{ℤ} Rep_{ℂ}(H)
id ⊗_{ℤ} ind_{C_{G}H→{1}} ⊗j↑≅
Λ ⊗_{ℤ} K_{p}^{C_{G}H}(EG × X^{H}) ⊗_{ℤ} K_{0}^{H}(∗)
μ↓
Λ ⊗_{ℤ} K_{p}^{C_{G}H×H}(EG × X^{H})
ind^{m_{H}} ↓≅
Λ ⊗_{ℤ} K_{p}^{G}(ind_{m_{H}} EG × X^{H})
id ⊗_{ℤ}K_{p}^{G}(ind_{m_{H}} pr_{2})↓
Λ ⊗_{ℤ} K_{p}^{G}(ind_{m_{H}} X^{H})
id ⊗_{ℤ}K_{p}^{G}(v_{H})↓
K_{p}^{G}(X)

Some explanations are in order. We have a left C_{G}H-action on EG × X^{H} by g(e, x) = (ge, gx) for g ∈ C_{G}H, e ∈ EG and x ∈ X^{H}. It extends to a C_{G}H × H-action by letting the factor H act trivially. The map pr_{1} : EG ×_{C_{G}H} X^{H} → C_{G}H\X^{H} is the canonical projection. It induces an isomorphism

Λ ⊗_{ℤ} K_{p}(pr_{1}) : Λ ⊗_{ℤ} K_{p}(EG ×_{C_{G}H} X^{H}) ⟶^{≅} Λ ⊗_{ℤ} K_{p}(C_{G}H\X^{H})

since each isotropy group of the C_{G}H-space X^{H} is finite and for any finite group L the projection induces an isomorphism Λ ⊗_{ℤ} H_{p}(BL) ⟶^{≅} Λ ⊗_{ℤ} H_{p}(∗) and hence by the Atiyah-Hirzebruch spectral sequence an isomorphism Λ ⊗_{ℤ} K_{p}(BL) ⟶^{≅} Λ ⊗_{ℤ} K_{p}(∗) for all p. The isomorphism j : K_{0}^{H}(∗) ⟶^{≅} Rep_{ℂ}(H) is the canonical isomorphism. The group homomorphism m_{H} : C_{G}H × H → G sends (g, h) to gh. Since its kernel acts freely on EG × X^{H}, the map ind_{m_{H}} is bijective. We denote by pr_{2} : EG × X^{H} → X^{H} the canonical projection. The G-map v_{H} : ind_{m_{H}} X^{H} = G ×_{m_{H}} X^{H} → X sends (g, x) to gx.

Notice that we obtain a contravariant ΛSub-module Λ ⊗_{ℤ} K_{p}(C_{G}?\X^{?}) by assigning to a finite subgroup H ⊂ G the Λ-module Λ ⊗_{ℤ} K_{p}(C_{G}H\X^{H}). We have already introduced the covariant Λ-module Λ ⊗_{ℤ} Rep_{ℂ}(?). Analogously to [13] one checks that the various maps \underline{ch}_{p}^{G}(X)(H) defined above induce a map of Λ-modules

ch_{p}^{G}(X) : Λ ⊗_{ℤ} K_{p}(C_{G}?\X^{?}) ⊗_{ΛSub} Λ ⊗_{ℤ} Rep_{ℂ}(?) → Λ ⊗_{ℤ} K_{p}^{G}(X).  (4.2)

Notice that for L ∈ F(X) and X = G/L the ΛSub-module Λ ⊗_{ℤ} K_{0}(C_{G}?\(G/L)^{?}) is isomorphic to the ΛSub-module Λ mor_{Sub} (?, L), which sends a finite subgroup H ⊂ G to the free Λ-module with base mor_{Sub} (H, L). By the Yoneda Lemma one obtaines a canonical isomorphism

Λ ⊗_{ℤ} K_{p}(C_{G}?\(G/L)^{?}) ⊗_{ΛSub} Λ ⊗_{ℤ} Rep_{ℂ}(?) ⟶^{≅} Λ ⊗_{ℤ} Rep_{ℂ}(L).

One easily checks that under this identification ch_{0}^{G}(G/L) becomes the canonical identification of Λ ⊗_{ℤ} Rep_{ℂ}(L) with Λ ⊗_{ℤ} K_{0}^{G}(G/L). Notice that K_{1}(C_{G}?\(G/L)^{?}) and K_{1}^{G}(G/L) are both trivial.

Hence ch_{p}^{G}(G/L) is bijective for all L ∈ F(X) and p = 0, 1. Because of Theorem 3.5 (d) the source of ch_{∗}^{G} is an equivariant homology theory on proper G-CW -complexes Y with F(Y) ⊂ F(X). One easily checks that ch_{∗}^{G} is compatible with the Mayer-Vietoris sequences. By induction over the number of equivariant cells and the Five-Lemma ch_{p}^{G}(Y) is bijective for any finite proper G-CW -complex Y with F(Y) ⊂ F(X). Notice that K_{p}^{G}(Y) is the colimit colim_{Z⊂Y} K_{p}^{G}(Z), where Z runs through all finite G-CW -subcomplexes Z of Y. The analogous statement holds for the source of ch_{∗}^{G}. Hence ch_{p}^{G}(Y) is bijective for all proper G-CW -complexes Y with F(Y) ⊂ F(X) and p = 0, 1. Now Theorem 0.7 follows from Theorem 3.5 (c).

5. The Baum-Connes Conjecture and the Trace Conjecture

In the sequel we denote for a proper G-CW -complex X by

asmb^{G} : K_{0}^{G}(X) → K_{0}(C_{r}^{∗}(G))  (5.1)

the assembly map which essentially assigns to an element in K_{0}^{G}(X) represented by an equivariant Kasparov cycle its index. Given a homomorphism ϕ : H → G of groups with finite kernel, there is an induction homomorphism ind_{ϕ} : K_{p}(C_{r}^{∗}(H)) → K_{p}(C_{r}^{∗}(G)) such that the following diagram commutes [18, Theorem 1]

K_{0}^{H}(X) ⟶^{asmb^{H}} K_{0}(C_{r}^{∗}(H))
ind^{ϕ}↓  ^{ind}^{ϕ}↓
K_{0}^{G}(ind_{ϕ} X) ⟶^{asmb^{G}} K_{0}(C_{r}^{∗}(G))

These induction homomorphisms, the assembly maps and the change of rings homomorphisms associated to the passage from C_{r}^{∗}(G) to N (G) are compatible with the external products

μ : K_{p}^{G}(X) × K_{q}^{G′}(X′) → K_{p+q}^{G×G′}(X × X′);
μ : K_{p}(C_{r}^{∗}(G)) × K_{q}(C_{r}^{∗}(G′)) → K_{p+q}(C_{r}^{∗}(G × G′));
μ : K_{p}(N (G)) × K_{q}(N (G′)) → K_{p+q}(N (G × G′))

for groups G and G′, a proper G-CW -complex X and a proper G′-CW -complex X′. We will use in the sequel the elementary fact that for any G-map f : X → Y of proper G-CW -complexes the

composition K_{0}^{G}(X) ⟶^{K_{0}^{G}(f)} K_{0}^{G}(Y) ⟶^{asmb^{G}} K_{0}(C_{r}^{∗}(G)) is asmb^{G} : K_{0}^{G}(X) → K_{0}(C_{r}^{∗}(G)). In the sequel the letter i denotes change of rings homomorphism for the canonical map C_{r}^{∗}(G) → N (G).

Let X be a proper G-CW -complex. We have introduced J = J^{G}(X) in (0.6). Define the homomorphism

ξ_{1} : ⊕_{(C)∈J} Λ ⊗_{ℤ} K_{0}(C_{G}C\X^{C}) ⊗_{Λ[W_{G}C]} im (θ_{C} : Λ ⊗_{ℤ} Rep_{ℂ}(C) → Λ ⊗_{ℤ} Rep_{ℂ}(C))
→ K_{0}(N (G)) (5.2)

by the composition of the equivariant Chern character of Theorem 0.7

ch_{0}^{G}(X) : ⊕_{(C)∈J} Λ ⊗_{ℤ} K_{0}(C_{G}C\X^{C}) ⊗_{Λ[W_{G}C]} im (θ_{C} : Λ ⊗_{ℤ} Rep_{ℂ}(C) → Λ ⊗_{ℤ} Rep_{ℂ}(C))

⟶^{≅} Λ ⊗_{ℤ} K_{0}^{G}(X),

the assembly map

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

and the change of rings homomorphism

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

This is the homomorphism which we want to understand. In particular we are interested in its image. We will identify it with a second easier to compute homomorphism

ξ_{2} : ⊕_{(C)∈J} Λ ⊗_{ℤ} K_{0}(C_{G}C\X^{C}) ⊗_{Λ[W_{G}C]} im (θ_{C} : Λ ⊗_{ℤ} Rep_{ℂ}(C) → Λ ⊗_{ℤ} Rep_{ℂ}(C))
→ Λ ⊗_{ℤ} K_{0}(N (G)), (5.3)

which is defined as follows. Let l : im(θ_{C}) → Rep_{ℂ}(C) be the inclusion. Let K_{0}(pr) : K_{0}(C_{G}C\X^{C}) → K_{0}(∗) be induced by the projection from C_{G}C\X^{C} to the one-point space ∗. We obtain a map

(i ∘ asmb^{{1}} ∘K_{0}(pr)) ⊗ l : K_{0}(C_{G}C\X^{C}) ⊗ im(θ_{C}) → K_{0}(N ({1})) ⊗ Rep_{ℂ}(C).

Define

α : K_{0}(N ({1})) ⊗ Rep_{ℂ}(C) → Rep_{ℂ}(C) [U] ⊗ [W] ↦ dim_{ℂ}(U) · [W].

Notice that α is essentially given by the external product and K_{0}(N (H)) = Rep_{ℂ}(H) holds by definition for any finite group H. Induction yields a map

ind_{C}^{G} : K_{0}(N (C)) → K_{0}(N (G)).

The composition of these three maps above induces for any finite cyclic subgroup C ⊂ G a homomorphism

ξ_{2}(C) : Λ ⊗_{ℤ} K_{0}(C_{G}C\X^{C}) ⊗_{Λ[W_{G}C]} im (θ_{C} : Λ ⊗_{ℤ} Rep_{ℂ}(C) → Λ ⊗_{ℤ} Rep_{ℂ}(C)) → K_{0}(N (G)).

Define ξ_{2} to be the direct sum ⊕_{(C)∈J} ξ_{2}(C) after the choice of a representative C ∈ (C) for each (C) ∈ J.

Theorem 5.4 Let X be a proper G-CW -complex. Then the maps ξ_{1} of (5.2) and ξ_{2} of (5.3) agree.

\underline{\underline{Proof}} : In the sequel maps denoted by the letter μ will be given by external products and pr denotes the projection from a space to the one-point space ∗. Fix a cyclic subgroup C ∈ F(X). Notice that the homomorphism m_{C} : C_{G}C × C → G (g, c) ↦ gc has a finite kernel so that induction is defined also on the level of the reduced group C^{∗}-algebra and the group von Neumann algebra. Denote by ν : Λ ⊗_{ℤ} K_{0}^{C_{G}C×C}(EG × X^{C}) → Λ ⊗_{ℤ} K_{0}^{G}(X) the composition of the maps id ⊗K_{0}^{G}(v_{C}), id ⊗K_{0}^{G}(ind_{m_{C}} pr_{2}) and ind_{m_{C}} appearing in the definition of \underline{ch}_{0}(X)(C). Then the following diagram commutes

Λ ⊗_{ℤ} K_{0}^{C_{G}C}(EG × X^{C}) ⊗_{ℤ} K^{C}(∗) ⟶^{id ⊗i∘asmb^{C_{G}C} ⊗j} Λ ⊗_{ℤ} K_{0}(N (C_{G}C)) ⊗_{ℤ} Rep_{ℂ}(C)
μ↓  ^{μ}↓
Λ ⊗_{ℤ} K_{0}^{C_{G}C×C}(EG × X^{C})  ⟶^{id ⊗i∘asmb^{C_{G}C×C}}  Λ ⊗_{ℤ} K_{0}(N (C_{G}C × C))
ν↓  ^{ind}^{m_{C}} ↓
Λ ⊗_{ℤ} K_{0}^{G}(X)  ⟶^{id ⊗i∘asmb^{G}}  Λ ⊗_{ℤ} K_{0}(N (G))

For any group G the map induced by the center-valued von Neumann dimension

dim_{N (G)}^{u} : K_{0}(N (G)) → Z(N (G))

is injective. Given a CW -complex Z and an element η ∈ K_{0}(Z), there is a closed manifold M with a map f : M → Z and an elliptic complex D^{∗} of differential operators of order 1 over M such that K_{0}(f) : K_{0}(M) → K_{0}(Z) maps the class [D^{∗}] ∈ K_{0}(M) to η [2]. In the case Z = BG the composition

K_{0}(M) ⟶^{K_{0}(f)} K_{0}(BG) ⟶^{(ind_{G→{1}})^{−1}} K_{0}^{G}(EG) ⟶^{asmb^{G}} K_{0}(C_{r}^{∗}(G)) ⟶^{i} K_{0}(N (G)) ⟶^{dim_{N (G)}^{u}} Z(N (G))

resp. the composition

K_{0}(M) ⟶^{K_{0}(pr)} K_{0}(∗) ⟶^{asmb^{{1}}} K_{0}(C^{∗}({1}) ⟶^{i} K_{0}(N ({1})) ⟶^{ind_{{1}}^{G}} K_{0}(N (G)) ⟶^{dim_{N (G)}^{u}} Z(N (G))

maps [D^{∗}] to the element index_{N (G)}^{u}(\overline{D}^{∗}) resp. index(D^{∗}) · 1_{N (G)}, where index_{N (G)}^{u}(\overline{D}^{∗}) resp. index(D^{∗}) has been defined in (1.7) resp. (1.1). We conclude from Theorem 0.4 and the injectivity of the map dim_{N (G)}^{u} of (1.6) that the following diagram commutes

K_{0}^{G}(EG) ⟶^{i∘asmb^{G}} K_{0}(N (G))
K_{0}(pr)∘ind_{G→{1}}^{−1}↓  ^{ind}_{{1}}^{G}↑
K_{0}(∗) ⟶^{i∘asmb^{{1}}} K_{0}(N ({1}))

Since there is a C_{G}C-map EG × X^{C} → EC_{G}C, we conclude from the diagram above applied to the case G = C_{G}C that the following diagram commutes

Λ ⊗_{ℤ} K_{0}^{C_{G}C}(EG × X^{C}) ⊗_{ℤ} K^{C}(∗) ⟶^{id ⊗i∘asmb^{C_{G}C} ⊗j} Λ ⊗_{ℤ} K_{0}(N (C_{G}C)) ⊗_{ℤ} Rep_{ℂ}(C)
id ⊗(K_{0}(pr)∘ind_{C_{G}C→{1}})⊗j↓  ↑^{id ⊗ ind_{{1}}^{C_{G}C} ⊗ id}

Λ ⊗_{ℤ} K_{0}(∗) ⊗_{ℤ} Rep_{ℂ}(C) id ⊗i∘asmb^{{1}} ⊗ id ⟶ Λ ⊗_{ℤ} K_{0}(N ({1}) ⊗_{ℤ} Rep_{ℂ}(C)

The composition

K_{0}(N ({1})⊗_{ℤ}Rep_{ℂ}(C) ⟶^{ind_{{1}}^{C_{G}C} ⊗ id} K_{0}(N (C_{G}C))⊗_{ℤ}Rep_{ℂ}(C) ⟶^{μ} K_{0}(N (C_{G}C×C)) ⟶^{ind_{m_{C}}} K_{0}(N (G))

agrees with the composition

K_{0}(N ({1})) ⊗ Rep_{ℂ}(C) ⟶^{α} Rep_{ℂ}(C) = K_{0}(N (C)) ⟶^{ind_{C}^{G}} K_{0}(N (G)).

We conclude that the following diagram commutes for any cyclic subgroup C ∈ F(X)

Λ ⊗_{ℤ} K_{0}^{C_{G}C}(EG × X^{C}) ⊗_{ℤ} K^{C}(∗) ⟶^{(id ⊗i∘asmb^{G})∘ν∘μ} Λ ⊗_{ℤ} K_{0}(N (G))
id ⊗(i∘asmb^{{1}} ∘K_{0}(pr)∘ind_{C_{G}C→{1}})⊗j↓  ↑^{id ⊗ ind_{C}^{G}}
Λ ⊗_{ℤ} K_{0}(N ({1})) ⊗_{ℤ} Rep_{ℂ}(C)  ⟶^{id ⊗α}  Λ ⊗_{ℤ} Rep_{ℂ}(C)

Hence the following diagram commutes for any cyclic subgroup C ∈ F(X)

Λ ⊗_{ℤ} K_{0}(C_{G}C\X^{C}) ⊗_{ℤ} Rep_{ℂ}(C) id ⊗(α∘(i∘asmb^{{1}} ∘K_{0}(pr))⊗id) ⟶ Λ ⊗_{ℤ} Rep_{ℂ}(C)

id ⊗\underline{ch}_{0}^{G}(X)(C)↓  ^{id ⊗ ind_{C}^{G}}↓
Λ ⊗_{ℤ} K_{0}^{G}(X)  ⟶^{id ⊗(i∘asmb^{G})}  Λ ⊗_{ℤ} K_{0}(N (G))

Now Theorem 5.4 (and hence also Theorem 0.8) follow.

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