Trace Class Operators, Regulators,

and Assembly Maps in K-Theory

Guillermo Cortiñas^{∗} and Gisela Tartaglia^{∗}

Received: July 18, 2013

Communicated by Max Karoubi

Abstract. Let G be a group and let KH be homotopy algebraic K-theory. We prove that if G satisfies the rational KH isomorphism conjecture for the group algebra L^{1}[G] with coefficients in the algebra of trace-class operators in Hilbert space, then it also satisfies the Ktheoretic Novikov conjecture for the group algebra over the integers, and the rational injectivity part of the Farrell-Jones conjecture with coefficients in any number field.

2010 Mathematics Subject Classification: Primary 19D55, Secondary 19F27, 19K99. Keywords and Phrases: Borel regulator, Homotopy algebraic Ktheory, Multiplicative K-theory, trace-class operators.

1 Introduction

Let G be a group, Fin the family of its finite subgroups, and E(G, Fin) the classifying space. Let L^{1} be the algebra of trace-class operators in an infinite dimensional, separable Hilbert space over the complex numbers. Consider the rational assembly map in homotopy algebraic K-theory

H_{p}^{G}(E(G, Fin), KH(L^{1})) ⊗ ℚ → KH_{p}(L^{1}[G]) ⊗ ℚ.  (1)

The rational KH-isomorphism conjecture ([1, Conjecture 7.3]) predicts that (1) is an isomorphism; it follows from a theorem of Yu ([14], [4]) that it is always injective. In the current article we prove the following.

∗The first named author was partially supported by MTM2012-36917-C03-02. Both authors were supported by CONICET, and partially supported by grants UBACyT 20020100100386 and PIP 11220110100800.

Documenta Mathematica 19 (2014) 439–455

Theorem 2. Assume that (1) is surjective. Let n ≡ p + 1 mod 2. Then:

i) The rational assembly map for the trivial family

H_{n}^{G}(E(G, {1}), K(ℤ)) ⊗ ℚ → K_{n}(ℤ[G]) ⊗ ℚ  (3)

is injective.

ii) For every number field F, the rational assembly map

H_{n}^{G}(E(G, Fin), K(F)) ⊗ ℚ → K_{n}(F [G]) ⊗ ℚ  (4)

is injective.

We remark that the K-theory Novikov conjecture asserts that part i) of the theorem above holds for all G, and that part ii) is equivalent to the rational injectivity part of the K-theory Farrell-Jones conjecture for number fields ([11, Conjectures 51 and 58 and Proposition 70]). The idea of the proof of Theorem 2 is to use an algebraic, equivariant version of Karoubi’s multiplicative K-theory. The latter theory assigns groups MK_{n}(𝔄) (n ≥ 1) to any unital Banach algebra 𝔄, which fit into a long exact sequence

HC_{n−1}^{top}(𝔄) → MK_{n}(𝔄) → K_{n}^{top}(𝔄) →^{Sch_{n}^{top}} HC_{n−2}^{top}(𝔄).

Here HC^{top} is the cyclic homology of the completed cyclic module C_{n}^{top}(𝔄) = 𝔄 ˆ⊗ … ˆ⊗𝔄 (n + 1 factors), ch^{top} is the Connes-Karoubi Chern character with values in its periodic cyclic homology HP_{∗}^{top}(𝔄), and S is the periodicity operator. Karoubi introduced a multiplicative Chern character

μ_{n} : K_{n}(𝔄) → MK_{n}(𝔄).  (5)

In particular if O is the ring of integers in a number field F one can consider the composite

K_{n}(O) → K_{n}(ℂ)^{hom(F,ℂ)} → MK_{n}(ℂ)^{hom(F,ℂ)}.  (6)

By comparing this map with the Borel regulator, Karoubi showed in [7] that (6) is rationally injective. It follows that

K_{n}(ℤ) → MK_{n}(ℂ)  (7)

is rationally injective. In the current paper we assign, to every unital ℂ-algebra A, groups κ_{n}(A) (n ∈ ℤ) which fit into a long exact sequence

HC_{n−1}(A/ℂ) → κ_{n}(A) → KH_{n}(L^{1} ⊗_{ℂ} A) →^{TrSch_{n}} HC_{n−2}(A/ℂ).

Here HC(/ℂ) is algebraic cyclic homology of ℂ-algebras, ch is the algebraic Connes-Karoubi Chern character and Tr is induced by the operator trace. We also introduce a character

τ_{n} : K_{n}(A) → κ_{n}(A) (n ∈ ℤ).  (8)

Trace Class Operators, Regulators, and Assembly Maps... 441

If 𝔄 is a finite dimensional Banach algebra and n ≥ 1 then κ_{n}(𝔄) = MK_{n}(𝔄) and (5) identifies with (8) (Proposition 20). Both κ and τ have equivariant versions, so that if X is a G-space and A is a ℂ-algebra, we have an assembly map

H_{n}^{G}(X, κ(A)) → κ_{n}(A[G]).

Let Fcyc be the family of finite cyclic subgroups. We show in Proposition 30 that the map

H_{n}^{G}(E(G, Fcyc), κ(ℂ)) → H_{n}^{G}(E(G, Fin), κ(ℂ))

is an isomorphism, and compute H_{n}^{G}(E(G, Fcyc), κ(ℂ)) ⊗ ℚ in terms of the finite cyclic subgroups of G. We use this and the rational injectivity of (7) to show, in Proposition 44, that the map

H_{n}^{G}(E(G, {1}), K(ℤ)) → H_{n}^{G}(E(G, Fin), K(ℂ)) →^{τ} H_{n}^{G}(E(G, Fin), κ(ℂ)) (9)

is rationally injective. It is well-known [11, Proposition 76] that the map

H_{n}^{G}(E(G, Fcyc), K(R)) ⊗ ℚ → H_{n}^{G}(E(G, Fin), K(R)) ⊗ ℚ

is an isomorphism for every unital ring R. In particular, we may substitute Fcyc for Fin in (4). We use this together with Proposition 30 and the rational injectivity of

K_{n}(F) → K_{n}(ℂ)^{hom(F,ℂ)} → MK_{n}(ℂ)^{hom(F,ℂ)}

(see Remark 24), to show in Proposition 48 that if m ≥ 1, Cyc_{m} is the family of cyclic subgroups whose order divides m, and ζ_{m} is a primitive m-root of 1, then the composite

\xymatrix{H_{n}^{G}(E(G, Cyc_{m}), K(F)) ⊗ ℚ \ar[r] \ar@{.>}[dr] & H_{n}^{G}(E(G, Cyc_{m}), K(ℂ))^{hom(F (ζ_{m}),ℂ)} ⊗ ℚ \ar[d]^{τ} \\  & H_{n}^{G}(E(G, Fcyc), κ(ℂ))^{hom(F (ζ_{m}),ℂ)} ⊗ ℚ}

(10)

is injective. Since the map colim_{m} E(G, Cyc_{m}) → E(G, Fcyc) is an equivalence, it follows that if the rational assembly map

H_{n}^{G}(E(G, Fcyc), κ(ℂ)) ⊗ ℚ → κ_{n}(ℂ[G]) ⊗ ℚ  (11)

is injective then so are both (3) and (4). We show in Corollary 41 that if (1) is surjective, then (11) is injective for n ≡ p + 1 mod 2. This proves Theorem 2. The rest of this paper is organized as follows. In Section 2 we define κ_{n}(A) and the map τ_{n} : K_{n}(A) → κ_{n}(A). By definition, if n ≤ 0, then κ_{n}(A) = KH_{n}(A⊗_{ℂ}L^{1}) and τ_{n} is the identity map (16). We show in Proposition 20 that if n ≥ 1 and 𝔄 is a finite dimensional Banach algebra, then κ_{n}(𝔄) = MK_{n}(𝔄) and τ_{n} = μ_{n}. Karoubi’s regulators and his injectivity results are recalled in Theorem 23. We use Karoubi’s theorem to prove, in Lemma 26, that if F is a number field, C a cyclic group of order m, n a multiple of m, and ζ_{n} a primitive n-root of 1, then the composite

K_{∗}(F [C]) → K_{∗}(F (ζ_{n})[C]) →

K_{∗}(ℂ[C])^{hom(F (ζ_{n}),ℂ)} →^{μ} MK_{∗}(ℂ[C])^{hom(F (ζ_{n}),ℂ)}

is rationally injective. The main result of Section 3 is Proposition 30, which computes H_{n}^{G}(E(G, Fcyc), κ(ℂ)) ⊗ ℚ in terms of group homology and of the groups κ_{∗}(ℂ[C]) for C ∈ Fcyc. The resulting formula is similar to existing formulas for equivariant K and cyclic homology, which are used in its proof ([3], [9], [10], [11],[12]). In Section 4 we show that the rational κ(ℂ)-assembly map is injective whenever the rational KH(L^{1})-assembly map is surjective (Corollary 41). For this we use the fact that for every m ≥ 1, the assembly map

H_{∗}^{G}(E(G, Cyc_{m}), HC(ℂ/ℂ)) → HC_{∗}(ℂ[G])

has a natural left inverse π_{m}, which makes the following diagram commute

\xymatrix{H_{∗}^{G}(E(G, Cyc_{m}), KH(L^{1})) ⊗ ℚ \ar[r] \ar[d]^{TrSch} & KH_{∗}(L^{1}[G]) ⊗ ℚ \ar[d]^{TrSch} \\ H_{∗−2}^{G}(E(G, Cyc_{m}), HC(ℂ/ℂ)) & HC_{∗−2}(ℂ[G]). \ar[l]_{π_{m}}}

Hence for every n we have an inclusion

TrSch(H_{n+1}^{G}(E(G, Cyc_{m}), KH(L^{1})) ⊗ ℚ) ⊂ π_{m}TrSch(KH_{n+1}(L^{1}[G]) ⊗ ℚ).

(12)

We show in Proposition 39 that the rational assembly map

H_{n}^{G}(E(G, Cyc_{m}), κ(ℂ)) ⊗ ℚ → κ_{n}(ℂ[G]) ⊗ ℚ

is injective if and only if the inclusion (12) is an equality. Corollary 41 is immediate from this. Section 5 is concerned with proving that (9) and (10) are injective (Propositions 44 and 48). Finally in Section 6 we show that if the identity holds in (12) for m = 1 then (3) is injective (Theorem 51) and that if it holds for m, then

H_{n}^{G}(E(G, Cyc_{m}), K(F)) ⊗ ℚ → K_{n}(F [G]) ⊗ ℚ

is injective for every number field F (Theorem 52).

2 The character τ : K(A) → κ(A)

2.1 Definition of τ

Let A be a ℂ-algebra and k ⊂ ℂ a subfield. Write C(A/k) for Connes’ cyclic module and HH(A/k), HC(A/k), HN(A/k) and HP (A/k) for the associated Hochschild, cyclic, negative cyclic and periodic cyclic chain complexes. When k = ℚ we omit it from our notation; thus for example, HH(A) = HH(A/ℚ). As usual, we write S, B and I for the maps appearing in Connes’ SBI sequence. To simplify notation we shall make no distinction between a chain complex and the spectrum the Dold-Kan corresponce associates to it. We write KH for Weibel’s homotopy algebraic K-theory and K^{nil} for the fiber of the comparison map K → KH. We have a map of fibration sequences [2, §11.3]

\xymatrix{K^{nil}(A) \ar[r] \ar[d]_{ν} & K(A) \ar[r] \ar[d] & KH(A) \ar[d]^{ch} \\ HC(A)[−1] \ar[r]_{B} & HN(A) \ar[r]_{I} & HP (A).}

Here ch is the Connes-Karoubi character. Write B for the algebra of bounded operators in an infinite dimensional, separable Hilbert space, and L^{1} ◁ B for the ideal of trace class operators. Recall from [6] that HP satisfies excision; in particular, the canonical map HP (L^{1} ⊗_{ℂ} A) → HP (B ⊗_{ℂ} A : L^{1} ⊗_{ℂ} A) is a quasi-isomorphism. We shall abuse notation and write Sch for the map that makes the following diagram commute

\xymatrix{KH(L^{1} ⊗_{ℂ} A)[+1] \ar[d]_{Sch} \ar[r]^{ch} & HP (L^{1} ⊗_{ℂ} A)[+1] \ar[d]^{≀} \\ HC(B ⊗_{ℂ} A : L^{1} ⊗_{ℂ} A)[−1] & HP (B ⊗_{ℂ} A : L^{1} ⊗_{ℂ} A)[+1]. \ar[l]^{S}}  (13)

By [5, Theorems 6.5.3 and 7.1.1], the map ν : K^{nil}(B ⊗_{ℂ} A : L^{1} ⊗_{ℂ} A) → HC(B ⊗_{ℂ} A : L^{1} ⊗_{ℂ} A)[−1] is an equivalence, and thus the map Sch fits into a fibration sequence

KH(L^{1} ⊗_{ℂ} A)[+1] ⟶^{Sch} HC(B ⊗_{ℂ} A : L^{1} ⊗_{ℂ} A)[−1] → K(B ⊗_{ℂ} A : L^{1} ⊗_{ℂ} A).

On the other hand the operator trace Tr : L^{1} → ℂ induces a map of cyclic modules

Tr : C(B ⊗_{ℂ} A : L^{1} ⊗_{ℂ} A) → C(A/ℂ)  (14)
Tr(b_{0} ⊗ a_{0} ⊗ ⋯ ⊗ b_{n} ⊗ a_{n}) = Tr(b_{0} ⋯ b_{n})a_{0} ⊗ ⋯ ⊗ a_{n}.

Note that Tr is defined on b_{0} ⋯ b_{n} since at least one of the b_{i} is in L^{1}. In particular Tr induces a chain map

Tr : HC(B ⊗_{ℂ} A : L^{1} ⊗_{ℂ} A) → HC(A/ℂ).  (15)

We define κ(A) as the homotopy cofiber of the composite of (15) and the map Sch of (13)

κ(A) := hocofi(KH(L^{1} ⊗_{ℂ} A)[+1] ⟶^{TrSch} HC(A/ℂ)[−1]).

Thus because, by definition, cyclic homology vanishes in negative degrees, we have

κ_{n}(A) = KH_{n}(L^{1} ⊗_{ℂ} A) (n ≤ 0).  (16)

By construction, there is an induced map K(B ⊗_{ℂ} A : L^{1} ⊗_{ℂ} A) → κ(A) which fits into a commutative diagram

[figure: KH(L^{1} ⊗_{ℂ} A)[+1]; Sch; KH(L^{1} ⊗_{ℂ} A)[+1]; HC(B ⊗_{ℂ} A : L^{1} ⊗_{ℂ} A)[−1]; HC(A/ℂ)[−1]; K(B ⊗_{ℂ} A : L^{1} ⊗_{ℂ} A); κ(A).]  (17)

A choice of a rank one projection p gives a map A → L^{1} ⊗_{ℂ} A, a ↦ p ⊗ a, and therefore a map K(A) → K(B ⊗_{ℂ} A : L^{1} ⊗_{ℂ} A). We shall be interested in the composite

τ : K(A) → K(B ⊗_{ℂ} A : L^{1} ⊗_{ℂ} A) → κ(A).  (18)

2.2 Comparison with Karoubi’s multiplicative Chern character

Suppose now that 𝔄 is a unital Banach algebra. Let Δ_{•}^{diff}𝔄 = C^{∞}(Δ_{•}, 𝔄) be the simplicial algebra of 𝔄-valued C^{∞}-functions on the standard simplices. Write KV^{diff}(𝔄) for the diagonal of the bisimplicial space [n] ↦ BGL(Δ_{n}^{diff}𝔄). We have

K_{n}^{top}(𝔄) = π_{n}KV^{diff}(𝔄) (n ≥ 1).

Consider the fiber F(𝔄) = hofiber(BGL^{+}(𝔄) → KV^{diff}(𝔄)). We have a homotopy fibration

ΩBGL^{+}(𝔄) → ΩKV^{diff}(𝔄) → F(𝔄).  (19)

Let ˆ⊗ be the projective tensor product of Banach spaces and let C^{top}(𝔄) be the cyclic module with C^{top}(𝔄)_{n} = 𝔄 ˆ⊗ … ˆ⊗𝔄 (n+1 factors). Write HC^{top}(𝔄) and HP^{top}(𝔄) for the cyclic and periodic cyclic complexes of C^{top}(𝔄). In [8] (see also [7, §7]), Max Karoubi constructs a map ch^{rel} : F(𝔄) → HC^{top}(𝔄)[−1] and defines his multiplicative K-groups as the homotopy groups

MK_{n}(𝔄) = π_{n}(hofiber(KV^{diff}(𝔄) → HC^{top}(𝔄)[−2])) (n ≥ 1).

He further defines the multiplicative Chern character as the induced map μ_{n} : K_{n}(𝔄) → MK_{n}(𝔄) (n ≥ 1).

Proposition 20. Let 𝔄 be a unital Banach algebra, and let n ≥ 1. Then there is a natural map κ_{n}(𝔄) → MK_{n}(𝔄) which makes the following diagram commute

\xymatrix{K_{n}(𝔄) \ar[r]^{τ_{n}} \ar[dr]_{μ_{n}} & κ_{n}(𝔄) \ar[d] \\  & MK_{n}(𝔄).}

If furthermore 𝔄 is finite dimensional, then κ_{n}(𝔄) → MK_{n}(𝔄) is an isomorphism.

Proof. Consider the simplicial ring

Δ_{•}𝔄 : [n] ↦ Δ_{n}𝔄 = 𝔄[t_{0}, …, t_{n}]/⟨1 − ∑_{i=0}^{n} t_{i}⟩.

Let KV (𝔄) be the diagonal of the bisimplicial set BGL(Δ_{•}𝔄). We have a homotopy commutative diagram

[figure: KV (L^{1} ⊗_{ℂ} 𝔄); HC(B ⊗_{ℂ} 𝔄 : L^{1} ⊗_{ℂ} 𝔄)[−2]; Tr; HC(𝔄/ℂ)[−2]; KV^{diff}(L^{1} ˆ⊗𝔄); HC^{top}(B ˆ⊗𝔄 : L^{1} ˆ⊗𝔄)[−2]; Tr; HC^{top}(𝔄)[−2]; KV^{diff}(𝔄); HC^{top}(𝔄)[−2]] (21)

By [5, Lemma 3.2.1 and Theorem 6.5.3(ii)] and [13, Proposition 1.5] (or [2, Proposition 5.2.3]), the natural map

KV_{n}(L^{1} ⊗_{ℂ} 𝔄) → KH_{n}(L^{1} ⊗_{ℂ} 𝔄)

is an isomorphism for n ≥ 1. It follows from this that for n ≥ 1, the group κ_{n}(𝔄) is isomorphic to π_{n} of the fiber of the composite of the first row of diagram (21). On the other hand, by Karoubi’s density theorem, the map KV^{diff}(𝔄) → KV^{diff}(L^{1} ˆ⊗𝔄) is an equivalence; inverting it and taking fibers and homotopy groups, we get a natural map κ_{n}(𝔄) → MK_{n}(𝔄) (n ≥ 1). The commutativity of the diagram of the proposition is clear. If now 𝔄 is finite dimensional, then 𝔄 ˆ⊗V = 𝔄 ⊗_{ℂ} V for any locally convex vector space V. Hence the map HC(𝔄/ℂ) → HC^{top}(𝔄) is the identity map. Furthermore, by [5, Theorem 3.2.1], the map KV (L^{1} ⊗_{ℂ} 𝔄) → KV^{diff}(L^{1} ˆ⊗𝔄) is an equivalence. It follows that κ_{n}(𝔄) → MK_{n}(𝔄) is an isomorphism for all n ≥ 1, finishing the proof.

Example 22. We have

κ_{n}(ℂ) = \begin{cases}ℂ^{∗} & n ≥ 1, odd. \\ ℤ & n ≤ 0, even. \\ 0 & otherwise. \end{cases}

2.3 Regulators

In view of Proposition 20 above we may substitute τ for μ in the theorem below.

Theorem 23. [7, Théorème 7.20] Let O be the ring of integers in a number field F. Write F ⊗ ℝ ≅ ℝ^{r_{1}} ⊕ ℂ^{r_{2}}; put r = r_{1} + r_{2}. Then the inclusion O ⊂ ℂ^{r} followed by the the map μ_{n} : K_{n}(ℂ)^{r} → MK_{n}(ℂ)^{r} induces a monomorphism K_{n}(O) ⊗ ℚ → MK_{n}(ℂ)^{r} ⊗ ℚ (n ≥ 1).

Remark 24. It follows from classical results of Quillen that the map K_{n}(O) → K_{n}(F) is a rational isomorphism for n ≥ 2. Thus K_{n}(F) → MK_{n}(ℂ)^{r} is rationally injective for n ≥ 2. Moreover, K_{1}(F) → MK_{1}(ℂ)^{r} is injective too, since the map τ_{1} : K_{1}(ℂ) → MK_{1}(ℂ) is the identity of ℂ^{∗}. Observe that the isomorphism F ⊗ ℝ ≅ ℝ^{r_{1}} ⊕ ℂ^{r_{2}} of Theorem 23 is not canonical; it implies choosing r_{2} nonreal embeddings F → ℂ out of the total 2r_{2}, so that no two of them differ by complex conjugation. On the other hand the map

ι : F → ℂ^{hom(F,ℂ)}, ι(x)_{σ} = σ(x)

is canonical. Moreover, the composite

reg_{n}(F) : K_{n}(F) → K_{n}(ℂ)^{hom(F,ℂ)} → MK_{n}(ℂ)^{hom(F,ℂ)}  (25)

is still a rational monomorphism. Indeed the map of the theorem is obtained by composing (25) with a projection MK_{n}(ℂ)^{hom(F,ℂ)} → MK_{n}(ℂ)^{r}.

Lemma 26. Let F be a number field, C a cyclic group of order m, n a multiple of m and ζ_{n} a primitive n-th root of 1. Then the composite map

K_{∗}(F [C]) → K_{∗}(F (ζ_{n})[C]) →^{ι}

K_{∗}(ℂ[C])^{hom(F (ζ_{n}),ℂ)} →^{μ} MK_{∗}(ℂ[C])^{hom(F (ζ_{n}),ℂ)}

is rationally injective.

Proof. Let G_{m} = Gal(F (ζ_{m})/F); if M is a G_{m}-module, write M^{G_{m}} for the fixed points. By [9, Lemma 8.4], the map F [C] → F (ζ_{m})[C] induces an isomorphism K_{∗}(F [C]) ⊗ ℚ → K_{∗}(F (ζ_{m})[C])^{G_{m}} ⊗ ℚ. In particular, K_{∗}(F [C]) → K_{∗}(F (ζ_{m})[C]) is rationally injective. Now if σ : F (ζ_{m}) → E is a field homomorphism, then K_{∗}(E[C]) = K_{∗}(E)^{m}, and the map K_{∗}(F (ζ_{m})[C]) → K_{∗}(E[C]) decomposes into a direct sum of m copies of the map K_{∗}(F (ζ_{m})) → K_{∗}(E). In particular this applies when E ∈ {F (ζ_{n}), ℂ}. In view of Theorem 23 and Remark 24, it follows that both K_{∗}(F (ζ_{m})[C]) → MK_{∗}(ℂ[C])^{hom(F (ζ_{m}),ℂ)} and K_{∗}(F (ζ_{n})[C]) → MK_{∗}(ℂ[C])^{hom(F (ζ_{n}),ℂ)} are rationally injective. Summing up, we have a commutative diagram

\xymatrix{K_{∗}(F [C]) \ar[d] \\ K_{∗}(F (ζ_{m})[C]) \ar[r] \ar[d] & MK_{∗}(ℂ[C])^{hom(F (ζ_{m}),ℂ)} \ar[d] \\ K_{∗}(F (ζ_{n})[C]) \ar[r] & MK_{∗}(ℂ[C])^{hom(F (ζ_{n}),ℂ)}}

We have shown that the first vertical map on the left and the two horizontal maps are rationally injective. Since the vertical map on the right is injective, we conclude that the composite of the left column followed by the bottom horizontal arrow is a rational monomorphism, finishing the proof.

3 Rational computation of equivariant κ-homology

Let G be a group and let OrG be its orbit category. For G/H ∈ OrG, let G(G/H) = G^{G}(G/H) be the transport groupoid. It follows from [4, §3] that the diagram (17) can be promoted to a commutative diagram of OrG-spectra whose columns are homotopy fibrations

\xymatrix{KH(L^{1} ⊗_{ℂ} A[G(G/H)])[+1] \ar[d]^{Sch} \ar[dr]^{TrSch} \\ HC(B ⊗_{ℂ} A[G(G/H)] : L^{1} ⊗_{ℂ} A[G(G/H)])[−1] \ar[r] \ar[d] & HC(A[G(G/H)]/ℂ)[−1] \ar[d] \\ K(B ⊗_{ℂ} A[G(G/H)] : L^{1} ⊗_{ℂ} A[G(G/H)]) \ar[r] & κ(A[G(G/H)]).}

If now X is any G-simplicial set, then taking G-equivariant homology yields a diagram whose columns are again homotopy fibrations

[figure: H^{G}(X, KH(L^{1} ⊗_{ℂ} A))[+1]; Sch; H^{G}(X, KH(L^{1} ⊗_{ℂ} A))[+1]; H^{G}(X, HC(B ⊗_{ℂ} A : L^{1} ⊗_{ℂ} A))[−1]; Tr; H^{G}(X, HC(A/ℂ))[−1]; H^{G}(X, K(B ⊗_{ℂ} A : L^{1} ⊗_{ℂ} A)); H^{G}(X, κ(A)).] (27)

Hence

H^{G}(X, κ(A)) = hocofi(H^{G}(X, KH(L^{1} ⊗_{ℂ} A))[+1] → H^{G}(X, HC(A/ℂ))[−1]). (28)

Documenta Mathematica 19 (2014) 439–455

Similarly, a choice of rank one projection induces a map of OrG-spectra

K(A[G(G/H)]) → K(B ⊗_{ℂ} A[G(G/H)] : L^{1} ⊗_{ℂ} A[G(G/H)]).

Taking equivariant homology we obtain a map

H^{G}(X, K(A)) → H^{G}(X, K(B ⊗_{ℂ} A : L^{1} ⊗_{ℂ} A)).

Composing this map with the bottom arrow in diagram (27) we obtain an equivariant character

τ : H^{G}(X, K(A)) → H^{G}(X, κ(A)).  (29)

In what follows we shall be interested in several families of finite subgroups of a given group. We write Fin and Fcyc for the family of finite subgroups and the subfamily of those finite subgroups that are cyclic. If m ≥ 1 we write Cyc_{m} for the family of those cyclic subgroups whose order divides m. If G is a group and F a family of subgroups, we write E(G, F) for the corresponding classifying space. If H ⊂ G is a subgroup in the family F, we write (H) for the conjugacy class of H and

(F) = {(H) : H ∈ F}

for the set of all conjugacy classes of subgroups of G in the family F. If G is a group and C ⊂ G is a cyclic subgroup, we write N_{G}C for its normalizer, Z_{G}C for its centralizer, and put

W_{G}C = N_{G}C/Z_{G}C.

If C ⊂ G is a finite cyclic group, and A(C) is its Burnside ring, then there is a canonical isomorphism A(C) ⊗ ℚ ≅ ℚ^{FinC}. We write θ_{C} ∈ A(C) ⊗ ℚ for the element corresponding to the characteristic function χ_{C} ∈ ℚ^{FinC}.

Proposition 30. Let G be a group. Then the map H_{∗}^{G}(E(G, Fcyc), κ(ℂ)) → H_{∗}^{G}(E(G, Fin), κ(ℂ)) is an isomorphism and

H_{n}^{G}(E(G, Fcyc), κ(ℂ)) ⊗ ℚ =

⨁_{p+q=n} ⨁_{(C)∈(Fcyc)} H_{p}(Z_{G}C, ℚ) ⊗_{ℚ[W_{G}C]} θ_{C} · κ_{q}(ℂ[C]) ⊗ ℚ (31)

Proof. If H is a finite subgroup, then the equivalence KH(L^{1}) ⟶^{∼} K^{top}(L^{1}) ←^{∼} K^{top}(ℂ) induces an equivalence KH(L^{1}[G(G/H)]) →^{∼} K^{top}(C^{∗}(G(G/H))). Hence if X is a (G, Fin)-complex, we have an equivalence

H^{G}(X, KH(L^{1})) ⟶^{∼} H^{G}(X, K^{top}(ℂ)).  (32)

Thus the map

H^{G}(E(G, Fcyc), KH(L^{1})) → H^{G}(E(G, Fin), KH(L^{1}))

is a weak equivalence because

H^{G}(E(G, Fcyc), K^{top}(ℂ)) → H^{G}(E(G, Fin), K^{top}(ℂ))

is ([11, Proposition 69]). Similarly,

H^{G}(E(G, Fcyc), HC(ℂ/ℂ)) → H^{G}(E(G, Fin), HC(ℂ/ℂ))

is an equivalence (see [12, §9] or [3, §7]). From (28) and what we have just proved, it follows that H^{G}(E(G, Fcyc), κ(ℂ)) → H^{G}(E(G, Fin), κ(ℂ)) is an equivalence. This shows the first assertion of the proposition. From (32), [10, Theorem 0.7] and [11, Theorem 172], we get

H_{n}^{G}(E(G, Fcyc), KH(L^{1})) ⊗ ℚ =

⨁_{p+q=n} ⨁_{(C)∈(Fcyc)} H_{p}(Z_{G}C, ℚ) ⊗_{ℚ[W_{G}C]} θ_{C} · K_{q}^{top}(ℂ[C]) ⊗ ℚ.

Next write con_{f}(G) for the conjugacy classes of elements of G of finite order, and Gen(C) for the set of all generators of C ∈ Fcyc. By using [12, Lemma 7.4] and the argument of the proof of [4, Proposition 2.2.1] we obtain

H_{n}^{G}(E(G, Fcyc), HC(ℂ/ℂ)) =

= ⨁_{p+q=n} ⨁_{(g)∈con_{f} (G)} H_{p}(Z_{G}(⟨g⟩), ℚ) ⊗ HC_{q}(ℂ/ℂ)

= ⨁_{p+q=n} ⨁_{(C)∈(Fcyc)} H_{p}(Z_{G}C, ℚ) ⊗_{ℚ[W_{G}(C)]} map(Gen(C), HC_{q}(ℂ/ℂ))

= ⨁_{p+q=n} ⨁_{(C)∈(Fcyc)} H_{p}(Z_{G}C, ℚ) ⊗_{ℚ[W_{G}(C)]} θ_{C} · HC_{q}(ℂ[C]/ℂ)

It follows from the proof of Proposition 20 that under the isomorphism (32) and the identity H_{∗}^{G}(−, HC(ℂ/ℂ)) = H_{∗}^{G}(−, HC^{top}(ℂ)) the map TrSch identifies with TrSch^{top}. Hence, by naturality, the map

(TrSch)_{n} : H_{n+1}^{G}(E(G, Fcyc), KH(L^{1})) → H_{n−1}^{G}(E(G, Fcyc), HC(ℂ/ℂ))

is induced by the maps

TrSch_{q}^{top} : K_{q+1}^{top}(ℂ[C]) → HC_{q−1}^{top}(ℂ[C]) = HC_{q−1}(ℂ[C]/ℂ).

The computation of H_{n}^{G}(E(G, Fin), κ(ℂ)) ⊗ ℚ is now immediate from this.

Remark 33. We have an equivalence of (G, Fcyc)-spaces

colim_{m} E(G, Cyc_{m}) ⟶^{≅} E(G, Fcyc)

where the colimit is taken with respect to the partial order of divisibility. Hence for every OrG-spectrum E,

H_{∗}^{G}(E(G, Fcyc), E) = colim_{m} H_{∗}^{G}(E(G, Cyc_{m}), E).

Moreover it is clear from the proof of Proposition 30 that for every m the map

H_{∗}^{G}(E(G, Cyc_{m}), κ(ℂ)) ⊗ ℚ → H_{∗}^{G}(E(G, Fcyc), κ(ℂ)) ⊗ ℚ

is the inclusion

⨁_{p+q=n} ⨁_{(C)∈(Cyc_{m})} H_{p}(Z_{G}C, ℚ) ⊗_{ℚ[W_{G}C]} θ_{C} · κ_{q}(ℂ[C]) ⊗ ℚ ↪

⨁_{p+q=n} ⨁_{(C)∈(Fcyc)} H_{p}(Z_{G}C, ℚ) ⊗_{ℚ[W_{G}C]} θ_{C} · κ_{q}(ℂ[C]) ⊗ ℚ.

4 Conditions equivalent to the rational injectivity of the κ assembly map

Let G be a group. As shown in the proof of Proposition (30), we have a direct sum decomposition

H_{n}^{G}(E(G, Fcyc), HC(ℂ/ℂ)) =

⨁_{p+q=n} ⨁_{(C)∈(Fcyc)} H_{p}(Z_{G}C, ℚ) ⊗_{ℚ[W_{G}(C)]} θ_{C} · HC_{q}(ℂ[C]/ℂ).

By the same proof, for each p, q we have

⨁_{(C)∈(Fcyc)} H_{p}(Z_{G}C, ℚ) ⊗_{ℚ[W_{G}(C)]} θ_{C} · HC_{q}(ℂ[C]/ℂ) =

⨁_{(g)∈con_{f} (G)} H_{p}(Z_{G}⟨g⟩, ℚ) ⊗ HC_{q}(ℂ/ℂ). (34)

On the other hand we also have a decomposition

HC_{n}(ℂ[G]/ℂ) = ⨁_{(g)∈con(G)} HC_{n}^{(g)}(ℂ[G]/ℂ).

The assembly map identifies

H_{n}^{G}(E(G, Fcyc), HC(ℂ/ℂ)) = ⨁_{(g)∈con_{f} (G)} HC_{n}^{(g)}(ℂ[G]/ℂ).

Thus there is a projection

π_{n}^{Fcyc} : HC_{n}(ℂ[G]/ℂ) → H_{n}^{G}(E(G, Fcyc), HC(ℂ/ℂ))

which is left inverse to the asembly map. By composing the map

TrSch : KH_{n+1}(L^{1}[G]) → HC_{n−1}(ℂ[G]/ℂ)

with the projection above, we obtain a map

π_{n−1}^{Fcyc}TrSch : KH_{n+1}(L^{1}[G]) ⊗ ℚ → H_{n−1}^{G}(E(G, Fcyc), HC(ℂ/ℂ)). (35)

Next, if m ≥ 1 then

H_{n}^{G}(E(G, Cyc_{m}), HC(ℂ/ℂ)) = ⨁_{(g)∈con_{f} (G),g^{m}=1} HC_{n}^{(g)}(ℂ[G]/ℂ).

Thus we also have a map

π_{n−1}^{Cyc_{m}}TrSch : KH_{n+1}(L^{1}[G]) ⊗ ℚ → H_{n−1}^{G}(E(G, Cyc_{m}), HC(ℂ/ℂ)). (36)

In the following proposition we use the following notation. We write

H_{n}^{G}(E(G, Cyc_{m}), κ(ℂ) ⊗ ℚ)^{+} :=

⨁_{p+q=n,q≥1} ⨁_{(C)∈(Cyc_{m})} H_{p}(Z_{G}C, ℚ) ⊗_{ℚ[W_{G}C]} θ_{C} · κ_{q}(ℂ[C]) ⊗ ℚ (37)

and

H_{n}^{G}(E(G, Cyc_{m}), κ(ℂ) ⊗ ℚ)^{−} :=

⨁_{p+q=n,q≤0} ⨁_{(C)∈(Cyc_{m})} H_{p}(Z_{G}C, ℚ) ⊗_{ℚ[W_{G}C]} θ_{C} · κ_{q}(ℂ[C]) ⊗ ℚ. (38)

Note that, by Proposition 30, H_{n}^{G}(E(G, Cyc_{m}), κ(ℂ)) ⊗ ℚ is the direct sum of (37) and (38).

Proposition 39. Let G be a group, n ∈ ℤ and m ≥ 1. The following are equivalent.

i) The rational assembly map

H_{n}^{G}(E(G, Cyc_{m}), κ(ℂ)) ⊗ ℚ → κ_{n}(ℂ[G]) ⊗ ℚ  (40)

is injective.

ii) The restriction of the rational assembly map to the summand (37) is injective.

iii) The image of the map (36) coincides with the image of

TrSch : H_{n+1}^{G}(E(G, Cyc_{m}), KH(L^{1})) ⊗ ℚ → H_{n−1}^{G}(E(G, Cyc_{m}), HC(ℂ/ℂ)).

Proof. It is clear that i) implies ii). Assume that ii) holds and consider the following commutative diagram with exact columns:

\xymatrix{H_{n+1}^{G}(E(G, Cyc_{m}), KH(L^{1})) ⊗ ℚ \ar[d]^{TrSch} \ar[r] & KH_{n+1}(L^{1}[G]) ⊗ ℚ \ar[d] \ar[dl]^{(36)} \\ H_{n−1}^{G}(E(G, Cyc_{m}), HC(ℂ/ℂ)) \ar[r] \ar[d] & HC_{n−1}(ℂ[G]) \ar[d] \\ H_{n}^{G}(E(G, Cyc_{m}), κ(ℂ)) ⊗ ℚ \ar[d] \ar[r] & κ_{n}(ℂ[G]) ⊗ ℚ \ar[d] \\ H_{n}^{G}(E(G, Cyc_{m}), KH(L^{1})) ⊗ ℚ \ar[r] & KH_{n}(L^{1}[G]) ⊗ ℚ.}

Let x be an element of the kernel of the map of part i), that is of the first map above bottom in the diagram above. Write x = x_{+} + x_{−}, with x_{+} in (37) and x_{−} in (38). The image of x under the vertical map must be zero, since by Yu’s theorem ([14], see also [4]), the bottom horizontal map is injective. By (16) and the proof of Proposition 30, this implies that x_{−} = 0, proving that ii) implies i). Next assume y is an element in the image of (36) which is not in the image of the vertical map TrSch in the diagram above. Then the image of y under the vertical map is a nonzero element of the kernel of the next horizontal map. Thus i) implies iii). The converse is also clear, using Yu’s theorem again.

Corollary 41. Let G be a group and let n, p ∈ ℤ with n ≡ p + 1 mod 2. Assume that the map

H_{p}^{G}(E(G, Fcyc), KH(L^{1})) ⊗ ℚ → KH_{p}(L^{1}[G]) ⊗ ℚ  (42)

is surjective. Then the map (40) is injective for every m ≥ 1.

Proof. By Yu’s theorem ([14],[4]) the map (42) is always injective; under our current assumptions, it is an isomorphism. Moreover, by [5, Theorem 6.5.3], the groups KH_{p}(L^{1}[G]) depend only on the parity of p. It follows that condition iii) of Proposition 39 holds for every m and every n ≡ p + 1 mod 2. This concludes the proof.

5 Rational injectivity of the equivariant regulators

Let G be a group. By composing the equivariant character (29) with the map induced by the inclusion ℤ ⊂ ℂ we obtain a map

H_{∗}^{G}(E(G, {1}), K(ℤ)) → H_{∗}^{G}(E(G, Fin), K(ℂ)) →^{τ} H_{∗}^{G}(E(G, Fin), κ(ℂ)).

(43)

Proposition 44. The map (43) is rationally injective.

Proof. We have

H_{n}^{G}(E(G, {1}), K(ℤ)) ⊗ ℚ = ⨁_{p+q=n} H_{p}(G, ℚ) ⊗ K_{q}(ℤ) ⊗ ℚ.  (45)

By Theorem 23, the regulators K_{q}(ℤ) → K_{q}(ℂ) → κ_{q}(ℂ) = MK_{q}(ℂ) induce a monomorphism from (45) to

⨁_{p+q=n} H_{p}(G, ℚ) ⊗ κ_{q}(ℂ) ⊗ ℚ.  (46)

The map (43) tensored with ℚ is the composite of the above monomorphism with the inclusion of (46) as a direct summand in (31).

Let F be a number field, G a group and m ≥ 1. Let ζ_{m} be a primitive m^{th} root of 1. The map E(G, Cyc_{m}) → E(G, Fcyc), together with the inclusion

F ⊂ F (ζ_{m}) →^{ι} ℂ^{hom(F (ζ_{m}),ℂ)}

and the character τ : K(ℂ) → κ(ℂ), induce a homomorphism

H_{∗}^{G}(E(G, Cyc_{m}), K(F)) → H_{∗}^{G}(E(G, Fcyc), κ(ℂ))^{hom(F (ζ_{m}),ℂ)}. (47)

Proposition 48. The map (47) is rationally injective.

Proof. By [9, Theorem 0.3], we have

H_{n}^{G}(E(G, Cyc_{m}), K(F)) =

⨁_{p+q=n} ⨁_{(C)∈(Cyc_{m})} H_{p}(Z_{G}C, ℚ) ⊗_{ℚ[Z_{G}C]} θ_{C} · K_{q}(F [C]) ⊗ ℚ. (49)

By Lemma 26 the maps K_{q}(F [C]) → κ_{q}(ℂ[C])^{hom(F (ζ_{m}),ℂ)} with C ∈ Cyc_{m} induce a rational monomorphism from (49) to

⨁_{p+q=n} ⨁_{(C)∈(Cyc_{m})} H_{p}(Z_{G}C, ℚ) ⊗_{ℚ[Z_{G}C]} θ_{C} · κ_{q}(ℂ[C])^{hom(F (ζ_{m}),ℂ)}.  (50)

The map (47) tensored with ℚ is the composite of the above monomorphism with the inclusion of (50) as a summand in (31).

6 Comparing conjectures and assembly maps

Theorem 51. Let G be a group. Assume that the equivalent conditions of Proposition 39 hold for G with m = 1. Then the assembly map

H_{n}^{G}(E(G, {1}), K(ℤ)) → K_{n}(ℤ[G])

is rationally injective. In particular this is the case whenever G satisfies the rational KH-isomorphism conjecture with L^{1}-coefficients.

Proof. Immediate from Proposition 44 and Corollary 41.

Theorem 52. Let G be a group and m ≥ 1. Assume that the equivalent conditions of Proposition 39 hold for G and m. Then for every number field F, the assembly map

H_{n}^{G}(E(G, Cyc_{m}), K(F)) → K_{n}(F [G])

is rationally injective. If moreover the condition holds for all m –as is the case, for example, if G satisfies the rational KH-isomorphism conjecture with L^{1}-coefficients- then G satisfies the rational injectivity part of the K-theory isomorphism conjecture with coefficients in any number field.

Proof. Immediate from Proposition 48 and Corollary 41.

References

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Dep. de Matemática-IMAS F. Cs. Exactas y Naturales Univ. de Buenos Aires Ciudad Universitaria Pab 1 1428 Buenos Aires Argentina 456

Documenta Mathematica 19 (2014)