Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 41 (2000), No. 1, 141-150.

Equivariant Higher K-theory for Compact Lie Group Actions

Aderemi O. Kuku

United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency International Centre for Theoretical Physics, Trieste, Italy.

Introduction

The aim of this paper is to construct an equivariant higher K-theory for compact Lie group actions in a way analogous to the ones constructed in [2], [3] for finite groups and [6] for profinite groups.

In Section 1, we discuss the category A(G) of homogeneous spaces on which Mackey functors are defined. In Section 2 we define the higher K-groups K_{n}^{G}(G/H, C), n ≥ 0 for any exact category C and show that K_{n}^{G}(−, C) : A(G) → ℤ-mod is a Mackey functor and that K_{n}^{G}(−, C) are K_{0}^{G}(−, C)-modules. In Section 3, we explore induction techniques in the style of [4] to show that K_{n}^{G}(G/G, \underline{M}(ℂ)) (all n ≥ 0) are hyperelementary computable where \underline{M}(ℂ) is the category of finite dimensional vector spaces over the complex numbers.

In a final section, we briefly discuss possible generalisations of the foregoing to the category of G-spaces of G-homotopy type of G-CW complexes.

Notations and Conventions

Let C be an exact category in the sense of Quillen [9]. Then for all n ≥ 0 we write K_{n}(C) for the Quillen K-groups π_{n+1}(BQC). If B is any small category, we denote by [B, C] the category of covariant functors from B to C. Unless otherwise stated, G denotes a compact Lie group, and maps between G-spaces are G-equivariant. If H is a closed subgroup of G, we write N_{G}(H) or just NH for the normaliser of H in G. Also we write R(G) for the complex representation ring of G.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2000 Heldermann Verlag

142 Kuku, A. O.: Equivariant Higher K-theory for Compact Lie Group Actions

1. Mackey functors on the category A(G) of homogeneous spaces

1.1. Let G be a compact Lie group, X a G-space. The component category π_{0}(G, X) is defined as follows: Objects of π_{0}(G, X) are homotopy classes of maps α : G/H → X where H is a closed subgroup of G. A morphism from [α] : G/H → X to [β] : G/K → X is a G-map σ : G/H → G/K such that βσ is G-homotopic to α.

Note that since Hom(G/H, X) ≃ X^{H} where φ → φ(eH), we could consider objects of π_{0}(G, H) as pairs (H, c) where c ∈ π_{0}(X^{H}) = the set of path components of X^{H}.

1.2. A G-ENR (Euclidean Neighbourhood Retract) is a G-space that is G-homeomorphic to a G-retract of some open G-subset of some G-module V. Let Z be a compact G-ENR, f : Z → X a G-map. For α : G/H → X in π_{0}(G, X), we identify α with the path component X_{α}^{H} into which G/H is mapped by α.

Put Z(f, α) = Z^{H} ∩ f^{−1}(X_{α}^{H}) := subspace of Z^{H} mapped under f into X_{α}^{H}. The action of N_{α}H/H on Z^{H} induces an action of N_{α}H/H on Z(f, α) i.e. Z(f, α) is an Aut(α)-space (see [4]). Note that Aut(α) = {σ : G/H → G/H|ασ ≃ α} and N_{α}(H)_{H}, the isotropy group of α ∈ π_{0}(X^{H}) is isomorphic to Aut(α).

1.3. Let {Z_{i}} be a collection of G-ENR, f_{i} : Z_{i} → X, G-maps. Say that f_{i} : Z_{i} → X is equivalent to f_{j} : Z_{j} → X if and only if for each α : G/H → X in π_{0}(G, X), the Euler characteristic χ(Z(f_{i}, α)/Aut(α)) = χ(Z(f_{j}, α)/Aut(α)).

Let U(G, X) be the set of equivalence classes [f : Z → X] where addition is given by [f_{0} : Z_{0} → X] + [f_{1} : Z_{1} → X] = [f_{0} + f_{1} : Z_{0} + Z_{1} → X]; the identity element is ϕ → X; and the additive inverse of [f : Z → A] is [f ∘ p : Z × A → Z → X], where A is a compact G-ENR with trivial G-action and χ(A) = −1 (see [4]).

Then U(G, X) is the free Abelian group generated by [α], α ∈ π_{0}(G, X) i.e. [f : Z → X] = Σn(α)[α], where G/H × E^{n} ⊂ Z is an open n-cell of Z, and the restriction of f to G/H × E^{n} defines an element [α] of U(G, X).

The cell is called an n-cell of type α. Let n(α) = Σ(−1)^{i}n(α, i) where n(α, i) = number of i-cells of type α (see [4]).

If X is a point, write U(G) for U(G, X).

1.4. For a compact Lie group G, the category A(G) is defined as follows: obA(G) := homogeneous spaces G/H. The morphisms in A(G)(G/H, G/K) are the elements of the Abelian group U(G, G/H × G/K) and have the form α : G/L → G/H × G/K which can be represented by diagram {G/H ←^{α} G/L →^{β} G/K}, so that U(G, G/H × G/K) = free Abelian group on the equivalence classes of diagrams G/H ←^{α} G/L →^{β} G/K where two such diagrams are equivalent if there exists an isomorphism σ : G/L → G/L′ such that the diagram

G/L
↗  ↘
G/H  ↓σ  G/K
↘  ↗
G/L′

commutes up to homotopy.

Composition of morphisms is given by a bilinear map

U(G, G/H_{1} × G/H_{2}) × U(G, G/H_{2} × G/H_{3}) → U(G, G/H_{1} × G/H_{3})

where the composition of (α, β_{1}) : A → G/H_{1} ×G/H_{2} and (β_{2}, γ) : B → G/H_{2} ×G/H_{3} yields a G-map (αᾱ, γγ̄) : C → G/H_{1} × G/H_{3}, where γ̄, ᾱ are maps γ̄ : C → B and ᾱ : C → A, respectively.

1.5. Remarks. (i) Each morphism G/H ⟵^{α} G/L ⟶^{β} G/K is the composition of special types of morphisms

G/H ⟵^{α} G/L ⟶^{id} G/L and G/L ⟵^{id} G/L ⟶^{β} G/K .

(ii) Let π_{0} (or G) be the homotopy category of the orbit category or (G), that is, the objects of π_{0} (or G) are the homogeneous G-spaces G/H and morphisms are homotopy classes [G/L → G/K] of G-maps G/L → G/K. We have a covariant functor π_{0} (or G) → A(G) given by [G/L →^{β} G/K] → (G/L ⟵^{id} G/L →^{β} G/K) and a contravariant functor π_{0} (or G) → A(G) given by

[G/H → G/L] → (G/H ← G/L ⟶^{id} G/L)

(iii) Addition is defined in A(G)(G/H, G/K) = U(G, G/H × G/K) by

(G/H ← G/L → G/K) + (G/H ← G/L′ → G/K)
= (G/H ← (G/L) ˙∪(G/L′) → G/K)

where (G/L) ˙∪(G/L′) is the topological sum of G/L and G/L′.

1.6. Let R be a commutative ring with identity. A Mackey functor M from A(G) to \underline{R-mod} is a contravariant additive functor. Note that M is additive if

M : A(G)(G/H, G/K) → \underline{R-mod}(M(G/K), M(G/H))

is an Abelian group homomorphism.

1.7. Remarks. M comprises of two types of induced morphisms

(i) If α : G/H → G/K is a G-map, regarded as an ordinary morphism α_{!} : G/H ⟵^{id} G/H ⟶^{α} G/K of A(G), we have an induced morphism

M(α_{!}) = M^{∗}(α) =: α^{∗} : M(G/K) → M(G/H)

(ii) If α in (i) is induced from H ⊂ K, i.e. α(gH) = gK, call α^{∗} the restriction morphism. (iii) If we consider α as a transfer morphism α^{!} : G/H ← G/K → G/K in A(G), then we have

M(α^{!}) =: M_{∗}(α) =: α_{∗} : M(G/H) → M(G/K)

and call α_{∗} the induced homomorphism associated to α.

1.8. Let M, N, L be Mackey functors on A(G). A pairing M × N → L is a family of bilinear maps M(S) × N(S) → L(S): (x, y) → x · y (S ∈ A(G)) such that for each G-map f : G/H = S → T = G/K we have

L^{∗}f(x, y) = (M^{∗}fx) · (N^{∗}fy) (x ∈ M(T), y ∈ N(T));
x · (N_{∗}fy) = L_{∗}f((M^{∗}fx) · y) (x ∈ M(T), y ∈ N(S));
(M_{∗}fx) · y = L_{∗}f(x · (N^{∗}fy)) (x ∈ N(S), y ∈ N(T)).

A Green functor V : A(G) → \underline{R-mod} is a Mackey functor together with a pairing V ×V → V such that for each object S, the map V (S) × V (S) → V (S) turns V (S) into an associative R-algebra such that f^{∗} preserves units.

If V is a Green functor, a left V -module is a Mackey functor M together with a pairing V × M → M such that M(S) is a left V (S)-module for every S ∈ A(G).

1.9. Remarks. The Mackey functor as defined in 1.6 is equivalent to the earlier definitions in [2], [3], [7] defined for finite and profinite groups G as functors from the category Ĝ of Gsets to \underline{R-mod}. Observe that if (M_{∗}, M^{∗}) = M is a Mackey functor (bifunctor) Ĝ → \underline{R-mod} (see [7]), we can get M̃ : A(G) → \underline{R-mod} by putting M̃(G/H) = M_{∗}(G/H) = M^{∗}(G/H) on objects while a morphism G/H ⟵^{α} G/L ⟶^{β} G/K in A(G) is mapped onto M(G/H) ⟵_{M_{∗}(α)} M(G/L) ⟵_{M^{∗}(β)} M(G/K) in \underline{R-mod}. Then M is compatible with composition

of morphisms.

Conversely, let M̃ : A(G) → \underline{R-mod} be given and for (α, β) ∈ A(G)(G/H, G/K), let M̃^{∗}(α) and M̄_{∗}(β) be as defined in 1.7. Then, we can extend M̃ additively to finite G-sets to obtain Mackey functors as defined in [2], [3], [7].

1.10. Universal example of a Green functor. Define V̄ (G/H) := U(G, G/H), U(H, G/G) := V̄ (H). Now, consider U(G, G/H) = U(G, G/G × G/H), as a morphism set in A(G). Then, the composition of morphisms

U(G, G/G × G/K) × U(G, G/H × G/K) → U(G, G/G × G/H)

defines an action of V̄ on morphisms.

1.11. Theorem. [4] V̄ : A(G) → \underline{ℤ-mod} is a Green functor, and any Mackey functor M : A(G) → \underline{ℤ-mod} is a V̄ -module.

2. An equivariant higher K-theory for G-actions

2.1. Let G be a compact Lie group, X a G-space. We can regard X as a category \underline{X} as follows. The objects of \underline{X} are elements of X and for x, x′ ∈ X, \underline{X}(x, x′) = \{g ∈ G|gx = x′\}.

2.2. Let X be a G-space, C an exact category in the sense of Quillen [9]. i.e. C is an additive category embeddable as a full subcategory of an Abelian category O such that C is equipped with a class E of exact sequences

0 → M′ → M → M′′ → 0  (I)

such that

(i) E is the class of sequences (I) in the C that are exact in O. (ii) C is closed under extensions in O that is, if (I) is an exact sequence in O and M′, M′′ ∈ C then M ∈ C.

Let [\underline{X}, C] be the category of functors \underline{X} → C. Then [\underline{X}, C] is an exact category where a sequence 0 → ζ′ → ζ → ζ′′ → 0 is exact in [\underline{X}, C] if and only if

0 → ζ′(x) → ζ(x) → ζ′′(x) → 0

is exact in C. In particular for X = G/H in A(G), [\underline{G/H}, C] is an exact category.

2.3. Example. The most important example of [\underline{G/H}, C] is when C is the category \underline{M}(ℂ) of finite dimensional vector spaces over the field ℂ of complex numbers. Here, the category [\underline{G/H}, \underline{M}(ℂ)] can be identified with the category of G-vector bundles on the compact G-space G/H where for any ζ ∈ [G/H, \underline{M}(ℂ)]. x ∈ G/H, ζ(x) ∈ \underline{M}(ℂ) is the fibre ζ̂_{x} of the vector bundle ζ̂ associated with ζ. Indeed, ζ̂ is completely determined by ζē where ē = eH (see [10]).

2.4. Definition. For X = G/H and all n ≥ 0, define K_{n}^{G}(X, C) as the n^{th} algebraic K-group of the exact category [\underline{X}, C] with respect to fibre-wise exact sequence introduced in 2.2.

2.5. Theorem. (i) For all n ≥ 0, K_{n}^{G}(−, C) : A(G) → ℤ-mod is a Mackey functor, (ii) K_{0}^{C}(−, C) : A(G) → \underline{ℤ-mod} is a Green functor and K_{n}^{G}(−, C) is a K_{0}^{G}(−, C)-module for all n ≥ 0.

Before proving 2.5, we first briefly discuss pairings and module structures on higher K-theory of exact categories.

2.6. Let E, E_{1}, E_{2} be three exact categories and E_{1} × E_{2} the product category. An exact pairing E_{1} × E_{2} → E : (M_{1}, M_{2}) → M_{1} ∘ M_{2} is a covariant functor from E_{1} × E_{2} → E such that E_{1} × E_{2}((M_{1}, M_{2}), (M′_{1}, M′_{2})) = E_{1}(M_{1}, M′_{1}) × E_{2}(M_{2}, M′_{2}) → E(M_{1} ∘ M_{2}, M′_{1} ∘ M′_{2}) is bi-additive and bi-exact, that is, for a fixed M_{2}, the functor E_{1} → E given by M_{1} → M_{1} ∘ M_{2} is additive and exact and for fixed M_{1}, the functor E_{2} → E : M_{2} → M_{1} ∘ M_{2} is additive and exact. It follows from [12] that such a pairing gives rise to a K-theoretic product K_{i}(E_{1}) × K_{j}(E_{2}) → K_{i+j}(E) and in particular to natural pairing K_{0}(E_{1}) ∘ K_{n}(E_{2}) → K_{n}(E) which could be defined as follows.

Any object M_{1} ∈ E induces an exact functor M_{1} : E_{2} → E : M_{2} → M_{1} ∘ M_{2} and hence a map K_{n}(M_{1}) : K_{n}(E_{2}) → K_{n}(E). If M′_{1} → M_{1} → M′′_{1} is an exact sequence in E, then we have an exact sequence of exact functors M_{1}^{′∗} → M_{1}^{∗} → M_{1}^{′′∗} from E_{2} to E such that for each object M_{2} ∈ E_{2} the sequence M′_{1}(M_{2}) → M_{1}^{∗}(M_{2}) → M_{1}^{′′∗}(M_{2}) is exact in E and hence by a result of Quillen [9] induces a relation K_{n}(M_{1}^{′∗}) + K_{n}(M_{1}^{′′∗}) = K_{n}(M_{1}^{∗}). So the map M_{1} → K_{n}(M_{1}) ∈ Hom(K_{n}(E_{2}), K_{n}(E)) induces a homomorphism K_{0}(E_{1}) → Hom(K_{n}(E), K_{n}(E)) and hence a pairing K_{0}(E_{1}) × K_{n}(E) → K_{n}(E).

If E_{1} = E_{2} = E and the pairing E × E is naturally associative (and commutative), then the associated pairing K_{0}(E) × K_{0}(E) → K_{0}(E) turns K_{0}(E) into an associative (and commutative) ring which may not contain the identity.

Now, suppose that there is a pairing E ∘ E_{1} → E_{1} which is naturally associative with respect to the pairing E ∘ E → E, then the pairing K_{0}(E) × K_{n}(E_{1}) → K_{n}(E_{1}) turns K_{n}(E_{1}) into a K_{0}(E)-module which may or may not be unitary. However, if E contains a unit i.e. an object E such that E ∘ M = M ∘ E are naturally isomorphic to M for each E-object M, then the pairing K_{0}(E) × K_{n}(E_{1}) → K_{n}(E_{1}) turns K_{n}(E_{1}) into a unitary K_{0}(E)-module.

Proof of 2.5. (i) It is clear from the definition of K_{n}^{G}(G/H, C) that for any G/H ∈ A(G),

K_{n}^{G}(G/H, C) ∈ \underline{ℤ-mod}. Now suppose that (G/H ⟵^{α} G/L ⟶^{β} G/K) ∈ A(G)(G/H, G/K),

then G/H ⟵^{α} G/L ⟶^{β} G/K goes to

K_{n}^{G}(G/H, C) ⟵ K_{n}^{G}(G/L, C) ⟵ K_{n}^{G}(G/K, C)

in \underline{R-mod}(K_{n}^{G}(G/K, C). If we write K_{n}^{G} for K_{n}^{G}(−, C), then

K_{n}^{G}(G/H ← (G/L) ˙∪(G/L′) → G/K)
= K_{n}^{G}(G/H ← G/L → G/K) + K_{n}^{G}(G/H ← G/L′ → G/K) .

Hence K_{n}^{G}(−, C) is a Mackey functor.

(ii) From the discussion in 2.6, it is clear that if we put E_{1} = E_{2} = E = [G/H, C], then

K_{0}^{G}(G/H, C) × K_{0}^{G}(G/H, C) → K_{0}^{G}(G/H, C)

turns K_{0}^{G}(G/H, C) into a commutative ring with identity. Also,

K_{0}^{G}(G/H, C) × K_{n}^{G}(G/H, C) → K_{n}^{G}(G/H, C)

turns

K_{n}^{G}(G/H, C) into K_{0}^{G}(G/H, C)-modules

Hence the result.

2.7. Examples. (i) In general [G/H, C] = category of H-representations in C. Hence [G/G, C] = category of G representations in C. If C = \underline{M}(ℂ), the category of finite dimensional vector spaces over the complex numbers ℂ, K_{0}^{G}(G/G, \underline{M}(ℂ)) is the complex representation ring denoted by R_{ℂ}(G) or simply R(G) in the literature. (ii) If C = \underline{M}(R) := category of finitely generated R-modules, where R is a Noetherian ring compatible with the topological structure of G, then K_{n}^{G}(G/H, \underline{M}(R) ≃ G_{n}(RH). (iii) If C = \underline{P} (R) = category of finitely generated projective R-modules, we have

K_{n}^{G}(G/H, \underline{P} (R)) = G_{n}(H, R) and

when R is regular, G_{n}(R, H) ≃ G_{n}(RH).

3. Induction theory for equivariant higher K-functors

In this section, we discuss the induction properties of the equivariant K-functors constructed in Section 2 leading to the proof of Theorem 3.10 below.

3.1. Definition. Let G be a compact Lie group. A finite family Σ = (G/H)_{j∈J} is called an inductive system. Such a system yields two homomorphisms p(Σ) (induction map) and i(Σ) (restriction maps) defined by

\begin{matrix}p(Σ) : ⊕_{j∈J} M(G/H_{j}) & → & M(G/G) \\ (x_{j}|_{j∈J}) & ↦ & Σ_{j∈J} p(H_{j})_{∗}(x_{j}) \\ i(Σ) : M(G/G) & → & ⊕_{j∈J} M(G/H_{j}) \\ x & ↦ & (p(H_{j})^{∗}x|j ∈ J) \end{matrix}

Note that p(H) denotes the unique morphism G/H → G/G. Σ is said to be projective if p(Σ) is surjective and Σ is said to be injective if i(Σ) is injective. Note that the identity [id] of U(G, G/K × G/H) has the form (see 1.3)

[id] = Σ_{α}n_{α}[α : G/L_{α} → G/K × G/H].  (I)

3.2. Let S(K, H) be the set of α over which the summation (I) is taken and let α = (α(1), α(2)) be the component of α, where α(1) : G/L_{α} → G/K; α(2) : G/L_{α} → G/H.

Define induction map

p(Σ, G/H) : ⊕_{j∈J}(⊕_{α∈S(H_{j},H)}M(G/L_{α})) → M(G/H)

by

(x(j, α)) ↦ (Σ_{j∈J}(Σ_{α∈S(H_{j},H)}n_{α}α(2)_{∗}x(j, α))

and restriction maps

i(Σ, G/H) : M(G/H) → ⊕_{j∈J}(⊕_{α∈S(H_{j},H)}M(G/L_{α})

by

x → (α(2)^{∗}x|(α ∈ (S(H_{j}, H) j ∈ J) .

3.3. Theorem. Let M = K_{n}^{G}(−, C), V = K_{0}^{G}(−, C) be respectively Mackey and Green functors A(G) → ℤ-mod defined in 2.5. If Σ is projective for V, then for each homogeneous space G/H, the induction map p(Σ, G/H) is split surjective and the restriction map i(Σ, G/H) is split injective.

Proof. Since p(Σ) is surjective for V, there exist elements x_{j} ∈V (G/H_{j}) such that Σ(H_{j})_{∗}x_{j} =1. Define a homomorphism

q(Σ, G/H) : M(G/H) → ⊕_{j}(⊕_{α}M(G/L_{α}))

by

q(Σ, G/H)x = α(1)^{∗}x_{j} · α(2)^{∗}x such that α ∈ S(H_{j}, h),

j ∈ J. Then p(Σ, G/H)q(Σ, G/H)

= Σ_{j}(Σ_{α}n_{α}α(2)_{∗}(α(1)^{∗}x_{j} · α(2)^{∗}x))
= Σ_{j}(Σ_{α}n_{α}α(2)_{∗}α(1)^{∗}x_{j})x
= Σp(H)^{∗}p(H_{j})_{∗}x_{j}) · x
= Σp(H)^{∗}(Σp(H_{j})_{∗}x_{j})x
= p(H)^{∗}(1)x = 1 · x = x .

So, p(Σ, G/H)q(Σ, G/H) is the identity. Hence q(Σ, G/H) is a splitting for p(Σ, G/H). We can also define a splitting j(Σ, G/H) for i(Σ, G/H) by j(Σ, G/H) : ⊕_{j}(⊕_{α}(MG/L_{α})) → M(G|H) where

x(j, α) ↦ Σ_{j}(Σ_{α}n_{α}α(2)_{∗}α(1)^{∗}x_{j} ∘ x(j, α)) .

3.4. Remarks. As will be seen below (3.6), V = K_{0}^{G}(−, C) has “defect sets” D(V) and so Σ = {G/H|H ∈ D(V)} is projective for V. It would then mean that an induction theorem for K_{0}^{G}(−, C) implies a similar theorem for K_{n}^{G}(−, C).

3.5. Definition. A finite set E of conjugacy classes (H) is an induction set for a Green functor V if ⊕V (G/H) → V (G/G) given by

(x(H)) → Σp(H)_{∗}x(H) is surjective .

Define E ≤ F iff for each (H) ∈ E, there exists (K) ∈ F such that (H) ≤ (K) i.e. H is subconjugate to K. Then ≤ is a partial ordering on induction sets.

3.6. Lemma. Every Green functor V possesses a minimal induction set D(V), called the defect set of V.

For proof see [4]. Hence K_{0}^{G}(−, C) has defect sets.

3.7. Let V be a Green functor and M be a V -module. Define homomorphisms

p_{1}, p_{2} : ⊕_{i,j∈J} ⊕_{α∈S(i,j)} M(G/L_{α}) → ⊕_{k∈J} M(G/H_{i})

by

p_{2}(x(i, j, α)) = ∑_{i∈J} ∑_{α∈S(i,j)} η_{α}α(2)_{∗}x(i, j, α)

and

p_{1}(x(i, j, α)) = ∑_{i∈J} ∑_{α∈S(i,j)} η_{α}α(1)_{∗}x(i, j, α)

where S(i, j) = S(H_{i}, H_{j}) and α ∈ S(i, j) is in the decomposition of [id] ∈ U(G, G/H_{i} × G/H_{j}).

3.8. Theorem. Let M = K_{n}^{G}(−, C). Then there exists an exact sequence

⊕_{i,j∈J}(⊕_{α∈S(i,j)}M(G/L_{α}) ⟶^{p_{2}−p_{1}} ⊕_{k∈J}M(G/H_{k}) ⟶^{p} M(G/G) → 0.

Proof. We have seen in 3.3 that p is surjective through the construction of a splitting homomorphism q such that pq=identity. We now construct a homomorphism q_{1} such that (p_{2} − p_{1})q_{1} + qp = id from which exactness follows.

Since p_{2} is defined as ⊕_{k∈J}p(ΣG|H_{k}) we define q_{1} = ⊕_{k∈J}q(G/H_{k}) and obtain as in the proof of 3.3 that p_{2}q_{1} = identity. One can also show that p_{1}q_{1} = qp. Hence the result.

3.9. Definition. A subgroup C of G is said to be cyclic if powers of a generator of C are dense in G. If p is a rational prime, then a subgroup K of G is called p-hyperelementary if there exists an exact sequence 1 ↦ C → K → P → 1 where P is a finite p-group and C a cyclic group such that the order of C/C_{0} is prime to p. Here C_{0} is the component of the identity in C. It is called hyperelementary if it is p-hyperelementary for some p.

Let H be the set of hyperelementary subgroups of G. We now have the following result which is the goal of this section and typifies results that can be obtained.

3.10. Theorem. Let M = K_{n}^{G}(−, \underline{M}(ℂ)). Then ⊕_{H}M(G/H) → M(G/G) is surjective (i.e. M satisfies hyper-elementary induction) i.e. M(G/G) can be computed in terms of p-hyperelementary subgroups of G).

Proof. It suffices to show that if V = K_{0}^{G}(−, \underline{M}(ℂ)) then ⊕_{H∈H} V (G/H) → V (G/G)(I) is

surjective since K_{n}^{G}(−, \underline{M}(ℂ) is a V -module.

Now it is clear from example 2.7 that K_{0}^{G}(G/G, \underline{M}(ℂ) is the complex representation ring R(G). Moreover, it is known (see [11]) that R(G) is generated as an Abelian group by modules induced from hyperelementary subgroups of G. This is equivalent to the surjectivity

of ⊕_{H∈H} V (G/H) → V (G/G).

4. Remarks on possible generalizations

4.1. Let B be a category with finite sums, a final object and finite pull-backs (and hence finite products).

A Mackey functor M : B → ℤ-mod is a bifunctor M = (M_{∗}, M^{∗}), M_{∗} covariant, M^{∗} contravariant such that M(X) = M_{∗}(X) = M^{∗}(X) for all X ∈ B and

(i) For any pullback diagram

\begin{matrix}A′ & ⟶^{p_{2}} & A_{2} \\ ↓ p_{1} & & ↓ f_{2} \\ A_{1} & ⟶^{f_{1}} & A \end{matrix} in B

the diagram

M(A′) ⟶^{p_{2}∗} M(A_{i})
↑ p_{1}^{∗}  ↑ f_{2}^{∗}
M(A_{i}) ⟶^{f_{2}∗} M(A)

(ii) M^{∗} transforms finite coproducts in B over finite products in \underline{ℤ-mod}.

4.2. Example. Now suppose that G is a compact Lie group. Let B be the category of Gspaces of the G-homotopy type of G-CW-complexes (e.g. G-ENR spaces, see [4] or [8]). Then B is a category with finite coproducts (topological sums), final object and finite pullbacks (fibred products) (see [1]). Hence a Mackey functor is defined on B along the lines of 4.1.

Hence, in a way analogous to what was done in [2] or [3] we could define for X, Y ∈ B, the notion of Y -exact sequences in the exact category [\underline{X}, C] (where C is an exact category) and obtain K_{n}^{G}(X, C, Y) as the n^{th} algebraic K-group of [\underline{X}, C] with respect to Y -exact sequences.

We could also have the notion of an element ζ ∈ [\underline{X}, C] being Y projective and obtain a full subcategory [\underline{X}, C]_{Y} of Y -projective functors in [\underline{X}, C] so that we could obtain P_{n}^{G}(X, C, Y) as the n^{th} algebraic K-group of [X, C]_{Y} with respect to split exact sequences and then show that K_{n}^{G}(−, C, Y), P_{n}^{G}(−, C, Y): B → ℤ-mod are Mackey functors and that K_{0}^{G}(−, C, Y) : B → ℤ-mod is a Green functor and K_{n}^{G}(−, C, Y), P_{n}^{G}(−, C, Y) are K_{0}^{G}(−, C, T)- modules in a way analogous to what was done in [2], [3].

It is hoped to explore these possibilities for further results in a future paper.

References

[1] Allday, G.; Puppe, V.: Cohomological Methods in Transformation Groups. Cambridge University Press 1993.

[2] Dress, A.; Kuku, A. O.: A convenient setting for equivariant higher algebraic K-theory. Lecture Notes in Math.966, Springer Verlag 1982, 59–68.

[3] Dress, A.; Kuku, A. O.: The Cartan map for equivariant higher algebraic K-groups. Comm. Algebra 9 (7) (1981), 727–746.

[4] tom Dieck, T.: Transformation Groups. Walter de Gruyter, Berlin 1987.

[5] Kuku, A. O.: K-theory of group-rings of finite groups over maximal orders in division algebras. J. Algebra 91 (1984), 18–31.

[6] Kuku, A. O.: Equivariant K-theory and the cohomology of profinite groups. Lecture Notes in Math. 1046, Springer Verlag 1984, 234–244.

[7] Kuku, A. O.: Axiomatic theory of induced representations of finite groups. Les cours du CIMPA, Nice 1985.

[8] Lück, W.: Transformation groups and Algebraic K-Theory. Lecture Notes in Math. 1408, Springer Verlag 1989.

[9] Quillen, D. G.: Higher algebraic K-theory I. Lecture Notes in Math. 341, Springer Verlag 1974, 77–139.

[10] Segal, G.: Representation ring of a compact Lie group. Inst. Hautes Études Sci. Publ. Math. 34 (1968), 113–128.

[11] Segal, G.: Equivariant K-theory. Inst. Hautes Études Sci. Publ. Math. 34 (1968).

[12] Waldhausen, F.: Algebraic K-theory of generalised free products. Ann. of Math. 108 (1978), 135–236.

Received April 3, 1998