DRESS INDUCTION AND THE BURNSIDE QUOTIENT GREEN RING

I. HAMBLETON, L. R. TAYLOR, AND E. B. WILLIAMS

Abstract. We define and study the Burnside quotient Green ring of a Mackey functor, introduced in our MSRI preprint [16]. Some refinements of Dress induction theory are presented, together with applications to computation results for K-theory and L-theory of finite and infinite groups.

1. Introduction

Induction theory began with Artin and Brauer’s work in representation theory, was continued by Swan [24] and Lam [18] for K-theory, and was put in its most abstract and elegant setting by Green [7] and Dress [3], [4]. The theory sets up a convenient framework for computing the value of a Mackey functor on some finite group G, given suitable generation results for a Green ring which acts on the Mackey functor (see [2], [19], [26], [27] for some of the subsequent developments in this subject).

The main examples in this theory are (i) the Swan Green ring SW (G, Z) (see [24]), which leads to the Brauer-Berman-Witt induction theorem for representations of finite groups, and computation results for Quillen K-theory K_{n}(RG), and (ii) the Dress Green ring GU(G, Z) (see [4]), which leads to computation results for the oriented surgery obstruction groups L_{n}(ZG) of C. T. C. Wall [29].

In Section 2 we define the Burnside quotient Green ring A_{M} for a Mackey functor M: D(G) → Ab, where D(G) denotes the Dress category of finite G-sets, and Ab the category of abelian groups. This Green ring A_{M} is the smallest quotient of the Burnside ring which is a Green ring, and still acts on the Mackey functor. As defined, it has many convenient naturality properties, and generation results for A_{M} will lead as usual to computation results for M. We define the concept of a Dress generating set X for a Green ring in Definition 3.5. The main result (see Theorem 3.6) is:

Theorem A. A finite G-set X is a Dress generating set for a Green ring G if and only if it is a Dress generating set for the Burnside quotient Green ring A_{G}.

The naturality of the Burnside quotient Green rings can now be used to obtain computability results for sub- or quotient Mackey functors (see Theorem 3.8). We also point out a useful refinement of Dress induction in Theorem 3.10. We use the Burnside quotient Green ring in Section 5 to study additive functors out of the categories RG-Morita defined in [15], where R is a commutative ring with unit. The main examples of such functors include K-theory, Hochschild homology and cyclic homology (see [15, 1.A.12]). We prove the following computability result (see Theorem 5.5):

Date: March 26, 2008.

Research partially supported by NSERC Research Grant A4000 and the NSF.

Theorem B. Any additive functor F: RG-Morita → Ab gives a Mackey functor on D(G) by composition with d: D(G) → RG-Morita. Any such Mackey functor is hyperelementary computable.

The Burnside quotient Green ring has been applied to study the permutation representations of finite groups in [13], free actions of finite groups on products of spheres in [8], and to the computation of Bass Nil-groups in [9]. This theory was surveyed and used in [14]. Our results also apply to the computation of K and L-theory for infinite groups, based on an idea of Farrell and Hsiang [6].

We introduce pre-Mackey and pseudo-Mackey functors in Section 6. A pre-Mackey functor is a just pre-bifunctor D(G) → Ab, and a pseudo-Mackey functor is a pre-Mackey functor which admits a finite filtration by Mackey functors. Such structures have been observed in a number of different contexts: the main examples include the higher Whitehead groups Wh_{n}(ZG), and the structure set of a compact manifold in surgery theory (see [30, Chap. 9]).

It turns out that the general scheme of Dress induction theory can be extended to pseudo-Mackey functors as well. In Section 7, we combine this idea with the Burnside quotient Green ring to study additive functors out of the category (RG, ω)-Morita (see [15, 1C]). We have the corresponding computability result (see Theorem 7.2):

Theorem C. Let F: (RG, ω)-Morita → Ab be an additive functor. Then the composite M = F ∘ d: D(G) → Ab is a pre-Mackey functor. Moreover:

(i) The 2-adic completion of M is 2-hyperelementary computable. (ii) If M is a Mackey functor, then M is hyperelementary computable.

As an application, we conclude from part (i) that the surgery obstruction groups L_{n}(ZG, ω), with arbitrary orientation character ω: G → {±1}, are 2-hyperelementary computable after 2-adic completion (see Example (7.3) for the meaning of computability in this setting). This computability result was certainly expected to be true, but the argument presented here seems to be the first actual proof in the non-oriented case. In the oriented case, where ω is trivial, part (ii) applies to L-theory and the computability is just the result of Dress [4, Theorem 1]. For non-trivial ω, the surgery obstruction group L_{n}(ZG, ω) is a Mackey functor if and only if it has exponent two (see [25], and see [29, 5.2.5] for an example where the L-groups do not have exponent two). In Lemma 7.1 we give a general necessary and sufficient condition on F for part (ii) to apply to M.

Acknowledgement. We were greatly indebted to L. Gaunce Lewis, Jr. for his meticulous reading of our 1990 MSRI preprint, and for his extensive helpful comments. He pointed out to us that the definition of the Burnside quotient Green ring appears in his unpublished notes, The theory of Green functors (1980). This paper is dedicated to his memory.

2. Dress Induction

We will first recall some definitions Dress used in his formulation of induction theory [4].

§2A. Mackey Functors. Let G be a finite group, and let D(G) denote the category whose objects are finite, left G-sets and whose morphisms are G-maps. A Mackey functor is a bifunctor M = (M_{∗}, M^{∗}): D(G) → Ab, where Ab denotes the category of abelian groups and groups homomorphisms, such that M_{∗}(S) = M^{∗}(S) for each object S ∈ D(G), and the following two properties hold:

(M1) For any pullback diagram of finite G-sets

\xymatrix{S \ar[r]^{Ψ} \ar[d]_{Φ} & S_{1} \ar[d]^{φ} \\ S_{2} \ar[r]^{ψ} & T}

the induced maps give an commutative diagram

\xymatrix{M(S) \ar[r]^{Ψ_{M}} & M(S_{1}) \\ M(S_{2}) \ar[r]^{ψ_{M}} \ar[u]^{Φ^{M}} & M(T) \ar[u]_{φ^{M}}}

Here we denote the covariant maps by ψ_{M} and the contravariant maps by φ^{M}.

(M2) The embeddings of S_{1} and S_{2} into the disjoint union S_{1} ⨆ S_{2} define an isomorphism M^{∗}(S_{1} ⨆ S_{2}) → M^{∗}(S_{1}) ⊕ M^{∗}(S_{2}). Let M(∅) = 0.

The property (M1) is the usual double coset formula, and (M2) gives additivity. We remark that for any bifunctor satisfying (M1), the composition M_{∗}(S_{1}) ⊕ M_{∗}(S_{2}) → M_{∗}(S_{1} ⨆ S_{2}) = M^{∗}(S_{1} ⨆ S_{2}) → M^{∗}(S_{1}) ⊕ M^{∗}(M_{2}) = M_{∗}(S_{1}) ⊕ M_{∗}(S_{2}) is just the identity matrix. It follows that any sub-bifunctor of a Mackey functor is Mackey.

Definition 2.1. If M and N are Mackey functors, then a homomorphism M → N of Mackey functors is a natural transformation of bifunctors Θ: M → N such that for each object S ∈ D(G) the function Θ_{S}: M(S) → N (S) is a homomorphism of abelian groups. It is easy to check that the kernel, ker Θ, the image, Im Θ, and the cokernel of Θ are all sub- or quotient Mackey functors of M or N.

§2B. Pairings and Green functors. If M, N, and L are Mackey functors, then a pairing is a family of bilinear maps

M(S) × N (S) → L(S)

indexed by the objects of D(G), such that for any G-map φ: S → T the following formulas hold:

φ^{L}(x · y) = φ^{M}(x) · φ^{N} (y) for x ∈ M(T), y ∈ N (T)
x · φ_{N} (y) = φ_{L}(φ^{M}(x) · y) for x ∈ M(T), y ∈ N (S)
φ_{M}(x) · y = φ_{L}(x · φ^{N}(y)) for x ∈ M(S), y ∈ N (T)

A Green ring is a Mackey functor G together with a pairing G ×G → G, and a collection of elements {1_{S} ∈ G(S)} such that the pairing defines an associative ring structure on each G(S) with unit 1_{S}, and φ^{G}(1_{T}) = 1_{S} for every G-map φ: S → T.

A homomorphism of Green rings Θ: G → K is a homomorphism of Mackey functors such that for each object S ∈ D(G) the function Θ_{S}: G(S) → K(S) is a unital ring homomorphism. If Θ_{S} is injective for each object S ∈ D(G), we say that G is a sub-Green ring of K. If Θ_{S} is surjective for each object S ∈ D(G), then we say that K is a quotient Green ring of G. Similarly, we define sub-quotient Green rings.

If M is a Mackey functor, then M is a Green module over a Green ring G if there exists a pairing G × M → M such that M(S) becomes a left G(S)-module from the pairing, and 1_{S} · x = x for all x ∈ M(S).

Example 2.2. If G → K is a homomorphism of Green rings, then K is a Green module over G under the pairing G × K → K induced by the homomorphism.

§2C. The Burnside ring. For any left G-set S, we let D_{S}(G) denote the category with objects (X, f), where X is a left G-set and f: X → S is a G-map. The morphisms F: (X_{1}, f_{1}) → (X_{2}, f_{2}) are G-maps F: X_{1} → X_{2} such that f_{2} ∘ F = f_{1}. We define a bifunctor

A: D(G) → Ab

by setting A(S) = K_{0}(D_{S}(G)). If φ: S → T is a G-map, then φ_{A}: A(S) → A(T) is the map induced on K_{0} by the composition (X, f) ↦ (X, φ ∘ f). The contravariant map φ^{A}: A(T) → A(S) is induced by the pullback construction applied to S ⟶^{φ} T ⟵^{f} Y, where (Y, f) is an object in D_{T} (G). The conditions (M1) and (M2) are easy to check, and A is a Mackey functor. There is also a pairing A × A → A defined by pullback: let (X_{1}, f_{1}) and (X_{2}, f_{2}) represent elements of A(S), and form the pullback X_{1} ⟶^{f_{1}} S ⟵^{f_{2}} X_{2} considered as G-set over S. This object in D_{S}(G) represents the product, and each A(S) becomes an associative ring with unit element represented by S ⟶^{id} S. The resulting Green ring is called the Burnside ring. Dress also remarks that the Burnside ring is the “universal” Green ring, since it acts on any Mackey functor M. The required pairing A × M → M is defined by pairing an element of A(S) represented by a G-set (X, f) over S, and an element x ∈ M(S), to get f_{M}(f^{M}(x)) ∈ M(S). It is not hard to check that M(S) is a unital A(S)-module under this bilinear pairing, so M is a Green module over A.

We remark that a homomorphism M → N of Mackey functors is compatible with the A-module action, so gives a map of A-Green modules.

If G is a Green ring, the same checks show that G is an A-algebra, implying in particular that a · (x · y) = (a · x) · y for all a ∈ A(S) and all x, y ∈ G(S). It follows that the map ι: A → G defined by a ↦ a · 1_{S}, for all a ∈ A(S), is a (unital) ring homomorphism. Indeed

(a · 1_{S}) · (b · 1_{S}) = a · (1_{S} · (b · 1_{S})) = a · (b · 1_{S}) = (a · b) · 1_{S}

for all a, b ∈ A(S), since G(S) is a A(S)-algebra. It is easy to check from the pairing formulas that ι: A → G is also a homomorphism of Green rings.

§2D. Ideals and quotient Green rings. There is a natural notion of a (left) Green ideal in a Green ring G, namely a sub-bifunctor I ⊂ G such that I(S) ⊂ G(S) is a left ideal in the ring G(S). Similarly, we have right ideals and two-sided ideals. If I ⊂ G is a two-sided Green ideal, then the quotient functor G/I, defined by S ↦ G(S)/I(S), is a Green ring under the quotient pairing inherited from G.

If G × M → M is a Green module structure on a Mackey functor M, then we define the Green ideal I_{M} ⊂ G as the sub-bifunctor of G with

I_{M}(S) = {a ∈ G(S) | φ_{G}(a) · y = 0, ψ^{G}(a) · z = 0}

for all φ: S → T, ψ: U → S, and all y ∈ M(T), z ∈ M(U). Note that elements of I_{M}(S) satisfy additional conditions (both “up” and “down”) beyond just acting trivially on M(S).

The pairing formulas show directly that I_{M}(S) is a two-sided ideal in the ring G(S), for every finite G-set S. We will check that I_{M} is a sub-bifunctor of G by looking at the operations induced by G-maps μ: V → S and λ: S → W on an arbitrary element a ∈ I_{M}(S).

First we consider λ_{G}(a) ∈ G(W). Let φ: W → T and ψ: U → W be any G-maps. We have

φ_{G}(λ_{G}(a)) · y = (φ ∘ λ)_{G}(a) · y = 0

by definition of I_{M}(S). Let

\xymatrix{X \ar[r]^{λ̃} \ar[d]_{ψ̃} & U \ar[d]^{ψ} \\ S \ar[r]^{λ} & W}

be the pullback square, and from (M1) we get

ψ^{G}(λ_{G}(a)) · z = λ̃_{G}(ψ̃^{G}(a)) · z = λ̃_{G}(ψ̃^{G}(a) · λ̃^{M}(z)) = 0

so λ_{G}(a) ∈ I_{M}(W).

Similarly, we must check that μ^{G}(a) ∈ I_{M}(V). Let φ: V → T and ψ: U → V be G-maps, and note that

φ_{G}(μ^{G}(a)) · y = φ_{M}(μ^{G}(a) · φ^{M}(y)) = 0

and ψ^{G}(μ^{G}(a)) · z = (μ ∘ ψ)^{G}(a) · z = 0.

We have now checked that I_{M} ⊂ G is a sub-bifunctor, and therefore I_{M} is a Mackey functor and a two-sided Green ideal in G. We define the quotient Green ring G_{M} = G/I_{M} to be the bifunctor whose value on objects is given by the quotient rings G_{M}(S) = G(S)/I_{M}(S). It is straightforward to check that G_{M} is a Green ring, since the formulas above show that the pairing G×G → G restricts to pairings I_{M}×G → I_{M} and G×I_{M} → I_{M} of Mackey functors. By construction, M is also a Green module over G_{M}.

Definition 2.3. Let M be a Mackey functor. The Burnside quotient Green ring of M is the Green ring A_{M} := A/I_{M}. Let ι_{M}: A → A_{M} denote the epimorphism of Green rings given by the natural quotient map.

Remark 2.4. For G a Green ring, the map ι: A → G defined above by a ↦ a · 1_{S} factors through ι_{G}: A → A_{G}, and we obtain a canonical induced homomorphism of Green rings A_{G} → G. The next result shows that in fact A_{G} = Im ι, which gives a quick alternate definition of A_{G} (for this observation, compare [3, p. 207], [22, p. 253], [14, p. 711], and [1, p. 236]).

Lemma 2.5. Let G be a Green ring. Then the canonical homomorphism of Green rings, A_{G} → G is injective.

Proof. For each G-set S, the natural transformation of bifunctors in the statement maps A_{G}(S) → G(S) by the ring homomorphism a ↦ a · 1_{S}, where a ∈ A(S) and 1_{S} ∈ G(S) is the unit. If a · 1_{S} = 0, and φ: S → T, ψ: U → S are G-maps, then it follows as above that φ_{A}(a) · 1_{T} = 0 and ψ^{A}(a) · 1_{U} = 0, Therefore {a ∈ A(S) | a · 1_{S} = 0} ⊂ I_{G}(S), and the ring homomorphism A_{G}(S) → G(S) is injective. □

We will explore Definition 2.3 by considering the Burnside quotient Green rings for filtrations of Mackey functors.

Definition 2.6. If M and N are Mackey functors, we say that M is a sub-functor of N (respectively N is a quotient functor of M) if there is a natural transformation Θ: M → N such that for each object S ∈ D(G) the function Θ_{S}: M(S) → N (S) is an injective (respectively, surjective) homomorphism of abelian groups. We say that M is a sub-quotient of N if there is a finite sequence of Mackey functors M = L_{0}, L_{1}, …, L_{r} = N such that each L_{i} is either a sub-functor or a quotient functor of L_{i+1}, for i = 0, …, r − 1. Note that the relation “M is a sub-quotient of N ” is a transitive relation.

Example 2.7. If Θ: G → K is a homomorphism of Green rings, then we can regard K as a Green module over G. Furthermore, ker Θ = I_{K} ⊂ G, and there is an induced homomorphism G_{K} → K of Green rings. If K is a quotient Green ring of G, then K = G_{K} = G/I_{K}.

Lemma 2.8. Let G be a Green ring and M a Green module over G. Then the Burnside quotient Green ring A_{M} is a quotient of A_{G}, and isomorphic to a sub-quotient of G.

Proof. Since A_{G} is a sub-Green ring of G, we just need to check that A_{M} is a quotient Green ring of A_{G} under the natural projection from A. This is equivalent to the statement that I_{G} ⊂ I_{M}. Let a ∈ I_{G}(S), and consider G-maps φ: S → T and ψ: U → S. For any y ∈ M(T),

φ_{A}(a) · y = φ_{A}(a) · (1_{T} · y) = (φ_{A}(a) · 1_{T}) · y = 0

since 1_{T} ∈ G(T). Similarly, for any z ∈ M(U),

ψ^{A}(a) · z = ψ^{A}(a) · 1_{U} · z = 0

and we see that a ∈ I_{M}(S). □

Lemma 2.9. Let M and N be Mackey functors, with M a sub-quotient of N. Then there is a surjective homomorphism of Green rings 𝔣: A_{N} → A_{M} such that 𝔣 ∘ ι_{N} = ι_{M}.

Proof. We will establish this result for sub-functors and quotient functors, and note that the general sub-quotient case follows by an inductive argument on the length of the chain joining M to N.

Suppose first that Θ: M → N is a natural transformation, with Θ_{S}: M(S) → N (S) injective for all finite G-sets S. Let a ∈ I_{N} (S) and let φ: S → T, ψ: U → S be G-maps. Then for any y ∈ M(T), Θ_{T} (φ_{A}(a) · y) = φ_{A}(a) · (Θ_{T} (y)) = 0 since Θ is a A-Green module map. Similarly, for any z ∈ M(U), Θ_{U}(ψ^{A}(a) · z) = ψ^{A}(a) · (Θ_{U}(z)) = 0. Since Θ_{T} and Θ_{U} are injective, it follows that a ∈ I_{M}(S), and I_{N} ⊂ I_{M} so that A_{N} maps onto A_{M}.

Next suppose that Θ: N → M is a natural transformation, with Θ_{S}: N (S) → M(S) surjective for all finite G-sets S. If a ∈ I_{N} (S), we check that φ_{A}(a)·y = 0 and ψ^{A}(a)·z = 0, for all y ∈ M(T) and all z ∈ M(U), by using the surjectivity of Θ_{T} and Θ_{U}, and the compatibility of Θ with the A-module structures on M and N. Therefore I_{N} ⊂ I_{M}. □

In general, if M is a sub-Mackey functor of N it is not true that I_{M} ⊂ I_{N}, so there is no natural map in the other direction from A_{M} onto A_{N}, but here is one more situation that works.

We say that M is a full lattice in N if there is a natural transformation Θ: M → N such that the induced maps Θ_{S}^{∗}: Hom(N (S), N (S)) → Hom(M(S), N (S)) are injective for all finite G-sets S. Note that M need not be a sub-Mackey functor of N for this condition to hold.

Lemma 2.10. Let M and N be Mackey functors, with M be a full lattice in N. Then there exists a surjective homomorphism of Green rings 𝔤: A_{M} → A_{N} such that 𝔤 ∘ ι_{M} = ι_{N}. If M is also a sub-functor of N, then 𝔤 is an isomorphism and the inverse to the 𝔣: A_{N} → A_{M} described previously.

Proof. Let φ: S → T be a G-map. For each a ∈ A(S) we can consider the action map y ↦ φ_{A}(a) · y as an element of Hom(N (T), N (T)). However if a ∈ I_{M}(a), this homomorphism is zero on the image of Θ_{T}, and therefore it vanishes identically. Similarly, we check that ψ^{A}(a) · z = 0 for all z ∈ N (U) and any G-map ψ: U → S. Therefore I_{M} ⊂ I_{N}. □

§2E. Amitsur complexes. Dress proves computation results for Mackey functors via the contractibility of certain chain complexes. Let X, Y be finite G-sets, and define a semi-simplicial set Am(X, Y) inductively. Let Am_{0}(X, Y) = Y and Am_{r}(X, Y) = X × Am_{r−1}(X, Y) for r ≥ 1. There are G-maps

d_{i}^{r}: Am_{r}(X, Y) → Am_{r−1}(X, Y)

for 0 ≤ i < r, defined by setting d_{0}^{r} as the projection X × Am_{r−1}(X, Y) → Am_{r−1}(X, Y), and for i > 0 by d_{i}^{r} = 1_{X} × d_{i−1}^{r−1}.

Definition 2.11. Let M be a Mackey functor. For given finite G-sets X, Y, the Amitsur complex M(Am(X, Y)) is the chain bi-complex whose chain group in dimension r is M(Am_{r}(X, Y)), with boundary operators ∂_{r} = ∑(−1)^{i}[d_{i}^{r}]_{M} and δ^{r} = ∑(−1)^{i}[d_{i}^{r}]^{M} for r ≥ 0. and zero otherwise.

This construction has certain naturality properties.

Lemma 2.12. Let M be a Mackey functor. The Amitsur complex gives a bifunctor

M(Am( , )): D(G) × D(G) → Chain(Ab)

where Chain(Ab) denotes the category of chain complexes of abelian groups.

For any Mackey functor M, and any finite G-set S, let M_{S} denote the Mackey functor defined by M_{S}(T) = M(S ×T), for any finite G-set T. There are natural transformations and Θ_{S}^{M}: M → M_{S} and Θ_{M}^{S}: M_{S} → M of Mackey functors induced by the projection maps S × T → T. Dress says that M is S-injective (respectively S-projective) if Θ_{S}^{M} is split-injective (respectively Θ_{M}^{S} is split surjective).

Lemma 2.13 ([4, Prop. 1.1′]). A Mackey functor M is S-injective if and only if it is S-projective.

Proof. Suppose that M is S-projective, so that Θ_{M}^{S} is split-injective. Let Φ: M → M_{S} be a natural transformation such that Θ_{M}^{S}∘Φ = Id_{M} (the identity natural transformation on M). If Δ: S → S × S denotes the diagonal map and p: S × T → T the second factor projection, we notice that

S × T ⟶^{Δ×1} S × S × T ⟶^{1×p} S × T

is just the identity map on S × T. It follows that

Θ_{M(T)}^{S} ∘ (Δ × 1)^{M} ∘ Φ_{S×T} ∘ Θ_{S}^{M(T)} = Id_{M(T)}

for any finite G-set T. One can check that the formula

Φ̃(T) := Θ_{M(T)}^{S} ∘ (Δ × 1)^{M} ∘ Φ_{S×T}

defines a natural transformation of bi-functors splitting Θ_{S}^{M} and hence M is S-injective. The converse is similar. □

Dress now proves that, for any finite G-set Y, both Amitsur complexes (M_{∗}(Am(S, Y), ∂) and (M^{∗}(Am(S, Y), δ) are contractible (we say M is S-computable), whenever M is Sinjective or S-projective. In particular, for Y = • there are exact sequences

\xymatrix{… \ar[r]^{∂_{3}} & M(S × S) \ar[r]^{∂_{2}} & M(S) \ar[r]^{∂_{1}} & M(•) \ar[r] & 0 \\ 0 \ar[r] & M(•) \ar[r]^{δ_{1}} & M(S) \ar[r]^{δ_{2}} & M(S × S) \ar[r]^{δ_{3}} & …}

which exhibit M(•) as a limit of induction or restriction maps respectively.

Here is the main theorem of Dress induction theory:

Proposition 2.14 ([4, Prop. 1.2]). Let G be a Green ring and S be a finite G-set. Then the following conditions are equivalent:

(i) The map φ_{G}: G(S) → G(•) associated to the projection φ: S → • is surjective. (ii) G is S-injective. (iii) All G-modules are S-injective.

This result focuses attention on the task of finding a suitable Green ring which acts on M, and then checking property (i). We remark that the Burnside ring A acts on any Mackey functor, but A is S-injective only if • ⊂ S. Hence the Burnside ring itself has no useful induction properties.

3. Dress generating sets

In the classical Mackey setting, computation is expressed in terms of families. A family of subgroups F of G is a collection of subgroups closed under conjugation and taking subgroups. For any finite G-set X let F(X) denote the family generated by the isotropy subgroups of X. For example, the family F(•) = {All}. Conversely, given a family F of subgroups, we can form the disjoint union X(F) of G-sets G/H, one for each conjugacy class of maximal elements in F, under the partial ordering from subgroup inclusion. For example, X({All}) = •.

We have seen that a good strategy for computing a Mackey functor M is to study the Green rings acting on M. We will apply this strategy to the Burnside quotient Green ring A_{M} of M.

Definition 3.1. Let G be a Green ring. A finite G-set X is a generating set for G if the natural map G(X) → G(•) is surjective (or equivalently, if 1_{•} ∈ Im{G(X) → G(•)}).

It is not true in general that a generating set for a Green ring G is also a generating set for the sub-Green ring A_{G}. To obtain generation for A_{G} it is usually necessary to enlarge the generating set.

For H a finite group and p a prime, let

C_{p}(H) = ⋂{H_{0} ◁ H | H/H_{0} is a p-group}

Notice that C_{p}(H) is a characteristic subgroup of p-power index in H, and C_{p}((C_{p}(H)) = C_{p}(H).

Definition 3.2. Let F be a family of subgroups of G and p a prime. Then hyper_{p}-F is the family consisting of all subgroups H in G such that C_{p}(H) ∈ F. If S is a G-set, then hyper_{p}-S is the corresponding G-set to hyper_{p}-F(S). This construction is due to Dress [4, p. 307].

It is easy to check that hyper_{p}-F is closed under taking subgroups and conjugation, so we obtain a family of subgroups. By construction, there is a G-map X → hyper_{p}-X for any X and hyper_{p}-hyper_{p}-X = hyper_{p}-X. One of Dress’s main results is the following:

Theorem 3.3 ([3, p. 207]). Let M be a Mackey functor. For any prime p, let K(Y) = ker(M(•) ⊗ Z_{(p)} → M(Y) ⊗ Z_{(p)}) and I(Y) = Im(M(hyper_{p}-Y) ⊗ Z_{(p)} → M(•) ⊗ Z_{(p)}), for any finite G-set Y. Then M(•) ⊗ Z_{(p)} = K(Y) + I(Y).

If Y is a finite G-set, we will use the notation ⟨Y ⟩ for the equivalence class of Y in the Dress category D(G). One useful consequence is:

Lemma 3.4. Let G_{0} be a sub-Green ring of G_{1}. For any prime p, and any finite G-set Y with ⟨Y ⟩ = ⟨hyper_{p}-Y ⟩, the natural map G_{0}(Y) ⊗ Z_{(p)} → G_{0}(•) ⊗ Z_{(p)} is surjective if and only if G_{1}(Y) ⊗ Z_{(p)} → G_{1}(•) ⊗ Z_{(p)} is surjective.

Proof. For any Green ring G and any finite G-set Y, the image of G(Y)⊗Z_{(p)} in G(•)⊗Z_{(p)} is an ideal. Hence either map is onto if and only if 1_{G_{i}(•)} is in the image. Since 1_{G_{0}(•)} goes to 1_{G_{1}(•)}, this proves the first implication.

For the converse, notice that the assumption G_{1}(Y) ⊗ Z_{(p)} → G_{1}(•) ⊗ Z_{(p)} is surjective implies that the Amitsur complex is contractible for the restriction maps induced by the transformation Y → •. In particular, G_{1}(•) ⊗ Z_{(p)} → G_{1}(Y) ⊗ Z_{(p)} is injective. Therefore G_{0}(•) ⊗ Z_{(p)} → G_{0}(Y) ⊗ Z_{(p)} is injective, and from Dress’s Theorem 3.3 we conclude that G_{0}(hyper_{p}-Y) ⊗ Z_{(p)} → G_{0}(•) ⊗ Z_{(p)} is surjective. □

Suppose that G is a Green ring which acts on a Mackey functor M. For many applications of induction theory, the “best” Green ring for M is the Burnside quotient Green ring A_{G}. This is a Green ring which acts on M, and by construction A_{G} is a sub-Green ring of G. In particular, the natural map A_{G} → G is an injection.

Definition 3.5. A finite G-set X is a Dress generating set for a Green ring G, provided that G(hyper_{p}-X) ⊗ Z_{(p)} → G(•) ⊗ Z_{(p)} is surjective for each prime p.

By Lemma 3.3, any finite G-set X such that the natural map G(•) → G(X) is injective is a Dress generating set for G. Notice that a Dress generating set for G is also a Dress generating set for any quotient Green ring of G. The following result is the main step in handling sub-Green rings.

Theorem 3.6. A finite G-set X is a Dress generating set for a Green ring G if and only if it is a Dress generating set for the Burnside quotient Green ring A_{G}.

Proof. We apply the result above to Y = hyper_{p}-X, for each prime p, and note that A_{G} is a sub-Green ring of G. □

The Burnside quotient Green ring can be used to compute Mackey functors obtained by sub-quotients.

Definition 3.7. A finite G-set X is a Dress generating set for a Mackey functor M, provided that X is a Dress generating set for the Burnside quotient Green ring A_{M} of M.

This is consistent with our previous Definition 3.5 for a Green ring.

Theorem 3.8. Let G be a Green ring and M, N Mackey functors.

(i) If M is a G-module and X is a Dress generating set for G, then X is a Dress generating set for M.

(ii) If N is a sub-quotient of M and X is a Dress generating set for M, then X is

a Dress generating set for N. (iii) If M is a full lattice in N and X is a Dress generating set for M, then X is a

Dress generating set for N.

Proof. Under the first assumption, A_{M} is a subquotient of G. In the other parts, A_{N} is a quotient of A_{M}.

□

We can translate this into a computability statement as follows:

Corollary 3.9. Let p be a prime and G be a Green ring. Suppose that F is a hyper_{p}- closed family of subgroups of G. Then G ⊗ Z_{(p)} is F-computable if and only if A_{G} ⊗ Z_{(p)} is F-computable.

The advantage of A_{G} over G is that A_{G} acts on Mackey functors which are subfunctors or quotient functors of M but G does not in general. For example, G never acts on A_{G} unless they are equal. We next point out another good feature of the Burnside quotient Green ring.

Theorem 3.10 ([8, Theorem 1.8]). Suppose that G is a Green ring which acts on a Mackey functor M, and F is a hyper_{p}-closed family of subgroups of G. If G ⊗ Z_{(p)} is F-computable, then every x ∈ M(G) ⊗ Z_{(p)} can be written as

x = ∑_{H∈F} a_{H} Ind_{H}^{G}(Res_{G}^{H}(x))

for some coefficients a_{H} ∈ Z_{(p)}, where the a_{H} are the same for all x.

Proof. Since G ⊗ Z_{(p)} is F-computable, we know that A_{G} ⊗ Z_{(p)} is also F-computable. Therefore, we can write 1 = ∑_{K∈F} b_{K} Ind_{K}^{G}(y_{K}), for some y_{K} ∈ A_{G}(K) ⊗ Z_{(p)} and b_{K} ∈ Z_{(p)}. For any x ∈ M(G) ⊗ Z_{(p)} we now have the formula

x = 1 · x = ∑_{K∈F} b_{K} Ind_{K}^{G}(y_{K} · Res_{G}^{K}(x))

But each y_{K} ∈ A_{G}(K)⊗Z_{(p)} can be represented by a sum ∑ c_{KH}[K/H], with c_{KH} ∈ Z_{(p)}, under the surjection A(K) → A_{G}(K). Therefore

x = ∑_{K∈F} b_{K} ∑_{H⊆K} c_{KH} Ind_{K}^{G}([K/H] · Res_{G}^{K}(x))
= ∑_{K∈F} b_{K} ∑_{H⊆K} c_{KH} Ind_{K}^{G}(Ind_{H}^{K}(Res_{K}^{H}(Res_{G}^{K}(x))))
= ∑_{K∈F} b_{K} ∑_{H⊆K} c_{KH} Ind_{H}^{G}(Res_{G}^{H}(x))

We now define a_{H} = ∑_{K∈F} b_{K}c_{KH}, and the formula becomes

x = ∑_{H∈F} a_{H} Ind_{H}^{G}(Res_{G}^{H}(x)) .

□

Example 3.11 (Representation theory). A Dress generating set for a Green ring G need not be a generating set for G. For example, let E denote the finite G-set, E = ∐ G/H, where we have one H for each p-elementary subgroup of G. It is known that E is a generating set for R_{C}(G), but not in general for the rational representation ring R_{Q}(G). On the other hand, complex representations are detected by characters, so any G-set with isotropy containing the cyclic family is a Dress generating set for R_{C}(G), or for the sub-Green ring R_{Q}(G) by Theorem 3.8 (ii). It follows that the hyperelementary family H gives a generating set X_{H} for R_{Q}(G). This implies the Brauer-Berman-Witt induction theorem for rational representations.

Example 3.12 (The Swan ring). The Swan ring is one of the main examples of Green rings in the classical setting of induction theory (see [24]). For any finite group, let SW (G, Z) denote the Grothendieck group of isomorphism classes of finitely-generated left ZG-modules, with [L] = [L′] + [L′′] whenever there is a short exact sequence

0 → L′ → L → L′′ → 0

of such ZG-modules. The operation L ⊗_{Z} L′ gives a ring structure on this Grothendieck group, so we obtain a commutative ring. The usual induction and restriction operations for such modules give the Swan ring the structure of a Mackey functor. We let

SW_{G}: D(G) → Ab

denote the Green ring (in the sense of Dress) defined by SW_{G}(G/H) := SW (H, Z), and extended to D(G) by additivity. Since SW (G, Z) is hyperelementary computable by Swan’s induction theorem (see [3, p. 211]), we see that any Mackey functor on which this Green ring acts is hyperelementary computable.

It follows that the Burnside quotient Green ring of the Swan ring, denoted A_{SW}, also has the hyperelementary set X_{H} as a Dress generating set (or more precisely, any G-set whose isotropy contains the cyclic family is a Dress generating set). In this case, A_{SW} (G/H) ⊂ SW_{G}(G/H) is the subring P (H, Z) ⊂ SW (H, Z) generated by the permutation modules Z[H/K], for all subgroups K ⊆ H.

4. Computation techniques

Dress generating sets can also be used to compute exact sequences of Mackey functors or filtrations of Mackey functors by sub-functors. We say that

M_{0} ⟶^{a} M_{1} ⟶^{b} M_{2}

is an exact sequence of Mackey functors if a and b are homomorphisms of Mackey functors, such that the sequence M_{0}(S) → M_{1}(S) → M_{2}(S) is exact for each finite G-set S. We define long exact sequences in a similar way.

Proposition 4.1. Suppose that M_{0} → M_{1} → M_{2} is an exact sequence of Mackey functors. Then X is a Dress generating set for M_{1} whenever X is a Dress generating set for M_{0} and M_{2}.

Proof. We may assume that M_{0} → M_{1} is injective, and that M_{1} → M_{2} is surjective, and the projections from A induce a natural transformation θ: A_{M_{1}} → A_{M_{0}} ⊕ A_{M_{2}} of Green rings. By exactness, I_{θ}(S) := ker θ_{S} is a nilpotent ideal (of nilpotence index 2). Let Ā_{M_{1}} = A_{M_{1}}/I_{θ} denote the quotient Mackey functor. Since this Mackey functor is mapped injectively by θ into A_{M_{0}} ⊕ A_{M_{2}}, we see that X is a Dress generating set for Ā_{M_{1}}. It follows that Ā_{M_{1}}(hyper_{p}-X)⊗Z_{(p)} → Ā_{M_{1}}(•)⊗Z_{(p)} is surjective for every prime p. But an element in A_{M_{1}}(•) ⊗ Z_{(p)} hitting 1_{•} ∈ Ā(•) ⊗ Z_{(p)} has the form 1 + u, where u ∈ I_{θ}(•) ⊗ Z_{(p)}. Since u^{2} = 0, 1 + u is invertible and (1 + u)^{−1} = 1 − u. If p_{X}: X → • denotes the projection map, and (p_{X})_{∗}(a) = 1 + u, then we have

(p_{X})_{∗}((p_{X})^{∗}(1_{•} − u) · a) = (1_{•} − u) · (p_{X})_{∗}(a) = 1_{•}

and hence X is a Dress generating set for A_{M_{1}}. □

Remark 4.2. In the proof of Proposition 4.1, we have shown for each prime p, there exists an element a ∈ A(hyper_{p}-X) ⊗ Z_{(p)} such that a ↦ 1_{•} in each of the Burnside quotient Green rings A_{M_{i}}(•) ⊗ Z_{(p)}, for i = 0, 1, 2. The same argument extends by induction to finite filtrations of a Mackey functor by sub-bifunctors.

Corollary 4.3. Let 0 = N_{0} ⊂ ⋯ ⊂ N_{r} = M be a filtration of a Mackey functor by sub-bifunctors. Then X is a Dress generating set for M if and only if X is a Dress generating set for each quotient N_{i}/N_{i−1}, for 1 ≤ i ≤ r.

A finite length chain complex of Mackey functors is a sequence (N_{i}, ∂_{i}) of Mackey functors N_{i}, 0 ≤ i ≤ r, and natural transformations ∂_{i}: N_{i} → N_{i−1}, 1 ≤ i ≤ r, such that (N (S), ∂_{∗}) is a chain complexes of abelian groups for each finite G-set S. A chain complex N of Mackey functors has homology groups H_{i}(N), 0 ≤ i ≤ r, which are subquotient Mackey functors of N_{i}.

Corollary 4.4. Suppose that N is a finite length chain complex of Mackey functors. If X is a Dress generating set for each N_{i}, 0 ≤ i ≤ r, then X is a Dress generating set for each of the homology Mackey functors H_{i}(N), 0 ≤ i ≤ r.

Another useful construction is “completion”.

Theorem 4.5. Let M be a Mackey functor, and let F denote a (possibly infinite) filtration

M = F_{0} ⊇ F_{1} ⊇ ⋯ ⊇ F_{r} ⊇ …

of M by sub-bifunctors. A finite G-set X is a Dress generating set for M̂_{F} = \underleftarrow{lim} M/F_{r} if and only if X is a Dress generating set for each quotient Mackey functor F_{r−1}/F_{r}, r ≥ 1.

Proof. Since each F_{r−1}/F_{r} is a sub-quotient of M̂_{F}, the necessity follows from the results above. For sufficiency, we first note by Corollary 4.3 that X is Dress generating set for each quotient M/F_{r}. It is enough to prove that X generates the inverse limit \underleftarrow{lim} A_{M/F_{r}} of the Burnside quotient Green rings for the sequence {M/F_{r}}. Suppose that X is a Dress generating set for each A_{M/F_{r}}, r ≥ 1, and set Y = hyper_{p}-X. If {a_{r}} is a sequence of elements in A_{M/F_{r}}(Y) hitting 1_{•}^{r}, we can use the contractibility of the Y -Amitsur complex for A_{M/F_{r+1}} inductively, to adjust each a_{r+1} by an element of A_{M/F_{r+1}}(Y × Y), so that a_{r+1} ↦ a_{r}. This gives us an element in the inverse limit \underleftarrow{lim} A_{M/F_{r}}(Y) hitting

1_{•} ∈ \underleftarrow{lim} A_{M/F_{r}}(•) := G(•), and hence the Green ring G acts on M̂_{F} with X as a Dress

generating set. Since A_{G} ↠ AM̂_{F} is surjective, it follows that M̂_{F} has X as a Dress generating set. □

Example 4.6. Here is an important special case. Let G be a Green ring acting on a Mackey functor M. If I ⊂ G is a Green ideal, we may filter M by the sub-bifunctors F_{r} = I^{r}M and then M̂_{F} is the I-adic completion of M.

In particular, for a given Mackey functor M we could take I = ⟨I_{M}, 2⟩, and then M̂_{I} is just the 2-adic completion of the Mackey functor M. Note that if M(X) is finitely generated, then M̂_{I}(X) ≅ M(X) ⊗ Ẑ_{2}.

5. Mackey functors and RG-Morita

In [15, 1.A.4] we introduced the category RG-Morita whose basic objects where finite groups H isomorphic to some subquotient of G, and whose morphisms were defined by a Grothendieck construction on the isomorphism classes of finite H_{2}-H_{1} bisets X, for which the order of the left stabilizer

H_{2}I(x) = {h ∈ H_{2} | hx = x}

is a unit in R, for all x ∈ X. Here R is a commutative ring with unit. We set X ∼ X′ if RX is isomorphic to RX′ as RH_{2}-RH_{1} bimodules. The balanced product X ×_{H_{2}} Y of an H_{3}-H_{2} biset X and an H_{2}-H_{1} biset Y is a H_{3}-H_{1} biset. This defines the composition for morphisms. The Add-construction [20, p. 194] is then applied to complete the definition. Many functors arising in algebraic K-theory and topology are actually functors out of RG-Morita, so it is of interest to recognize when these are Mackey functors.

To relate Mackey functors and RG-Morita, we will need the G-Burnside category, A(G), whose objects are subgroups H ⊂ G, and where Hom_{A(G)}(H_{1}, H_{2}) is the Grothendieck construction applied to the isomorphism classes of finite bifree H_{2} -H_{1} bisets (meaning both left and right actions are free). Because of the Grothendieck construction, A(G) is an Ab–categories: the morphism sets are abelian groups and the compositions are bilinear [20, I.8, p. 28]. Let

u: A(G) → A_{•}(G)

denote the associated universal free additive category, and the universal inclusion (see [20, VII.2, problem 6, p. 194]).

The morphisms in A(G) are defined by the Grothendieck construction with addition operation the disjoint union of bisets. By convention, the empty biset ∅ represents the zero element. Composition comes from the balanced product:

H_{3}X_{H_{2}} ∘_{H_{2}}X_{H_{1}} = (_{H_{3}}X_{H_{2}}) ×_{H_{2}} (_{H_{2}}X_{H_{1}}).

The reader should check that this is well–defined on isomorphism classes of bisets and “bilinear” in that

(_{H_{3}}X_{H_{2}} ⨆_{H_{3}}Y_{H_{2}}) ∘_{H_{2}}X_{H_{1}} ≅ (_{H_{3}}X_{H_{2}} ∘_{H_{2}}X_{H_{1}}) ⨆ (_{H_{3}}Y_{H_{2}} ∘_{H_{2}}X_{H_{1}})

with a similar formula for disjoint union on the right. The morphisms in A_{•}(G) are matrices of morphisms in A(G).

Definition 5.1. We define a contravariant involution τ: A(G) → A(G), by the identity on objects, and on morphisms it is the map induced on the Grothendieck construction by the function which takes the finite bifree H_{2} -H_{1} biset _{H_{2}}X_{H_{1}} to the finite bifree H_{1} -H_{2} biset _{H_{1}}X_{H_{2}} which is X as a set and h_{1} · x · h_{2} is defined to be h_{2}^{−1}xh_{1}^{−1}.

The reader needs to check that isomorphic bisets are isomorphic after reversing the order, and should also check that the transpose conjugate of a disjoint union is isomorphic to the disjoint union of the conjugate transposes of the pieces. This means that τ is a functor which induces a homomorphism of Hom–sets. It is clearly an involution, not just up to natural equivalence. Since τ is a homomorphism on Hom-sets, it induces an additive contravariant involution τ^{•}: A_{•}(G) → A_{•}(G), called conjugate transpose, which commutes with the functor u. By definition, τ^{•} acts on a matrix of morphisms by applying τ to each entry, and then transposing the matrix. There is a functor

a: A_{•}(G) → RG-Morita

given by the inclusion on objects and morphisms (but the equivalence relation on morphisms is different in RG-Morita).

There is a functor A_{•}(G) → R-Morita, called the R-group ring functor, where R-Morita has objects R-algebras and morphisms defined by stable isomorphism classes of bimodules (see [15, 1.A.1]). This functor factors through RG-Morita: it sends H ↦ RH on objects, and X ↦ RX on morphisms.

We will define the following diagram of categories and functors:

(5.2)  \xymatrix{ & A_{•}(G) \ar[d]^{a} \ar[dr]^{R-group ring} \\ D(G) \ar[r]^{d} \ar@{-->}[ur]^{j} & RG-Morita \ar[r] & R-Morita}

To complete the definition of the functors in this diagram, we need to introduce another category. Let D_{∗}(G) denote the category whose objects are pairs (X, b), consisting of a finite G-space X and an ordered collection b = (b_{1}, ⋯, b_{n}) of base-points, one for each G-orbit of X. The morphisms are the G-maps (not necessarily base-point preserving). There is a functor

μ: D_{∗}(G) → D(G)

defined by forgetting the base-points. Since every object of D(G) is isomorphic to the image μ(X, b) of an object of D_{∗}(G), and μ induces a bijection on morphism sets, it follows that μ gives an equivalence between the categories D_{∗}(G) and D(G), with inverse functor μ′ [20, IV.4, Theorem 1, p. 91].

We can now define two functors

(j_{•}, j^{•}): D_{∗}(G) → A_{•}(G).

The covariant functor j_{•} is the additive extension of the functor which sends an object (G/H, eH) to the isotropy subgroup H, and sends the G map f: G/H → G/K to the biset _{K}K_{g^{−1}Hg}, where f(eH) = gK. If we change the coset representative and write f(eH) = g_{1}K, then the map

(5.3)  ψ: _{K}K_{g^{−1}Hg} → _{K}K_{g_{1}^{−1}Hg_{1}}

defined by ψ(k) = k(g^{−1}g_{1}) gives an bijection of K-K bisets. Note that 1_{G/H}: G/H → G/H goes to_{H}H_{H} which is the identity. Check that if f_{1} : G/H_{1} → G/H_{2} and f_{2} : G/H_{2} → G/H_{3} are G-maps and if f_{1}(eH_{1}) = g_{1}H_{2} and f_{2}(eH_{2}) = g_{2}H_{3} then f_{2}∘f_{1}(eH_{1}) = (g_{1}g_{2})H_{3} and

H_{3}(H_{3})_{g_{2}^{−1}H_{2}g_{2}} ×_{H_{2}} _{H_{2}}(H_{2})_{g_{1}^{−1}H_{1}g_{1}}

is isomorphic to _{H_{3}}(H_{3})_{(g_{1}g_{2})^{−1}H_{1}(g_{1}g_{2})} by the map (h_{3}, h_{2}) ↦ h_{3}g_{2}^{−1}h_{2}g_{2}.

The contravariant functor j^{•} agrees with j_{•} on objects, but sends the G map f: G/H → G/K to the biset _{g^{−1}Hg}K_{K} where f(eH) = gK. Rather than check identity and composition directly, just note that _{g^{−1}Hg}K_{K} is isomorphic to τ(_{K}K_{g^{−1}Hg}) by the function which sends k to k^{−1}, so j^{•} = τ ∘ j_{•} and hence j^{•} is a contravariant functor.

Definition 5.4. We define the bifunctor

j: D(G) → A_{•}(G)

as the composition j = (j^{•}, j_{•}) ∘ μ′. Let

d = a ∘ j: D(G) → RG-Morita

denote the composition in diagram (5.2). □

For any additive functor F: A_{•}(G) → Ab, the composition F ∘ j: D(G) → Ab is a Mackey functor (see [17]). Our main application is the following:

Theorem 5.5. Any additive functor F: RG-Morita → Ab gives a Mackey functor on D(G) by composition with d: D(G) → RG-Morita. Any such Mackey functor is hyperelementary computable.

Proof. The functor d: D(G) → RG-Morita factors through A_{•}(G), so we obtain Mackey functors by composition. We will show that any such Mackey functor, M, is a Green module over the Burnside quotient Green ring A_{SW} of the Swan ring, and then apply Example 3.12. Let L = Z[H/K] denote a permutation module, for some subgroups K ⊂ H of G, and let X denote an H-H-biset, which is free as a left H-set. Then H/K × X is again an H-H-biset by the formula h_{1}(hK, x)h_{2} = (h_{1}hK, h_{1}xh_{2}), for all h, h_{1}, h_{2} ∈ H and all x ∈ X. Note that R[H/K × X] = L ⊗_{Z} RX as RH-RH bimodules, so this construction applied to X = _{H}H_{H}, sending Z[H/K] ↦ H/K × H, gives a welldefined homomorphism

P (H, Z) → Hom_{RG-Morita}(H, H)

from the Grothendieck group of permutation modules, for each subgroup H ⊂ G. The adjoints of these homomorphisms give a pairing A_{SW} × M → M, and the Green module properties follow easily from bimodule identities (compare [22, 11.2]). Since A_{SW} is hyperelementary computable, we conclude that any Mackey functor out of R-Morita is hyperelementary computable. □

Example 5.6 (Controlled topology). The bounded categories C_{M,G}(R) of [10, §4], and the continuously controlled categories B_{G}(X × [0, 1); R) of [11, §6] are functors out of A_{•}(G), for any finite group G, and hence any additive functor from these categories to abelian groups gives a Mackey functor on D(G).

Example 5.7 (Farrell-Hsiang induction). There is a useful extension of induction theory to (possibly) infinite groups, due to Farrell and Hsiang [5]. Given any representation pr: Γ → G, with G finite, we get a new R-group ring functor A(G) → R-Morita by sending G/H ↦ R[Γ_{H}], where Γ_{H} = pr^{−1}(H) is the pre-image of H in Γ. We have a generating set for the morphisms Hom_{A(G)}(H_{1}, H_{2}) consisting of the bisets H_{2} ×_{K} H_{1}, where K ⊂ H_{2} × H_{1} is a subgroup (see [15, 1.A.9]). We send the biset H_{2} ×_{K} H_{1} to the bimodule R[Γ_{H_{2}}]⊗_{R[Γ_{K}]}R[Γ_{H_{1}}]. By composition with any additive functor F: R-Morita → Ab, we again obtain Mackey functors. Since the Swan ring acts on R-Morita (by tensor product as above), any such Mackey functor is a a Green module over the Swan ring, and we obtain hyperelementary computation as before. The main examples are listed in [15, 1.A.12], including Quillen K-theory K_{n}(R[Γ]).

Remark 5.8. An alternate (and slightly sharper) formulation of this example could be given by defining RΓ-Morita for any discrete group Γ: the objects are finite groups H isomorphic to some subquotient H ≅ Γ_{1}/Γ_{0} of Γ, where Γ_{0} ◁ Γ_{1} and Γ_{1} is finite index in Γ. The morphisms are H_{2}-H_{1} bisets as before. Then from any representation pr: Γ → G, where G is finite, we get a functor d: D(G) → RΓ-Morita and Theorem 5.5 holds in this new setting.

Example 5.9 (Cohomotopy). The bifunctor which assigns to H ⊂ G the 0th stable cohomotopy group π^{0}(BH) is a Green ring, which is easily checked to have the Sylow family as a contracting family. Hence any Mackey functor derived from the cohomology of a space and its finite coverings has the Sylow family as a contracting family. More explicitly, let X, x_{0} be a space (semi-locally 1-connected) and a representation π_{1}(X, x_{0}) → G, define X(H) for any H ⊂ G as the corresponding cover. Given any homology theory E_{∗} or cohomology theory, E^{∗}, the assignment H ↦ E_{∗}(X(H)) or H ↦ E^{∗}(X(H)) are Mackey functors which are modules over the Green ring H ↦ π^{0}(BH).

Lam, [18, Ex. 2.1 and Remark 2.4, p. 108-109], shows that cohomology (ordinary or Tate) with twisted coefficients H^{i}(?; M) is a Mackey functor on D(G) where M is a fixed G-module. Since the cohomotopy Green ring acts on this Mackey functor, it is Sylow computable. If pr: Γ → G is a homomorphism and M is a Γ-module, then H^{i}(pr^{−1}(?); M) is also a Mackey functor on D(G) with the cohomotopy Green ring acting. An interesting example of this situation is Galois cohomology.

6. Pseudo-Mackey functors and pseudo-complexes

We wish to apply the computation strategy described above to a more general situation. The main examples of interest are the higher Whitehead groups Wh_{n}(ZG) and the nonoriented surgery obstruction groups L_{n}(ZG, ω).

Definition 6.1. A covariant pre-functor f: D → E between two categories is just a function S ↦ ob(f)(S) on objects, and a function

hom(f): Hom_{D}(S_{1}, S_{2}) → Hom_{E}(ob(f)(S_{1}), ob(f)(S_{2}))

on Hom-sets. A functor is a pre-functor which preserves identities and compositions. Similarly, we define a contravariant pre-functor, and a pre-bifunctor then consists of a pair (f_{∗}, f^{∗}) of pre-functors, where f_{∗} is covariant, f^{∗} is contravariant, and ob(f_{∗}) = ob(f^{∗}). We call these Mackey pre-functors if D = D(G) and E = Ab.

A pre-natural transformation T: f_{1} → f_{2} is a function

S ↦ T (S) ∈ Hom_{E}(ob(f_{1})(S), ob(f_{2})(S)) .

A natural transformation of (covariant) pre-functors is a pre-natural transformation T: f_{1} → f_{2} such that the diagram

[figure: ob(f_{1})(S_{1}); hom(f_{1})(ϕ); T (S_{1}); ob(f_{1})(S_{2}); T (S_{2}); ob(f_{2})(S_{1}); hom(f_{2})(ϕ); ob(f_{2})(S_{2})]

commutes for all pairs of objects S_{1}, S_{2} ∈ D and all ϕ ∈ Hom_{D}(S_{1}, S_{2}). There is a similar definition for (pre-)natural transformations of contravariant pre-functors, and a natural transformation of pre-bifunctors is a single function which is natural transformation for both the covariant and contravariant parts of the bifunctor. A pre-pairing between three Mackey pre-functors M, N and L is a collection of functions μ(S): M(S)×N (S) → L(S). Finally, if M → N is an injective natural transformation of Mackey pre-functors, then we say that M is a sub Mackey pre-functor of N.

Note that if M: D(G) → Ab is a Mackey pre-functor, we can apply M to any of the Amitsur complexes Am(X, Y), and obtain ∂_{r} and δ^{r} maps as usual, but we can not be sure that ∂_{r} ∘ ∂_{r+1} = 0 or δ^{r+1} ∘ δ^{r} = 0. We call M(Am(X, Y)) a pre-Amitsur complex. This construction gives a pre-functor D(G) × D(G) → Chain(Ab).

Definition 6.2. A Mackey pre-functor M is called a pseudo-Mackey functor provided that there exists a finite collection of sub Mackey pre-functors 0 = N_{0} ⊂ N_{1} ⊂ ⋯ ⊂ N_{r} = M and the quotient pre-bifunctors N_{i}/N_{i−1} are actually Mackey functors, for 1 ≤ i ≤ r. The collection {N_{i}/N_{i−1} | 1 ≤ i ≤ r} will be called the associated graded Mackey functor to M.

A natural transformation M → N of pseudo-Mackey functors is a natural transformation of Mackey pre-functors which preserves the filtrations. Notice that the Burnside ring A “acts” on a Mackey pre-functor via the usual formula (which gives a pre-pairing). The action of A on a pseudo-Mackey functor M preserves the filtration, and the induced action on the sub-quotients N_{i}/N_{i−1} is the usual action.

We say that a finite G-set X is a Dress generating set for a pseudo-Mackey functor M provided X is a Dress generating set for each of the Mackey functors N_{i}/N_{i−1} in its associated filtration. This agrees with our previous definitions if M is a Mackey functor filtered by Mackey sub-functors. Notice that the image of the natural map of Green rings A → ⨁_{i=1}^{r} A_{N_{i}/N_{i−1}} is a Green ring with X as a Dress generating set. It follows that there exists an element a ∈ A(hyper_{p}-X) ⊗ Z_{(p)}, for each prime p, whose image in A(•) ⊗ Z_{(p)} acts as 1_{•} on each sub-quotient N_{i}/N_{i−1}(•) ⊗ Z_{(p)}, 1 ≤ i ≤ r.

Lemma 6.3. Suppose that M_{0} → M_{1} and M_{1} → M_{2} are natural transformations of Mackey pre-functors, such that M_{0}(Y) → M_{1}(Y) → M_{2}(Y) is exact for every finite Gset Y. If M_{0} and M_{2} are pseudo-Mackey functors, then M_{1} is a pseudo-Mackey functor. Moreover, if X is a Dress generating set for M_{0} and M_{2}, then X is a Dress generating set for M_{1}.

Proof. The pre-image of the associated filtration for M_{2} gives a filtration N_{0} ⊂ N_{1} ⊂ ⋯ ⊂ N_{r} = M_{1}, with M_{0} ⊂ N_{i} for 0 ≤ i ≤ r. Since a sub-bifunctor of a Mackey functor is Mackey, we see that the quotient pre-functors N_{i}/N_{i−1} are actually Mackey functors (and they all have Dress generating set X by Theorem 3.8). Now we extend this filtration by adjoining the associated filtration for M_{0}. Since each of the sub-quotients in this extended filtration have Dress generating set X, the result follows. □

We also get a computational result for pseudo-Mackey functors. The Amitsur precomplex (M_{∗}(Am(X, Y), ∂_{∗}) is now a pseudo-complex, meaning that the boundary maps ∂_{∗} are filtration-preserving (and the associated graded is an actual complex). It will be called pseudo-contractible if it is equipped with degree +1 filtration-preserving natural transformations

s_{r}: M(Am_{r}(X, Y)) → M(Am_{r+1}(X, Y))

of pre-functors, for r ≥ 0, which contract the Amitsur complexes for the associated graded Mackey functors to M. The collection s_{∗} = {s_{r}} is called a pseudo-contraction. We make a similar definition for the cochain Amitsur complex and the degree -1 cochain pseudocontractions σ^{r}.

We can construct pseudo-contractions by using any element a ∈ A(X) such that a acts as 1_{•} ∈ on each sub-quotient N_{i}/N_{i−1}(•) ⊗ Z_{(p)}, 1 ≤ i ≤ r, to build “chain homotopies” s_{r}(a) and “cochain homotopies” σ^{r}(a). These are pseudo-contractions in the above sense.

Proposition 6.4. Let M be a pseudo-Mackey functor, and X, Y finite G-sets. If (M_{∗}(Am(X, Y), ∂_{∗}) is pseudo-contractible with pseudo-contraction s_{∗}, then there are canonical filtration-preserving natural transformations (∂′_{∗}, s′_{∗}) for which (M_{∗}(Am(X, Y), ∂′_{∗}) is a chain complex, and s′_{∗} is a chain contraction. If the pseudo-complex was already a complex, ∂′_{∗} = ∂_{∗}, and if in addition s_{∗} was already a contraction, then s′_{∗} = s_{∗}.

Proof. Let (C_{i}, ∂_{i}, s_{i}) be our data, where ∂_{i} and s_{i} are natural transformations. We assume that for i < r, ∂_{i} ∘ ∂_{i+1} = 0, ∂_{i+1} = ∂_{i+1} ∘ s_{i} ∘ ∂_{i+1}, and s_{i−1} ∘ ∂_{i} + ∂_{i+1} ∘ s_{i} = 1_{C_{i}}. For r ≤ 0 these identities clearly hold. We proceed to show how these conditions may be achieved for i = r by modifying ∂_{r+1} and s_{r} (if necessary). Throughout the inductive construction, we do not change the maps induced by (∂_{∗}, s_{∗}) on the Amitsur complex for the associated graded Mackey functor to M. We also note that the process does not change the given ∂_{1} : C_{1} → C_{0}, but may change s_{0} in the first step.

First, let ∂′_{r+1} = ∂_{r+1} − s_{r−1} ∘ ∂_{r} ∘ ∂_{r+1}. Then ∂_{r} ∘ ∂′_{r+1} = 0 and if ∂_{r} ∘ ∂_{r+1} = 0 we have ∂′_{r+1} = ∂_{r+1}. Note that both ∂′_{r+1} and ∂_{r+1} preserve the induced filtration from M, and induce the same map on the Amitsur complexes for the associated graded Mackey functor to M.

Next, we modify s_{r}. Let ψ_{r} = s_{r−1} ∘ ∂_{r} + ∂′_{r+1} ∘ s_{r}. By construction, ψ_{r} preserves the filtration and induces the identity on the associated graded. Hence, ψ_{r} = 1_{C_{r}} + u, where u is nilpotent, and ψ_{r} is invertible. Since ∂_{r} ∘ ψ_{r} = ∂_{r}, we can set s′_{r} = s_{r} ∘ ψ_{r}^{−1} and obtain s_{r−1} ∘ ∂_{r} + ∂′_{r+1} ∘ s′_{r} = 1_{C_{r}} by pre-composing with ψ_{r}. Notice that if s_{r} was already part of a chain contraction, then we do not alter it. It follows that ∂′_{r+1} = ∂′_{r+1} ∘ s′_{r} ∘ ∂′_{r+1} and the induction step is complete. The naturality of ∂′_{r} and ψ_{r} follow inductively from the explicit formulas. The naturality of ψ_{r} implies the naturality of s′_{r} for use at the next step of the induction. Since no choices were involved in the construction of (∂′_{∗}, s′_{∗}), the new maps are canonically determined by the original data (∂_{∗}, s_{∗}). □

Remark 6.5. After this process, the new contractible complex gives an expression for M(Y) as a direct summand of M(X × Y), with respect to the original induction map ∂_{1} : M(X × Y) → M(Y), and the new “restriction” map s′_{0} : M(Y) →: M(X × Y), since ∂_{1} ∘ s′_{0} = id. In this situation, we say that M(Y) is computed from the family F(X). If M was actually a Mackey functor, computability is this sense would agree with the notion previously defined. Similar remarks apply to the contravariant version M^{∗}(Am(X, Y), δ^{∗}).

We will also need a slight extension of this result. A filtered pre-complex (C, ∂) is a pre-complex of abelian groups equipped with a filtration

C = F_{0}C ⊃ F_{1}C ⊃ F_{2}C ⊃ …

where each F_{i}C is a pre-subcomplex of (C, ∂), meaning that ∂_{r}(F_{i}C_{r}) ⊆ F_{i}C_{r−1}, for all i, r. We say that (C, ∂) is a pseudo-complex if the additional relation ∂_{r} ∘ ∂_{r+1} = 0 holds, for all r, on each subquotient F_{i}C/F_{i+1}C. We say that a pseudo-complex has a pseudo-contraction s_{∗} = (s_{r}) provided that s_{r}(F_{i}C_{r}) ⊆ F_{i}C_{r+1}, and s_{∗} induces an actual contraction on each subquotient complex F_{i}C/F_{i+1}C.

A pseudo-complex (C, ∂) has a natural completion

(C, ∂) ⟶ \underleftarrow{lim} C/F_{i}C := (Ĉ, ˆ∂)

given by the inverse limit pre-complex with respect to the natural projections C → C/F_{i}C, i ≥ 0. A pseudo-contraction s_{∗} of (C, ∂) induces a pre-contraction ŝ_{∗} of (Ĉ, ˆ∂).

Proposition 6.6. Let (C, ∂) be a pseudo-complex with filtration {F_{i}C | i ≥ 0}. If (C, ∂) admits a filtered pseudo-contraction s_{∗}, then there exists canonical data for which (Ĉ, ∂′_{∗}, s′_{∗}) is a contracted chain complex. If the pseudo-complex was already a complex, ∂′_{∗} = ˆ∂_{∗}, and if in addition s_{∗} was already a contraction, then s′_{∗} = ŝ_{∗}.

Proof. The proof follows the same outline as for Proposition 6.4, but we notice that the map ψ_{r} = 1_{C_{r}} + u has the additional property that u^{i+1} = 0 on the quotient C_{r}/F_{i}C_{r}.

This follows by induction from the exact sequences

0 → F_{i+1}C/F_{i}C → F_{0}C/F_{i+1}C → F_{0}C/F_{i}C → 0

of pseudo-contractible complexes. Then ψ_{r} induces an invertible map on C_{r}/F_{i}C_{r}, for each i ≥ 0. We define s′_{r} = s_{r} ∘ ψ_{r}^{−1} on C_{r}/F_{i}C_{r} as before. By induction, we have constructed contraction data (C/F_{i}C, ∂′, s′), for each i ≥ 0. In addition, this contraction data is compatible with the projections C/F_{i+1}C → C/F_{i}, and hence induce contraction data (Ĉ, ∂′, s′) for the inverse limit complex. □

Remark 6.7. Once again, this process doesn’t change ∂_{1}, so the new contractible complex gives an expression for Ĉ_{0} as a direct summand of Ĉ_{1}, with respect to completion of the original boundary map ∂_{1} : C_{1} → C_{0}.

Example 6.8 (Whitehead groups). Define the Whitehead groups, W h_{n}(ZG), as the homotopy groups of the spectrum which is the cofibre of the Loday assembly map

BG^{+} ∧ K(Z) → K(ZG).

The Loday assembly map is a map of bifunctors, [21, Main Thm, p. 223], and the Whitehead groups are bifunctors. Furthermore, the W h_{n}, n ≤ 3, are Mackey functors, but it is not obvious from this description that the other higher Whitehead groups are actually Mackey functors. However, from the long exact sequence in homotopy theory we see that they are pseudo-Mackey functors. From Example 3.12, Example 5.9 and Proposition 6.4, we see that the W h_{n}(ZG) are computed by the hyperelementary family. Similarly, the W h_{n}(ZG) ⊗ Z_{(p)} are computed by the p-hyperelementary family (see [24], [18], and [21] for partial results in this direction).

Example 6.9 (Tate cohomology). The Tate cohomology of W h_{n} or Quillen’s K_{n} are bifunctors which are subquotients of W h_{n} or K_{n}, and hence are computed by the hyperelementary family. The localization maps W h_{n} → W h_{n}⊗Z_{(2)} and K_{n} → K_{n}⊗Z_{(2)} induce isomorphisms on Tate cohomology. Hence the Tate cohomology is computed by the 2hyperelementary family. Given any pseudo-Mackey subfunctor of W h_{n} or K_{n} which is invariant under the involution, we can form the Tate cohomology and this Tate cohomology functor is computed by any family which contains the 2-hyperelementary family. □

7. Surgery obstruction groups

In [4, Theorem 1], Dress claims computability results for “any of the L-functors defined by C. T. C. Wall” in [29]. However, the non-oriented L-groups L_{n}(ZG, ω) are not always Mackey functors, and so the techniques described by Dress in [4] do not appear to be adequate to prove the result in this generality. The point is that an inner automorphism by an element g ∈ G with ω(g) = −1 induces multiplication by −1 (which may not be the identity) on L_{n}(ZG, ω) (see [25]). One of the main applications of our more general techniques is to supply a proof that non-oriented L-theory is hyperelementary computable, in the sense that L_{n}(ZG, ω) is the limit of restrictions or inductions involving hyperelementary subgroups of G.

Fix a finite group G, and the geometric antistructure for which θ = id and b = e ∈ G, [15, 1.B.3]. Let ω: G → {±1} be a fixed orientation homomorphism, and for each subgroup H ⊂ G let ω_{H} = ω|_{H}. We define the following categories:

(i) A(G, ω), with objects finite groups H isomorphic to some subgroup of G, and morphisms given by a Grothendieck group construction on finite biset forms (X, ω_{X}) (see [15, p. 256] for the definition). We construct A_{•}(G, ω) by taking the additive completion.

(ii) (R, −)-Morita, with objects and morphisms as defined in [15, 1.B.2], and the quotient category (R, −)-Witt from [15, 1.C.2], for any commutative ring R with unit.

(iii) (RG, ω)-Morita, with objects H isomorphic to some subquotient K/N of G, with N ⊂ ker ω, and morphisms given by the Grothendieck group construction on finite biset forms (X, ω_{X}), modulo an equivalence relation, as defined in [15, 1.B.3]. We can define the analogous quotient category (RG, ω)-Witt by setting metabolic forms (see [15, p. 254]) to zero in the morphisms.

Notice that by forgetting the orientation map ω we get functors into the categories discussed in Section 5. The construction of Definition 5.4 gives a pre-bifunctor

j: D(G) → A_{•}(G, ω)

extending the pre-functor Or(G) → A_{•}(G, ω) out of the orbit category, defined on objects by G/H ↦ H and on morphisms by sending the G-map f: G/H → G/K, given by f(eH) = gK, to the biset form (_{K}K_{g^{−1}Kg}, ω_{K}). This definition depends on the choice of coset representative g for the morphism f in Or(G), since this time, if x ∈ K and ω(x) = −1, the two morphisms eH ↦ gK and eH ↦ gxK are sent to different biset forms.

Lemma 7.1. In A_{•}(G, ω), the morphism [_{H}H_{x^{−1}Hx}, ω_{H}] = ω(x) · id for all x ∈ H. If F: A_{•}(G, ω) → Ab is an additive functor, then

F ∘ j: A_{•}(G, ω) → Ab

is a Mackey pre-functor, which is a Mackey functor if and only if all the inner automorphism morphisms F (_{H}H_{x^{−1}Hx}, ω_{H}) = id, for all x ∈ H.

Proof. The identity morphism in A_{•}(G, ω) is represented by the biset form (_{H}H_{H}, ω_{H}). The map ψ: _{H}H_{H} → _{H}H_{xHx^{−1}} of biset forms defined by ψ(h) = hx^{−1}, see (5.3), induces an isometry of biset forms (_{H}H_{x^{−1}Hx}, ω_{H}) ≅ (_{H}H_{H}, ω(x) · ω_{H}) and hence

[_{H}H_{x^{−1}Hx}, ω_{H}] = ω(x) · id

in the Grothendieck group of morphisms of A_{•}(G, ω).

The property (M1) depends on conjugations acting trivially, or in other words,should induce F (ψ) = id for all x ∈ H (including those with ω(x) = −1). □

The R-group ring functor of [15, 1.B.4] induces a functor from A_{•}(G, ω) to (RG, ω)-Morita or further into (R, −)-Witt. The required formulas are in section 1.B of [15], including the remark that since our morphisms are formed via a Grothendieck construction, we are entitled to equate metabolics on isomorphic modules. There is a functor a: A_{•}(G, ω) → (RG, ω)-Morita as before, and we let

d: D(G) → (RG, ω)-Morita

be the pre-bifunctor d = a ∘ j. There is a homomorphism from the Dress ring

GU(H, Z) → Hom_{(R,−)-Morita}(H, H)

given by tensor product (see [4] where it is asserted that GU(G, Z) acts on L-theory, or [12, p. 143] for explicit formulas). Dress [4] showed that the the hyperelementary family contracts the Dress ring. We observe that the same formulas give an action of the Burnside quotient Green ring A_{GU} on (RG, ω)-Morita.

Theorem 7.2. Let F: (RG, ω)-Morita → Ab be an additive functor. Then F ∘d: D(G) → Ab is a pre-Mackey functor, and the 2-adic completion of any such pre-Mackey functor is 2-hyperelementary computable. If M = F ∘ d is a Mackey functor, then M is hyperelementary computable.

Proof. In the oriented case (ω ≡ 1) the pre-functor M is actually a Mackey functor, by Proposition 7.1. More generally, whenever M = F ∘ d is a Mackey functor the result follows as in Theorem 5.5, since M is a Green module over A_{GU}. By [4, Theorem 3], and Theorem 3.6, the Burnside quotient Green ring of the Dress ring is hyperelementary computable.

In the non-oriented case, we define a filtration F_{i} = 2^{i}F, i ≥ 0, with F_{0} = F, and note that the sub-quotients (F_{i}/F_{i+1}) ∘ d are Mackey functors. Now we let (C, ∂) denote the filtered Amitsur pseudo-complex for F_{∗} ∘ d with respect to 2-hyperelementary induction, and the result follows from Proposition 6.6. Notice that the passage from a pseudocontractible pseudo-complex to a contractible complex does not change the first boundary map, so F ∘ d is 2-adically detected (generated) by the given restriction (induction) maps to the 2-hyperelementary subgroups. □

Example 7.3 (Non-oriented L-theory). The main example for us is the surgery obstruction group L_{n}(ZG, ω). It is a foundational result of Wall [28] that the surgery obstruction groups for finite groups are finitely-generated, with 2-primary torsion exponent. Theorem 7.2 computes L_{n}(ZG, ω) ⊗ Ẑ_{2} as a limit (and as a colimit) over the 2-hyperelementary subgroups H ⊂ G, H ∈ H. These limits use the standard induction or restriction maps, e.g. for induction we have the surjective map

∂_{1} : ⨁_{H∈H} L_{n}(ZH, ω) ⊗ Ẑ_{2} → L_{n}(ZG, ω) ⊗ Ẑ_{2}

and our contraction data gives the relation subgroup ker ∂_{1} = Im ∂′_{2}.

We conclude that L_{n}(ZG, ω) is also effectively 2-hyperelementary computable: the torsion subgroup is isomorphic to that of L_{n}(ZG, ω) ⊗ Ẑ_{2}, and the divisibility of the signatures is computable since the kernel and cokernel of the natural transformation

L_{n}(ZG, ω) → L_{n}(ℝG, ω)

of pseudo-Mackey functors are both 2-primary torsion groups (see [28, 7.3, 7.4]). The groups L_{n}(ℝG, ω) were computed explicitly by Wall [29, 2.2.1] in terms of the irreducible characters of G. The proof of computability given here applies in the oriented case (ω ≡ 1), but in that case the L-group is a Mackey functor and the argument is essentially the same as the one given by Dress. Other important examples were listed in [15, 1.B.8].

Example 7.4 (L-theory with decorations). Let R be a commutative ring with unit, and consider any L-group L_{n}^{B}(RG, ω) for RG with anti-involution given by ω: G → {±1} with decoration in any involution invariant sub-bifunctor, B, of K_{i}(ZG) or W h_{i}(ZG), i ≤ 1 (see [12] for a summary of the definitions). It was checked in [12, Thm. 5.3, Cor. 5.5 and Ex. 5.14] that the corresponding round L-theories are functors out of (ZG, ω)- Morita. Hence these L-theories are pseudo-Mackey functors and are contracted by the hyperelementary family. It was also checked in [12, Prop 5.6, Cor. 5.7 and Ex. 5.14] that the corresponding ordinary L-theories are functors out of (ZG, ω)-Morita, so the same computation result holds.

Example 7.5 (Localization). Dress [4] shows that the Dress ring is contracted by any family containing the 2-hyperelementary and p-elementary families. More precisely, he showed that the 2-localization of the Dress ring is contracted by the 2-hyperelementary family, and the p-localization, p odd, is contracted by the p-elementary family.

Proposition 6.4 and a standard “mixing” argument shows that this smaller family suffices to contract the L^{B} functors described above. For sub-bifunctors B closed under the action of the Dress ring, this was proved by Dress [4] and Wall [29]. A similar argument shows that the odd-dimensional L^{B}-groups are contracted by the 2-hyperelementary family alone.

Example 7.6 (Symmetric, hyper-quadratic and lower L-theory). The Ranicki symmetric and hyper-quadratic L-theories [23] are also functors out of (ZG, ω)-Morita and hence are contracted by the hyperelementary family. The hyper-quadratic theory is a 2-torsion group with an exponent so it is contracted by the 2-hyperelementary family (as above, we note that the 2-localization map induces an isomorphism on this functor and use the 2local contraction of functors out of (Z, −)-Morita by the 2-hyperelementary family). The lower L-theories for a ring with anti-structure can be defined in terms of the L-theory of the ring with some Laurent variables adjoined [23] and hence are functors out of (ZG, ω)- Morita. Therefore L-theories with decorations in sub-Mackey functors of K_{i} for i < 0 are contracted by the hyperelementary family. The higher L-theories of Weiss and Williams [31] should also be amenable to these techniques.

Example 7.7 (Farrell-Hsiang induction). The technique of [5, §§1-2] was originally introduced to apply induction theory to the L-groups of an infinite group Γ. Let pr: Γ → G be a homomorphism to a finite group G, and define an orientation character for Γ by the composition ω ∘ pr, where ω: G → {±1} is an orientation character for G. Then L_{∗}^{B}(RΓ, ω) is an additive functor (RG, ω)-Witt → Ab, which defines a pseudo-Mackey functor as above. To check this, note that we again have a generating set for the morphisms consisting of the bisets X = H_{2} ×_{K} H_{1}, where K ⊂ H_{2} × H_{1} is a subgroup. To produce the needed biform on X, we adapt the formulas in [15, 1.B] with θ_{X} = id. If ω ≡ 1, it follows that these L-groups can be computed in terms of the L-theory of the various subgroups Γ_{H} = pr^{−1}(H), H ⊂ G. In particular, it is enough to use the hyperelementary subgroups H of G.

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Department of Mathematics & Statistics McMaster University Hamilton, ON L8S 4K1, Canada

E-mail address: ian@math.mcmaster.ca

Department of Mathematics University of Notre Dame Notre Dame, IN 46556, USA

E-mail address: taylor.2@nd.edu

Department of Mathematics University of Notre Dame Notre Dame, IN 46556, USA

E-mail address: williams.4@nd.edu