The Burnside Ring and Equivariant Stable Cohomotopy for Infinite Groups

Wolfgang Lück^{∗} Fachbereich Mathematik Universität Münster Einsteinstr. 62 48149 Münster Germany

November 30, 2005

Abstract

After we have given a survey on the Burnside ring of a finite group, we discuss and analyze various extensions of this notion to infinite (discrete) groups. The first three are the finite-G-set-version, the inverselimit-version and the covariant Burnside group. The most sophisticated one is the fourth definition as the zero-th equivariant stable cohomotopy of the classifying space for proper actions. In order to make sense of this definition we define equivariant stable cohomotopy groups of finite proper equivariant CW -complexes in terms of maps between the sphere bundles associated to equivariant vector bundles. We show that this yields an equivariant cohomology theory with a multiplicative structure. We formulate a version of the Segal Conjecture for infinite groups. All this is analogous and related to the question what are the possible extensions of the notion of the representation ring of a finite group to an infinite group. Here possible candidates are projective class groups, Swan groups and the equivariant topological K-theory of the classifying space for proper actions. Key words: Burnside ring, equivariant stable cohomotopy, infinite groups. Mathematics Subject Classification 2000: 55P91, 19A22.

0 Introduction

The basic notions of the Burnside ring and of equivariant stable cohomotopy have been defined and investigated in detail for finite groups. The purpose of this article is to discuss how these can be generalized to infinite (discrete) groups. The guideline will be the related notion of the representation ring which allows several generalizations to infinite groups, each of which reflects one aspect of the original notion for finite groups. Analogously we will present several possible generalizations of the Burnside ring for finite groups to infinite (discrete) groups. There seems to be no general answer to the question which generalization is the right one. The answer depends on the choice of the background problem such as universal additive properties, induction theory, equivariant stable homotopy theory, representation theory, completion theorems and so on. For finite groups the representation ring and the Burnside ring are related to all these topics simultaneously and for infinite groups the notion seems to split up into different ones which fall together for finite groups but not in general.

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

The following table summarizes in the first column the possible generalizations to infinite groups of the representation ring R_{F} (G) with coefficients in a field F of characteristic zero. In the second column we list the analogous generalizations for the Burnside ring. In the third column we give key words for their main property, relevance or application. Explanations will follow in the main body of the text.

R_{F} (G)  A(G)  key words

K_{0}(F G)

\underline{A}(G) universal additive invariant, equivariant Euler characteristic

Sw^{f}(G; F)

\overline{A}(G) induction theory, Green functors

R_{cov,F} (G) := colim_{H∈Sub _{FIN} (G)}R_{F} (H)

A_{cov}(G) := colim_{H∈Sub _{FIN} (G)}A(H) collecting all values for finite subgroups with respect to induction

R_{inv,F} (G) := invlim_{H∈Sub _{FIN} (G)}R_{F} (H)

A_{inv}(G) := invlim_{H∈Sub _{FIN} (G)}A(H) collecting all values for finite subgroups with respect to restriction

K_{G}^{0}(\underline{E}G)

A_{ho}(G) := π_{G}^{0}(\underline{E}G) completion theorems, equivariant vector bundles

K_{0}^{G}(\underline{E}G) π_{0}^{G}(\underline{E}G) representation theory, Baum-Connes Conjecture, equivariant homotopy theory The various rings are linked by the following diagram of ring homomorphisms

\xymatrix{A_{ho}(G) = π_{G}^{0}(\underline{E}G) \ar[r]^{^{edge^{G}}} \ar[dd]^{ψ_{G}^{0}(\underline{E}G)} & A_{inv}(G) \ar[d]^{P_{inv}^{G}} & \overline{A}(G) \ar[l]^{T^{G}} \ar[d]^{_{\overline{P}_{G}}} \\  & R_{inv,ℚ}(G) \ar[d]^{c} & Sw^{f}(G; ℚ) \ar[l]^{S^{G,ℚ}} \ar[d]^{c} \\ K_{G}^{0}(\underline{E}G) \ar[r]^{_{edge_{G}}} & R_{inv,ℂ}(G) & Sw^{f}(G; ℂ) \ar[l]^{S^{G,ℂ}}}

where c denotes the obvious change of coefficients homomorphisms and the other maps will be explained later.

We will also define various pairings which are summarized in the following diagram which reflects their compatibilities.

[figure: \overline{A}(G) × \underline{A}(G); μ_{A}^{G}; \overline{P}^{G}×\underline{P}^{G}; T^{G}×id; \underline{A}(G); _{\underline{P}_{G}}; Q_{A}^{G}; Sw^{f}(G; ℚ) × K_{0}(ℚG); μ_{K}^{G}; S^{G,ℚ}; K_{0}(ℚG); Q_{K}^{G}; A_{inv}(G) × \underline{A}(G); ν_{A}^{G}; P_{inv}^{G}×(P_{cov}^{G}∘(V^{G})^{−1}); ℤ; id; R_{inv,ℚ}(G) × R_{cov,ℚ}(G); ν_{R}^{G}; W^{G,ℚ}; ℤ] (0.1)

In Section 1 we give a brief survey about the Burnside ring A(G) of a finite group G in order to motivate the generalizations. In Sections 2, 3 and 4 we treat the finite-G-set-version of the Burnside Ring \overline{A}(G), the inverse-limit-version of the Burnside ring A_{inv}(G) and the covariant Burnside group \underline{A}(G). These definitions are rather straightforward. The most sophisticated version of the Burnside ring for infinite groups is the zero-th equivariant stable cohomotopy π_{G}^{0}(\underline{E}G) of the classifying space \underline{E}G for proper G-actions. It will be constructed in Section 6 after we have explained the notion of an equivariant cohomology theory with multiplicative structure in Section 5. One of the main result of this paper is

Theorem 6.5 Equivariant stable cohomotopy π_{?}^{∗} defines an equivariant cohomology theory with multiplicative structure for finite proper equivariant CW - complexes. For every finite subgroup H of the group G the abelian groups π_{G}^{n}(G/H) and π_{H}^{n} are isomorphic for every n ∈ ℤ and the rings π_{G}^{0}(G/H) and π_{H}^{0} = A(H) are isomorphic.

An important test in the future will be whether the version of the Segal Conjecture for infinite groups discussed in Section 8 is true.

The papers is organized as follows:

1.  Review of the Burnside Ring for Finite Groups
2.  The Finite-G-Set-Version of the Burnside Ring
3.  The Inverse-Limit-Version of the Burnside Ring
4.  The Covariant Burnside Group
5.  Equivariant Cohomology Theories
6.  Equivariant Stable Cohomotopy in Terms of Real Vector Bundles
7.  The Homotopy Theoretic Burnside Ring
8.  The Segal Conjecture for Infinite Groups

References

1 Review of the Burnside Ring for Finite Groups

In this section we give a brief review of the definition, properties and applications of the Burnside ring for finite groups in order to motivate our definitions for infinite groups.

Definition 1.1. (Burnside ring of a finite group). The isomorphism classes of finite G-sets form a commutative associative semi-ring with unit under disjoint union and cartesian product. The Burnside ring A(G) is the Grothendieck ring associated to this semi-ring.

As abelian group the Burnside ring A(G) is the free abelian group with the set {G/H | (H) ∈ ccs(G)} as basis, where ccs(G) denotes the set of conjugacy classes of subgroups of G. The zero element is represented by the empty set, the unit is represented by G/G. The interesting feature of the Burnside ring is its multiplicative structure.

Given a group homomorphism f: G_{0} → G_{1} of finite groups, restriction with f defines a ring homomorphism f^{∗}: A(G_{1}) → A(G_{0}). Thus A(G) becomes a contravariant functor from the category of finite groups to the category of commutative rings. Induction defines a homomorphism of abelian groups f_{∗}: A(G_{0}) → A(G_{1}), [S] ↦ [G_{1} ×_{f} S], which is not compatible with the multiplication. Thus A(G) becomes a becomes a covariant functor from the category of finite groups to the category of abelian groups.

1.1 The Character Map and the Burnside Ring Congruences

Let G be a finite group. Let ccs(G) be the set of conjugacy classes (H) of subgroups H ⊆ G. Define the character map

char^{G}: A(G) → ∏_{(H)∈ccs(G)} ℤ  (1.2)

by sending the class of a finite G-set S to the numbers {|S^{H}| | (H) ∈ ccs(G)}. This is an injective ring homomorphism whose image can be described by the so called Burnside ring congruences which we explain next.

In the sequel we denote for a subgroup H ⊆ G by N_{G}H its normalizer {g ∈ G | g^{−1}Hg = H}, by C_{G}H = {g ∈ G | gh = hg for h ∈ H} its centralizer, by W_{G}H its Weyl group N_{G}H/H and by [G : H] its index. Let p_{H}: N_{G}H → W_{G}H be the canonical projection. Denote for a cyclic group C by Gen(C) the set of its generators. We conclude from [46, Proposition 1.3.5]

Theorem 1.3 (Burnside ring congruences for finite groups). An element {x(H)} ∈ ∏_{(H)∈ccs(G)} ℤ lies in the image of the injective character map char^{G} defined in (1.2) if and only if we have for every (H) ∈ ccs(G)

∑_{(C)∈ccs(W_{G}H) C cyclic} | Gen(C)| · [W_{G}H : N_{W_{G}H}C] · x(p_{H}^{−1}(C)) ≡ 0 mod |W_{G}H|.

Example 1.4 (A(ℤ/p). Let p be a prime and let G be the cyclic group ℤ/p of order p. Then A(G) is the free abelian group generated by [G] and [G/G]. The multiplication is determined by the fact that [G/G] is the unit and [G] · [G] = p · [G]. There is exactly one non-trivial Burnside ring congruence, namely the one for H = {1} which in the notation of Theorem 1.3 is

x(1) ≡ x(G) mod p.

1.2 The Equivariant Euler Characteristic

Next we recall the notion of a G-CW -complex.

Definition 1.5 (G-CW -complex). Let G be a group. A G-CW -complex X is a G-space together with a G-invariant filtration

∅ = X_{−1} ⊆ X_{0} ⊂ X_{1} ⊆ … ⊆ X_{n} ⊆ … ⊆ ⋃_{n≥0} X_{n} = X

such that X carries the colimit topology with respect to this filtration (i.e., a set C ⊆ X is closed if and only if C ∩ X_{n} is closed in X_{n} for all n ≥ 0) and X_{n} is obtained from X_{n−1} for each n ≥ 0 by attaching equivariant n-dimensional cells, i.e., there exists a G-pushout

∐_{i∈I_{n}} G/H_{i} × S^{n−1} \underrightarrow{∐_{i∈In} q_{i}^{n}} X_{n−1}
↓  ↓
∐_{i∈I_{n}} G/H_{i} × D^{n} \overrightarrow{∐_{i∈In} Q_{i}^{n}} X_{n}

A G-CW -complex X is called finite if it is built by finitely many equivariant cells G/H × D^{n} and is called cocompact if G\X is compact. The conditions finite and cocompact are equivalent for a G-CW -complex. Provided that G is finite, X is compact if and only if X is cocompact A G-map f: X → Y of G-CW -complexes is called cellular if f(X_{n}) ⊆ Y_{n} holds for all n.

Definition 1.6 (Equivariant Euler Characteristic). Let G be a finite group and X be a finite G-CW -complex. Define its equivariant Euler characteristic

χ^{G}(X) ∈ A(G)

by

χ^{G}(X) := ∑_{n=0}^{∞}(−1)^{n} · ∑_{i∈I_{n}}[G/H_{i}]

after choices of the G-pushouts as in Definition 1.5.

This definition is independent of the choice of the G-pushouts by the next result. The elementary proofs of the next two results are left to the reader. We denote by X^{H} and X^{>H} respectively the subspace of X consisting of elements x ∈ X whose isotropy group G_{x} satisfies H ⊆ G_{x} and H ⊊ G_{x} respectively.

Lemma 1.7. Let G be a finite group.

(i) Let X be a finite G-CW -complex. Then

χ^{G}(X) = ∑_{(H)∈ccs(G)} χ (W_{H}G\(X^{H}, X^{>H})) · [G/H],

where χ denotes the classical (non-equivariant) Euler characteristic;

(ii) If X and Y are G-homotopy equivalent finite G-CW -complexes, then

χ^{G}(X) = χ^{G}(Y);

(iii) If

X_{0} ⟶^{i_{0}} X_{1}
i_{2}↓  ↓
X_{2} ⟶ X

is a G-pushout of finite G-CW -complexes such that i_{1} is an inclusion of finite G-CW -complexes and i_{2} is cellular, then

χ^{G}(X) = χ^{G}(X_{1}) + χ^{G}(X_{1}) − χ^{G}(X_{0});

(iv) If X and Y are finite G-CW -complexes, then X × Y with the diagonal G-action is a finite G-CW -complex and

χ^{G}(X × Y) = χ^{G}(X) · χ^{G}(Y);

(v) The image of χ^{G}(X) under the character map char^{G} of (1.2) is given by the collection of classical (non-equivariant) Euler characteristics {χ(X^{H}) | (H) ∈ ccs(G)}.

An equivariant additive invariant for finite G-CW -complexes is a pair (A, a) consisting of an abelian group and an assignment a which associates to every finite G-CW -complex X an element a(X) ∈ A such that a(∅) = 0, G-homotopy invariance and additivity hold, i.e., the obvious versions of assertions (ii) and (iii) appearing in Lemma 1.7 are true. An equivariant additive invariant (U, u) is called universal if for every equivariant additive invariant (A, a) there is precisely one homomorphism of abelian groups ϕ: U → A such that ϕ(u(X)) = a(X) holds for every finite G-CW -complex X. Obviously (U, u) is (up to unique isomorphism) unique if it exists.

Theorem 1.8 (The universal equivariant additive invariant). Let G be a finite group. The pair (A(G), χ^{G}) is the universal equivariant additive invariant for finite G-CW -complexes.

1.3 The Equivariant Lefschetz Class

The notion of an equivariant Euler characteristic can be extended to the notion of an equivariant Lefschetz class as follows.

Definition 1.9. Let G be a finite group and X be a finite G-CW -complex. We define the equivariant Lefschetz class of a cellular G-selfmap f: X → X

Λ^{G}(f) ∈ A(G)

by

Λ^{G}(f) = ∑_{(H)∈ccs(G)} Λ (W_{G}H\(f^{H}, f^{>H})) · [G/H],

where Λ(W_{G}H\(f^{H}, f^{>H})) ∈ ℤ is the classical Lefschetz number of the endomorphism W_{G}H\(f^{H}, f^{>H}) of the pair of finite CW -complexes W_{G}H\(X^{H}, X^{>H}) induced by f.

Obviously Λ^{G}(id: X → X) agrees with χ^{G}(X). The elementary proof of the next result is left to the reader.

Lemma 1.10. Let G be a finite group.

(i) If f and g are G-homotopic G-selfmaps of a finite G-CW -complex X, then

Λ^{G}(f) = Λ^{G}(g);

(ii) Let

X_{0} ⟶^{i_{1}} X_{1}
i_{2}↓  ↓
X_{2} ⟶ X

be a G-pushout of finite G-CW -complexes such that i_{1} is an inclusion of finite G-CW -complexes and i_{2} is cellular. Let f_{i}: X_{i} → X_{i} for i = 0, 1, 2 and f: X → X be G-selfmaps compatible with this G-pushout. Then

Λ^{G}(f) = Λ^{G}(f_{1}) + Λ^{G}(f_{2}) − Λ^{G}(f_{0});

(iii) Let X and Y be finite G-CW -complexes and f: X → X and g: Y → Y be G-selfmaps. Then

Λ^{G}(f × g) = χ^{G}(X) · Λ^{G}(g) + χ^{G}(Y) · Λ^{G}(f);

(iv) Let f: X → Y and g: Y → X be G-maps of finite G-CW -complexes. Then

Λ^{G}(f ∘ g) = Λ^{G}(g ∘ f);

(v) The image of Λ^{G}(f) under the character map char^{G} of (1.2) is given by the collection of classical (non-equivariant) Lefschetz numbers {Λ(f^{H}) | (H) ∈ ccs(G)}.

One can also give a universal property characterizing the equivariant Lefschetz class (see [22]).

The equivariant Lefschetz class has also the following homotopy theoretic meaning.

Definition 1.11. A G-homotopy representation X is a finite-dimensional G-CW -complex such that for each subgroup H ⊆ G the fixed point set X^{H} is homotopy equivalent to a sphere S^{n(H)} for n(H) the dimension of the CW - complex X^{H}.

An example is the unit sphere SV in an orthogonal representation V of G. Denote by [X, X]^{G} the set of G-homotopy classes of G-maps X → X. The proof of the next theorem can be found in [24, Theorem 3.4 on page 139] and is a consequence of the equivariant Hopf Theorem (see for instance [46, page 213], [49, II.4], [21]).

Theorem 1.12. Let X be a G-homotopy representation of the finite group G. Suppose that

(i) Every subgroup H ⊆ G occurs as isotropy group of X;

(ii) dim(X^{G}) ≥ 1;

(iii) The group G is nilpotent or for every subgroup H ⊆ G we have dim(X^{>H})+ 2 ≤ dim(X^{H}).

Then the following map is an bijection of monoids, where the monoid structure on the source comes from the composition and the one on the target from the multiplication

deg^{G}: [X, X]^{G} ⟶^{≅} A(G), [f] ↦ (Λ^{G}(f) − 1) · (χ^{G}(X) − 1).

We mention that the image of the element deg^{G}(f) for a self-G-map of a G-homotopy representation under the character map char^{G} of (1.2) is given by the collection of (non-equivariant) degrees {deg(f^{H}) | (H) ∈ ccs(G)}.

1.4 The Burnside Ring and Equivariant Stable Cohomotopy

Let X and Y be two finite pointed G-CW -complexes. Pointed means that we have specified an element in its 0-skeleton which is fixed under the G-action. If V is a real G-representation, let S^{V} be its one-point compactification. We will use the point at infinity as base point for S^{V}. If V is an orthogonal representation, i.e., comes with an G-invariant scalar product, then S^{V} is G-homeomorphic to the unit sphere S(V ⊕ ℝ). Given two pointed G-CW -complexes X and Y with base points x and y, define their one-point-union X ∨ Y to be the pointed G-CW -complex X × {y} ∪ {x} × Y ⊆ X × Y and their smash product X ∧ Y to be the pointed G-CW -complex X × Y/X ∨ Y.

We briefly introduce equivariant stable homotopy groups following the approach due to tom Dieck [49, II.6].

If V and W are two complex G-representations, we write V ≤ W if there exists a complex G-representation U and a linear G-isomorphism ϕ: U⊕V → W. If ϕ: U ⊕ V → W is a linear G-isomorphism, define a map

b_{V,W} : [S^{V} ∧ X, S^{V} ∧ Y]^{G} → [S^{W} ∧ X, S^{W} ∧ Y]^{G}

by the composition

[S^{V} ∧ X, S^{V} ∧ Y]^{G} ⟶^{u_{1}} [S^{U} ∧ S^{V} ∧ X, S^{U} ∧ S^{V} ∧ Y]^{G}

⟶^{u_{2}} [S^{U⊕V} ∧ X, S^{U⊕V} ∧ Y]^{G} ⟶^{u_{3}} [S^{W} ∧ X, S^{W} ∧ Y]^{G},

where the map u_{1} is given by [f] ↦ [id_{S^{U}} ∧f], the map u_{2} comes from the obvious G-homeomorphism S^{U⊕V} ⟶^{≅} S^{U} ∧ S^{V} induced by the inclusion V ⊕ W → S^{V} ∧ S^{W} and the map u_{3} from the G-homeomorphism S^{ϕ}: S^{U⊕V} ⟶^{≅} S^{W}. Any two linear G-isomorphisms ϕ_{0}, ϕ_{1} : V_{1} → V_{2} between two complex G-representations are isotopic as linear G-isomorphisms. (This is not true for real G-representations.) This implies that the map b_{V,W} is indeed independent of the choice of U and ϕ. One easily checks that b_{V_{2},V_{1}} ∘ b_{V_{0},V_{1}} = b_{V_{0},V_{2}} holds for complex G-representations V_{0}, V_{1} and V_{2} satisfying V_{0} ≤ V_{1} and V_{1} ≤ V_{2}.

Let I be the set of complex G-representations with underlying complex vector space ℂ^{n} for some n. (Notice that the collections of all complex Grepresentations does not form a set.) Define on the disjoint union

∐_{V ∈I}[S^{V} ∧ X, S^{V} ∧ Y]^{G}

an equivalence relation by calling [f] ∈ [S^{V} ∧ X, S^{V} ∧ Y]^{G} and [g] ∈ [S^{W} ∧ X, S^{W} ∧ Y]^{G} equivalent if there exists a representation U ∈ I with V ≤ U and W ≤ U such that b_{V,U}([f]) = b_{W,U}([g]) holds. Let ω_{0}^{G}(X, Y) for two pointed G-CW -complexes X and Y be the set of equivalence classes.

If V is any complex G-representation (not necessarily in I) and f: S^{V} ∧X → S^{V} ∧ Y is any G-map, there exists an element W ∈ I with V ≤ W and we get an element in ω_{0}^{G}(X, Y) by b_{V,W} ([f]). This element is independent of the choice of W and also denoted by [f] ∈ ω_{0}^{G}(X, Y).

One can define the structure of an abelian group on the set ω_{0}^{G}(X, Y) as follows. Consider elements x, y ∈ ω_{0}^{G}(X, Y). We can choose an element of the shape ℂ ⊕ U in I for ℂ equipped with the trivial G-action and G-maps f, g: S^{ℂ⊕U} ∧ X → S^{ℂ⊕U} ∧ Y representing x and y. Now using the standard pinching map ∇: S^{ℂ} → S^{ℂ} ∨ S^{ℂ} one defines x + y as the class of the G-map

S^{ℂ⊕U} ∧ X ⟶^{≅} S^{ℂ} ∧ S^{U} ∧ X ⟶^{∇∧id_{S^{U}} ∧ id_{X}} (S^{ℂ} ∨ S^{ℂ}) ∧ S^{U} ∧ X

⟶^{≅} (S^{ℂ}∧S^{U} ∧X)∨(S^{ℂ}∧S^{U} ∧X) ⟶^{≅} (S^{ℂ⊕U} ∧X)∨(S^{ℂ⊕U} ∧X) ⟶^{f∨g} S^{ℂ⊕U} ∧Y.

The inverse of x is defined by the class of

S^{ℂ⊕U} ∧ X ⟶^{≅} S^{ℂ} ∧ S^{U} ∧ X ⟶^{d∧f} S^{ℂ} ∧ S^{U} ∧ Y ⟶^{≅} S^{ℂ⊕U} ∧ Y

where d: S^{ℂ} → S^{ℂ} is any pointed map of degree −1. This is indeed independent of the choices of U, f and g.

We define the abelian groups

\begin{aligned}ω_{n}^{G}(X, Y) & = & ω_{0}^{G}(S^{n} ∧ X, Y) & n ≥ 0; \\ ω_{n}^{G}(X, Y) & = & ω_{0}^{G}(X, S^{−n}Y) & n ≤ 0; \\ ω_{G}^{n}(X, Y) & = & ω_{−n}^{G}(X, Y) & n ∈ ℤ; \end{aligned}

Obviously ω_{n}^{G}(X, Y) is functorial, namely contravariant in X and covariant in Y.

Let X and Y be (unpointed) G-CW -complexes. Let X_{+} and Y_{+} be the pointed G-CW -complexes obtained from X and Y by adjoining a disjoint base point. Denote by {•} the one-point-space. Define abelian groups

\begin{aligned}π_{n}^{G}(Y) & = & ω_{n}^{G}(\{•\}_{+}, Y_{+}) & n ∈ ℤ; \\ π_{G}^{n}(X) & = & ω_{G}^{n}(X_{+}, \{•\}_{+}) & n ∈ ℤ; \\ π_{n}^{G} & = & π_{n}^{G}(\{•\}) & n ∈ ℤ; \\ π_{G}^{n} & = & π_{−n}^{G} & n ∈ ℤ. \end{aligned}

The abelian group π_{0}^{G} = π_{G}^{0} becomes a ring by the composition of maps. The abelian groups π_{n}^{G}(Y) define covariant functors in Y and are called the equivariant stable homotopy groups of Y. The abelian groups π_{G}^{n}(X) define contravariant functors in X and are called the equivariant stable cohomotopy groups of X.

We emphasize that our input in π_{G}^{n} and π_{n}^{G} are unpointed G-CW -complexes. This is later consistent with our constructions for infinite groups, where all G-CW -spaces must be proper and therefore have empty G-fixed point sets and cannot have base points.

Theorem 1.12 implies the following result due to Segal [41].

Theorem 1.13. The isomorphism deg^{G} appearing in Theorem 1.12 induces an isomorphism of rings

deg^{G}: π_{G}^{0} ⟶^{≅} A(G).

For a more sophisticated and detailed construction of and more information about the equivariant stable homotopy category we refer for instance to [17], [23].

1.5 The Segal Conjecture for Finite Groups

The equivariant stable cohomotopy groups π_{G}^{n}(X) are modules over the ring π_{G}^{0} = A(G), the module structure is given by composition of maps. The augmentation homomorphism ϵ^{G}: A(G) → ℤ is the ring homomorphism sending the class of a finite set S to |S| which is just the component belonging to the trivial subgroup of the character map defined in (1.2). The augmentation ideal I_{G} ⊆ A(G) is the kernel of the augmentation homomorphism ϵ^{G}.

For an (unpointed) CW -complex X we denote by π_{s}^{n}(X) the (non-equivariant) stable cohomotopy group of X_{+}. This is in the previous notation for equivariant stable cohomotopy the same as π_{{1}}^{n}(X) for {1} the trivial group. If X is a finite G-CW -complex, we can consider π_{s}^{n}(EG ×_{G} X). Since π_{G}^{n}(X) is a A(G)-module, we can also consider its I_{G}-adic completion denoted by π_{G}^{n}(X)_{I_{G}}. The following result is due to Carlsson [8].

Theorem 1.14 (Segal Conjecture for finite groups). The Segal Conjecture for finite groups G is true, i.e., for every finite group G and finite G-CW - complex X there is an isomorphism

π_{G}^{n}(X)_{I_{G}} ⟶^{≅} π_{s}^{n}(EG ×_{G} X).

In particular we get in the case X = {•} and n = 0 an isomorphism

A(G)_{I_{G}}  ⟶^{≅} π_{s}^{0}(BG).  (1.15)

Thus the Burnside ring is linked via its I_{G}-adic completion to the stable cohomotopy of the classifying space BG of a finite group G.

Example 1.16 (Segal Conjecture for ℤ/p). Let G be the cyclic group ℤ/p of order p. We have computed A(G) in Example 1.4. If we put x = [G] − p · [G/G], then the augmentation ideal is generated by x. Since

x^{2} = ([G] − p)^{2} = [G]^{2} − 2p · [G] + p^{2} = (−p) · x,

we get x^{n} = (−p)^{n−1}x and hence I_{G}^{n} = p^{n−1} · I_{G} for n ∈ ℤ, n ≥ 1. This implies

A(G)_{I_{G}} = invlim_{n→∞}ℤ ⊕ I_{G}/I_{G}^{n} = ℤ × ℤ_{p},

where ℤ_{p} denotes the ring of p-adic integers.

1.6 The Burnside Ring as a Green Functor

Let R be an associative commutative ring with unit. Let FGINJ be the category of finite groups with injective group homomorphisms as morphisms. Let M: FGINJ → R-MODULES be a bifunctor, i.e., a pair (M_{∗}, M^{∗}) consisting of a covariant functor M_{∗} and a contravariant functor M^{∗} from FGINJ to R-MODULES which agree on objects. We will often denote for an injective group homomorphism f: H → G the map M_{∗}(f): M(H) → M(G) by ind_{f} and the map M^{∗}(f): M(G) → M(H) by res_{f} and write ind_{H}^{G} = ind_{f} and res_{G}^{H} = res_{f} if f is an inclusion of groups. We call such a bifunctor M a Mackey functor with values in R-modules if

(i) For an inner automorphism c(g): G → G we have M_{∗}(c(g)) = id: M(G) → M(G);

(ii) For an isomorphism of groups f: G ⟶^{≅} H the composites res_{f} ∘ ind_{f} and ind_{f} ∘ res_{f} are the identity;

(iii) Double coset formula We have for two subgroups H, K ⊂ G

res_{G}^{K} ∘ ind_{H}^{G} = ∑_{KgH∈K\G/H} ind_{c(g) : H∩g^{−1}Kg→K} ∘ res_{H}^{H∩g^{−1}Kg},

where c(g) is conjugation with g, i.e., c(g)(h) = ghg^{−1}.

Let ϕ: R → S be a homomorphism of associative commutative rings with unit. Let M be a Mackey functor with values in R-modules and let N and P be Mackey functors with values in S-modules. A pairing with respect to ϕ is a family of maps

m(H): M(H) × N(H) → P (H), (x, y) ↦ m(H)(x, y) =: x · y,

where H runs through the finite groups and we require the following properties for all injective group homomorphisms f: H → K of finite groups:

(x_{1} + x_{2}) · y = x_{1} · y + x_{2} · y for x_{1}, x_{2} ∈ M(H), y ∈ N(H);
x · (y_{1} + y_{2}) = x · y_{1} + x · y_{2} for x ∈ M(H), y_{1}, y_{2} ∈ N(H);
(rx) · y = ϕ(r)(x · y)  for r ∈ R, x ∈ M(H), y ∈ N(H);
x · sy = s(x · y)  for s ∈ S, x ∈ M(H), y ∈ N(H);
res_{f}(x · y) = res_{f}(x) · res_{f}(y) for x ∈ M(K), y ∈ N(K);
ind_{f}(x) · y = ind_{f} (x · res_{f}(y)) for x ∈ M(H), y ∈ N(K);
x · ind_{f}(y) = ind_{f} (res_{f}(x) · y) for x ∈ M(K), y ∈ N(H).

A Green functor with values in R-modules is a Mackey functor U together with a pairing with respect to id: R → R and elements 1_{H} ∈ U(H) for each finite group H such that for each finite group H the pairing U(H) × U(H) → U(H) induces the structure of an R-algebra on U(H) with unit 1_{H} and for any morphism f: H → K in FGINJthe map U^{∗}(f): U(K) → U(H) is a homomorphism of R-algebras with unit. Let U be a Green functor with values in R-modules and M be a Mackey functor with values in S-modules. A (left) U-module structure on M with respect to the ring homomorphism ϕ: R → S is a pairing such that any of the maps U(H) × M(H) → M(H) induces the structure of a (left) module over the R-algebra U(H) on the R-module ϕ^{∗}M(H) which is obtained from the S-module M(H) by rx := ϕ(r)x for r ∈ R and x ∈ M(H).

Theorem 1.17. (i) The Burnside ring defines a Green functor with values in ℤ-modules;

(ii) If M is a Mackey functor with values in R-modules, then M is in a canonical way a module over the Green functor given by the Burnside ring with respect to the canonical ring homomorphism ϕ: ℤ → R.

Proof. (i) Let f: H → G be an injective homomorphism of groups. Define ind_{f}: A(H) → A(G) by sending the class of a finite H-set S to the class of the finite G-set G ×_{f} S. Define res_{f}: A(G) → A(H) by considering a finite G-set as an H-set by restriction with f. One easily verifies that the axioms of a Green functor with values in ℤ-modules are satisfied. (ii) We have to specify for any finite group G a pairing m(G): A(G) × M(G) → M(G). This is done by the formula

m(G) (∑_{i} n_{i} · [G/H_{i}], x) := ∑_{i} n_{i} · ind_{H_{i}}^{G} ∘ res_{G}^{H_{i}}(x).

One easily verifies that the axioms of a module over the Green functor given by the Burnside ring are satisfied.

Theorem 1.17 is the main reason why the Burnside ring plays an important role in induction theory. Induction theory addresses the question whether one can compute the values of a Mackey functor on a finite group by its values on a certain class of subgroups such as the family of cyclic or hyperelementary groups. Typical examples of such Mackey functors are the representation ring R_{F} (G) or algebraic K and L-groups K_{n}(RG) and L_{n}(RG) of groups rings. The applications require among other things a good understanding of the prime ideals of the Burnside ring. For more information about induction theory for finite groups we refer to the fundamental papers by Dress [11], [12] and for instance to [46, Chapter 6]. Induction theory for infinite groups is developed in [5].

As an illustration we give an example how the Green-functor mechanism works.

Example 1.18 (Artin’s Theorem). Let R_{ℚ}(G) be the rational representation ring of the finite group G. For any finite cyclic group C one can construct an element

θ_{C} ∈ R_{ℚ}(C)

which is uniquely determined by the property that its character function sends a generator of C to |C| and every other element of C to zero.

Let G be a finite group. Let ℚ be the trivial 1-dimensional rational Grepresentation. It is not hard to check by a calculation with characters that

|G| · [ℚ] = ∑_{\begin{matrix}C⊂G \\ C cyclic \end{matrix}} ind_{C}^{H} θ_{C}.  (1.19)

Assigning to a finite group G the rational representation ring R_{ℚ}(G) inherits the structure of a Green functor with values in ℤ-modules by induction and restriction. Suppose that M is a Mackey functor with values in ℤ-modules which is a module over the Green functor R_{ℚ}. Then for every finite group G the cokernel of the map

⨁_{\begin{matrix}C⊂G \\ C cyclic \end{matrix}} ind_{C}^{G}: ⨁_{\begin{matrix}C⊂G \\ C cyclic \end{matrix}} M(C) → M(G)

is annihilated by multiplication with |G|. This follows from the following calculation for x ∈ M(G) based on (1.19) and the axioms of a Green functor and a module over it

|G| · x = (|G| · [ℚ]) · x = ∑_{\begin{matrix}C⊂G \\ C cyclic \end{matrix}} ind_{C}^{H}(θ_{C}) · x = ∑_{\begin{matrix}C⊂G \\ C cyclic \end{matrix}} ind_{C}^{H}(θ_{C} · res_{H}^{C} x).

Examples for M are algebraic K- and L-groups K_{n}(RG) and L_{n}(RG) for any ring R with ℚ ⊆ R. We may also take M to be R_{F} for any field F of characteristic zero and then the statement above is Artin’s Theorem (see [43, Theorem 26 on page 97].

1.7 The Burnside Ring and Rational Representations

Let R_{ℚ}(G) be the representation ring of finite-dimensional rational G-represen-tation. Given a finite G-set S, let ℚ[S] be the rational G-representation given by the ℚ-vector space with the set S as basis. The next result is due to Segal [40].

Theorem 1.20. (The Burnside ring and the rational representation ring for finite groups). Let G be a finite group. We obtain a ring homomorphism

P^{G}: A(G) → R_{ℚ}(G), [S] ↦ [ℚ[S]].

It is rationally surjective. If G is a p-group for some prime p, it is surjective. It is bijective if and only if G is cyclic.

1.8 The Burnside Ring and Homotopy Representations

We have introduced the notion of a G-homotopy representation in Definition 1.11. The join of two G-homotopy representations is again a G-homotopy representation. We call two G-homotopy representations X and Y stably G-homotopy equivalent if for some G-homotopy representation Z the joins X ∗ Z and Y ∗ Z are G-homotopy equivalent. The stable G-homotopy classes of G-homotopy representations together with the join define an abelian semi-group. The Ghomotopy representation group V (G) is the associated Grothendieck group. It may be viewed as the homotopy version of the representation ring. Taking the unit sphere yields a group homomorphism R_{ℝ}(G) → V (G).

The dimension function of a G-homotopy representation X

dim(X) ∈ ∏_{(H)∈ccs(G)} ℤ

associates to the conjugacy class (H) of a subgroup H ⊆ G the dimension of X^{H}. The question which elements in ∏_{(H)∈ccs(G)} ℤ occur as dim(X) is studied for instance in [47], [48], [49, III.5] and [50]. Define V (G, dim) by the exact sequence

0 → V (G, dim) → V (G) ⟶^{dim} ∏_{(H)∈ccs(G)} ℤ.

Let Pic(A(G)) be the Picard group of the Burnside ring, i.e., the abelian group of projective A(G)-modules of rank one with respect to the tensor product. The next result is taken from [50, 6.5].

Theorem 1.21 (V (G, dim) and the Picard group of A(G)). There is an isomorphism

V (G, dim) ⟶^{≅} Pic(A(G)).

Further references about the Burnside ring of finite groups are [7], [10],[15], [16], [19], [20], [36], [45], [52].

2 The Finite-G-Set-Version of the Burnside Ring

From now on G can be any (discrete) group and need not be finite anymore. Next we give a first definition of the Burnside ring for infinite groups.

Definition 2.1. (The finite-G-set-version of the Burnside ring). The isomorphism classes of finite G-sets form a commutative associative semi-ring with unit under the disjoint union and the cartesian product. The finite-G-set-version of the Burnside ring \overline{A}(G) is the Grothendieck ring associated to this semi-ring.

To avoid any confusion, we emphasize that finite G-set means a finite set with a G-action. This definition is word by word the same as given for a finite group in Definition 1.1.

Given a group homomorphism f: G_{0} → G_{1} of groups, restriction with f defines a ring homomorphism f^{∗}: A(G_{1}) → A(G_{0}). Thus A(G) becomes a contravariant functor from the category of groups to the category of commutative rings. Provided that the image of f has finite index, induction defines a homomorphism of abelian groups f_{∗}: A(G_{0}) → A(G_{1}), [S] ↦ [G_{1} ×_{f} S], which is not compatible with the multiplication.

2.1 Character Theory and Burnside Congruences for the Finite-G-Set-Version

The definition of the character map (1.2) also makes sense for infinite groups and we denote it by

\overline{char}^{G}: \overline{A}(G) → ∏_{(H)∈ccs(G)} ℤ, [S] ↦ (|S^{H}|)_{(H)}.  (2.2)

Given a group homomorphism f: G_{0} → G_{1}, define a ring homomorphism

f^{∗}: ∏_{(H_{1})∈ccs(G_{1})} ℤ → ∏_{(H_{0})∈ccs(G_{0})} ℤ  (2.3)

by sending {x(H_{1})} to {x(f(H_{0}))}. One easily checks

Lemma 2.4. The following diagram of commutative rings with unit commutes for every group homomorphism f: G_{0} → G_{1}

\overline{A}(G_{1}) ⟶^{\overline{char}^{G_{1}}} ∏_{(H_{1})∈ccs(G_{1})} ℤ
f^{∗}↓  ↓^{f^{∗}}
\overline{A}(G_{0}) ⟶_{\overline{char}^{G_{0}}} ∏_{(H_{0})∈ccs(G_{0})} ℤ

Theorem 2.5 (Burnside ring congruences for \overline{A}(G)). The character map \overline{char}^{G} is an injective ring homomorphism.

Already the composition

\overline{A}(G) ⟶^{\overline{char}^{G}} ∏_{(H)∈ccs(G)} ℤ ⟶^{pr} ∏_{\begin{matrix}(H)∈ccs(G) \\ [G:H]<∞ \end{matrix}} ℤ

for pr the obvious projection is injective.

An element x = {x(H)} ∈ ∏_{(H)∈ccs(G)} ℤ lies in the image of the character

map \overline{char}^{G} defined in (2.2) if and only if it satisfies the following two conditions:

(i) There exists a normal subgroup K_{x} ⊆ G of finite index such that x(H) = x(H · K_{x}) holds for all H ⊆ G, where H · K_{x} is the subgroup {hk | h ∈ H, k ∈ K_{x}};

(ii) We have for every (H) ∈ ccs(G) with [G : H] < ∞:

∑_{(C)∈ccs(W_{G}H) C cyclic} | Gen(C)|·[W_{G}H : N_{W_{G}H}C]·x(p_{H}^{−1}(C)) ≡ 0 mod |W_{G}H|,

where p_{H}: N_{G}H → W_{G}H is the obvious projection.

Proof. Obviously \overline{char}^{G} is a ring homomorphism.

Suppose that x ∈ \overline{A}(G) lies in the kernel of \overline{char}^{G}. For any finite G-set the intersection of all its isotropy groups is a normal subgroup of finite index in G. Hence we can find an epimorphism p_{x}: G → Q_{x} onto a finite group Q_{x} and \overline{x} ∈ A(Q_{x}) such that x lies in the image of p_{x}^{∗}: \overline{A}(Q_{x}) → \overline{A}(G). Since the map

p_{x}^{∗}: ∏_{(H)∈ccs(Q_{x})} ℤ → ∏_{(K)∈ccs(G)} ℤ

is obviously injective and the character map \overline{char}^{Q_{x}} is injective by Theorem 1.3, we conclude x = 0. Hence \overline{char}^{G} is injective.

Suppose that y lies in the image of \overline{char}^{G}. Choose x ∈ \overline{A}(G) with \overline{char}^{G}(x) = y. As explained above we can find an epimorphism p_{x}: G → Q_{x} onto a finite group Q_{x} and \overline{x} ∈ A(Q_{x}) such that p_{x}^{∗}: \overline{A}(Q_{x}) → \overline{A}(G) maps \overline{x} to x. Then Condition (i) is satisfied by Lemma 2.4 if we take K_{x} to be the kernel of p_{x}. Condition (ii) holds for x since the proof of Theorem 1.3 carries though word by word to the case, where G is possibly infinite but H ⊆ G is required to have finite index in G and hence W_{G}H is finite.

We conclude that \overline{char}^{G}(x) = 0 if and only if pr ∘\overline{char}^{G}(x) = 0 holds. Hence pr ∘\overline{char}^{G} is injective.

Now suppose that x = {y(H)} ∈ ∏_{(H)∈ccs(G)} ℤ satisfies Condition (i) and Condition (ii). Let Q_{x} = G/K_{x} and let p_{x}: G → Q_{x} be the projection. In the sequel we abbreviate Q = Q_{x} and p = p_{x}. Then Condition (i) ensures that x lies in the image of the injective map

p^{∗}: ∏_{(H)∈ccs(Q)} ℤ → ∏_{(K)∈ccs(G)} ℤ.

Let y ∈ ∏_{(H)∈ccs(Q)} ℤ be such a preimage. Because of Lemma 2.4 it suffices to prove that y lies in the image of the character map

char^{Q}: A(Q) → ∏_{(H)∈ccs(Q)} ℤ.

By Theorem 1.3 this is true if and only if for every subgroup K ⊆ Q the congruence

∑_{(C)∈ccs(W_{Q}K) C cyclic} | Gen(C)|·[W_{Q}K : N_{W_{Q}K}C]·y (p_{K}^{Q})^{−1}(C)) ≡ 0 mod |W_{Q}K|

holds, where p_{K}^{Q}: N_{Q}K → W_{G}K is the projection. Fix a subgroup K ⊆ Q. Put H = p^{−1}(K) ⊆ G. The epimorphism p: G → Q induces an isomorphism \overline{p}: W_{G}H ⟶^{≅} W_{Q}K. Condition (ii) applied to x and H yields

∑_{(C)∈ccs(W_{G}H) C cyclic} | Gen(C)| · [W_{G}H : N_{W_{G}H}C] · x(p_{H}^{−1}(C)) ≡ 0 mod |W_{G}H|.

For any cyclic subgroup C ⊆ W_{G}H we obtain a cyclic subgroup \overline{p}(C) ⊆ W_{Q}K and we have

| Gen(C)| = | Gen(\overline{p}(C))|;
[W_{G}H : N_{W_{G}H}C] = [W_{Q}K : N_{W_{Q}K} \overline{p}(C)];
x (p_{H}^{−1}(C)) = y ((p_{K}^{Q})^{−1}(\overline{p}(C))) .

Now the desired congruence for y follows. This finishes the proof of Theorem 2.5.

Example 2.6 (\overline{A} of the integers). Consider the infinite cyclic group ℤ. Any subgroup of finite index is of the form nℤ for some n ∈ ℤ, n ≥ 1. As an abelian group \overline{A}(Z) is generated by the classes [ℤ/nℤ] for n ∈ ℤ, n ≥ 1. The condition (ii) appearing in Theorem 2.5 reduces to the condition that for every subgroup nℤ for n ∈ ℤ, n ≥ 1 the congruence

∑_{m∈ℤ,m≥1,m|n} ϕ (\frac{n}{m}) · x(m) ≡ 0 mod n

holds, where ϕ is the Euler function, whose value ϕ(k) is | Gen(ℤ/kℤ)|. The condition (i) reduces to the condition that there exists n_{x} ∈ ℤ, n_{x} ≥ 1 such that for all m ∈ ℤ, m ≥ 1 we have x(mℤ) = x(gcd(m, n_{x})ℤ), where gcd(m, n_{x}) is the greatest common divisor of m and n_{x}.

Remark 2.7 (The completion Â(G) of \overline{A}(G)). We call a G-set almost finite if each isotropy group has finite index and for every positive integer n the number of orbits G/H in S with [G : H] ≤ n is finite. A G-set S is almost finite if and only if for every subgroup H ⊆ G of finite index the H-fixed point set S^{H} is finite and S is the union ⋃\begin{matrix}(H)∈ccs(G) \\ [G:H]<∞ \end{matrix} S^{H}. Of course every finite G-set S is almost finite.

The disjoint union and the cartesian product with the diagonal G-action of two almost finite G-sets is again almost finite. Define Â(G) as the Grothendieck ring of the semi-ring of almost finite G-sets under the disjoint union and the cartesian product. There is an obvious inclusion of rings \overline{A}(G) → Â(G). We can define as before a character map

ĉhar^{G}: Â(G) → ∏_{\begin{matrix}(H)∈ccs(G) \\ [G:H]<∞ \end{matrix}} ℤ, [S] ↦ (|S^{H}|)_{(H)}.  (2.8)

We leave it to the reader to check that ĉhar^{G} is injective, and that an element x in ∏\begin{matrix}(H)∈ccs(G) \\ [G:H]<∞ \end{matrix} ℤ lies in its image if and only if x satisfies condition (ii) appearing

in Theorem 2.5.

Dress and Siebeneicher [13] analyze Â(G) for profinite groups G and put it into relation with the Witt vector construction. They also explain that Â(G) is a completion of \overline{A}(G). The ring Â(ℤ) is studied and put in relation to the necklace algebra, λ-rings and the universal ring of Witt vectors in [14].

2.2 The Finite-G-Set-Version and the Equivariant Euler Characteristic and the Equivariant Lefschetz Class

The results of Sections 1.2 and 1.3 carry over to \overline{A}(G) if one considers only finite G-CW -complexes X whose isotropy group all have finite index in G. But this is not really new since for any such G-CW -complex X there is a subgroup H ⊆ G, namely the intersection of all isotropy groups, such that H is normal, has finite index in G and acts trivially on X. Thus X is a finite Q-CW -complex for the finite group Q = G/H and all these invariant are obtained from the one over Q by applying the obvious ring homomorphism A(Q) = \overline{A}(Q) → \overline{A}(G) to the invariants already defined over the finite group Q.

2.3 The Finite-G-Set-Version as a Green Functor

The notions and results of Subsection 1.6 carry over to the finite-G-set-version \overline{A}(G) for an infinite group G, we replace the category FGINJ by the category GRIFIwhose objects are groups and whose morphisms are injective group homomorphisms whose image has finite index in the target. However, for infinite groups this does not seem to be the right approach to induction theory. The approach presented in Bartels-Lück [5] is more useful. It is based on classifying spaces for families and aims at reducing the family of subgroups, for instance from all finite subgroups to all hyperelementary finite subgroups or from all virtually cyclic subgroups to the family of subgroups which admit an epimorphism to a hyperelementary group and whose kernel is trivial or infinite cyclic.

2.4 The Finite-G-Set-Version and the Swan Ring

Let R be a commutative ring. Let Sw^{f}(G; R) be the abelian group which is generated by the RG-isomorphisms classes of RG-modules which are finitely generated free over R with the relations [M_{0}]−[M_{1}]−[M_{2}] = 0 for any short exact RG-sequence 0 → M_{0} → M_{1} → M_{2} → 0 of such RG-modules. It becomes a commutative ring, the so called Swan ring Sw^{f}(G; R), by the tensor product ⊗_{R}. If G is finite and F is a field, then Sw^{f}(G; F) is the same as the representation ring R_{F} (G) of (finite-dimensional) G-representations over K.

Let G_{0}(RG) be the abelian group which is generated by the RG-isomorphism classes of finitely generated RG-modules with the relations [M_{0}]−[M_{1}]−[M_{2}] = 0 for any short exact RG-sequence 0 → M_{0} → M_{1} → M_{2} → 0 of such RGmodules. There is an obvious map

ϕ: Sw^{f}(G; R) → G_{0}(RG)

of abelian groups. It is an isomorphism if G is finite and R is a principal ideal domain. This follows from [9, Theorem 38.42 on page 22].

We obtain a ring homomorphism

\overline{P}^{G}: \overline{A}(G) → Sw^{f}(G; ℚ), [S] ↦ [R[S]],  (2.9)

where R[S] is the finitely generated free R-module with the finite set S as basis and becomes a RG-module by the G-action on S.

Theorem 1.20 does not carry over to \overline{A}(G) for infinite groups. For instance, the determinant induces a surjective homomorphism

det: Sw(ℤ; ℚ) → ℚ^{∗}, [V] ↦ det(l_{t}: V → V),

where l_{t} is left multiplication with a fixed generator t ∈ ℤ. Given a finite ℤ-set S, the map l_{t}: ℚ[S] → ℚ[S] satisfies (l_{t})^{n} = id for some n ≥ 1 and hence the image of the composition det ∘\overline{P}_{ℤ} is contained in \{±1\}. Therefore the map \overline{P}^{ℤ} of (2.9) is not rationally surjective.

2.5 Maximal Residually Finite Quotients

Let G be a group. Denote by G_{0} the intersection of all normal subgroups of finite index. This is a normal subgroup. Let p: G → G/G_{0} be the projection. Recall that G is called residually finite if for every element g ∈ G with g ≠ 1 there exists a homomorphism onto a finite group which sends g to an element different from 1. If G is countable, then G is residually finite if and only if G_{0} is trivial. The projection p: G → G_{mrf} := G/G_{0} is the projection onto the maximal residually finite quotient of G, i.e., G_{mrf} is residually finite and every epimorphism f: G → Q onto a residually finite group Q factorizes through p

into a composition G ⟶^{p} G_{mrf} ⟶^{\overline{f}} Q. If G is a finitely generated subgroup of GL_{n}(F) for some field F, then G is residually finite (see [35], [51, Theorem 4.2]). Hence for every finitely generated group G each G-representation V with coefficients in a field F is obtained by restriction with p: G → G_{mrf} from a G_{mrf}-representation. In particular every G-representation with coefficient in a field F is trivial if G is finitely generated and G_{mrf} is trivial.

One easily checks that

p^{∗}: \overline{A}(G_{mrf}) ⟶^{≅} \overline{A}(G)

is an isomorphism. In particular we have \overline{A}(G) = ℤ if G_{mrf} is trivial. If G is finitely generated, then

p^{∗}: Sw^{f}(G_{mrf}; F) ⟶^{≅} Sw^{f}(G; F)

is an isomorphism. In particular we have Sw^{f}(G; F) = ℤ if G is finitely generated and G_{mrf} is trivial.

Example 2.10 (\overline{A}(ℤ/p^{∞}) and Sw^{f}(ℤ/p^{∞}; ℚ)). Let ℤ/p^{∞} be the Prüfer group, i.e., the colimit of the directed system of injections of abelian groups ℤ/p → ℤ/p^{2} → ℤ/p^{3} → · · ·. It can be identified with both ℚ/ℤ_{(p)} and ℤ[1/p]/ℤ. We want to show that the following diagram is commutative and consists of isomorphisms

\overline{A}(\{1\}) = ℤ ⟶_{≅}^{p^{∗}}  \overline{A}(ℤ/p^{∞})
\overline{P}^{\{1\}}↓≅  \overline{P}^{ℤ/p^{∞}} ↓≅
Sw^{f}({1}; ℚ) = ℤ ⟶_{≅}^{p^{∗}} Sw^{f}(ℤ/p^{∞}; ℚ)

where p: ℤ/p^{∞} → {1} is the projection. Obviously the diagram commutes and the left vertical arrow is bijective. Hence it remains to show that the horizontal arrows are bijective.

Let f: ℤ/p^{∞} → Q be any epimorphism onto a finite group. Since ℤ/p^{∞} is abelian, Q is a finite abelian group. Since any element in ℤ/p^{n} has p-power order, we conclude from the definition of ℤ/p^{∞} as a colimit that Q is a finite abelian p-group. Since ℚ is p-divisible, the quotient Q must be p-divisible. Therefore Q must be trivial. Hence (ℤ/p^{∞})_{mrf} is trivial and the upper horizontal arrow is bijective.

In order to show that the lower horizontal arrow is bijective, it suffices to show that every (finite-dimensional) rational ℤ/p^{∞}-representation V is trivial. It is enough to show that for every subgroup ℤ/p^{m} its restriction res_{i_{m}} V for the inclusion i_{m}: ℤ/p^{m} → ℤ/p^{∞} is trivial. For this purpose choose a positive integer n such that dim_{ℚ}(V) < (p−1)·p^{n}. Consider the rational ℤ/p^{m+n}-representation res_{i_{m+n}} V. Let p_{k}^{m+n}: ℤ/p^{m+n} → ℤ/p^{k} be the canonical projection. Let ℚ(p^{k}) be the rational ℤ/p^{k}-representation given by adjoining a primitive p^{k}-th root of unity to ℚ. Then the dimension of ℚ(p^{k}) is (p − 1) · p^{k−1}. A complete system of representatives for the isomorphism classes of irreducible rational ℤ/p^{m+n}- representations is {res_{p_{k}^{m+n}} ℚ(p^{k}) | k = 0, 1, 2, … m + n}. Since dim_{ℚ}(V) < (p−1)·p^{n}, there exists a rational ℤ/p^{n}-representation with W with res_{i_{m+n}} V ≅ res_{p_{n}^{m+n}} W. Hence we get an isomorphism of rational ℤ/p^{m}-representations

res_{i_{m}} V ≅ res_{i_{m,m+n}} ∘ res_{p_{n}^{m+n}} W

where i_{m,m+n}: ℤ/p^{m} → ℤ/p^{m+n} is the inclusion. Since the composition p_{n}^{m+n} ∘ i_{m,m+n} is trivial, the rational ℤ/p^{m}-representation res_{i_{m}} V is trivial.

It is not true that

Sw^{f}(ℤ/p^{∞}; ℂ) → Sw^{f}({1}; ℂ) = ℤ

is bijective because Sw^{f}(ℤ/p^{∞}; ℂ) has infinite rank as abelian group (see Example 3.16).

3 The Inverse-Limit-Version of the Burnside Ring

In this section we present the inverse-limit-definition of the Burnside ring for infinite groups.

The orbit category Or(G) has as objects homogeneous spaces G/H and as morphisms G-maps. Let Sub(G) be the category whose objects are subgroups H of G. For two subgroups H and K of G denote by conhom_{G}(H, K) the set of group homomorphisms f: H → K, for which there exists an element g ∈ G with gHg^{−1} ⊂ K such that f is given by conjugation with g, i.e., f = c(g): H → K, h ↦ ghg^{−1}. Notice that c(g) = c(g′) holds for two elements g, g′ ∈ G with gHg^{−1} ⊂ K and g′H(g′)^{−1} ⊂ K if and only if g^{−1}g′ lies in the centralizer C_{G}H = {g ∈ G | gh = hg for all h ∈ H} of H in G. The group of inner automorphisms of K acts on conhom_{G}(H, K) from the left by composition. Define the set of morphisms

mor_{Sub (G)}(H, K) := inn(K)\ conhom_{G}(H, K).

There is a natural projection pr: Or(G) → Sub(G) which sends a homogeneous space G/H to H. Given a G-map f: G/H → G/K, we can choose an element g ∈ G with gHg^{−1} ⊂ K and f(g′H) = g′g^{−1}K. Then pr(f) is represented by c(g): H → K. Notice that mor_{Sub (G)}(H, K) can be identified with the quotient mor_{Or (G)}(G/H, G/K)/C_{G}H, where g ∈ C_{G}H acts on mor_{Or (G)}(G/H, G/K) by composition with R_{g^{−1}} : G/H → G/H, g′H ↦ g′g^{−1}H. We mention as illustration that for abelian G the set of morphisms mor_{Sub (G)}(H, K) is empty if H is not a subgroup of K, and consists of precisely one element given by the inclusion H → K if H is a subgroup in K.

Denote by Or_{FIN} (G) ⊂ Or(G) and Sub_{FIN} (G) ⊂ Sub(G) the full subcategories, whose objects G/H and H are given by finite subgroups H ⊂ G.

Definition 3.1. (The inverse-limit-version of the Burnside ring). The inverse-limit-version of the Burnside ring A_{inv}(G) is defined to be the commutative ring with unit given by the inverse limit of the contravariant functor

A(?): Sub_{FIN} (G) → RINGS, H ↦ A(H).

Since inner automorphisms induce the identity on A(H), the contravariant functor appearing in the definition above is well-defined.

Consider a group homomorphism f: G_{0} → G_{1}. We obtain a covariant functor Sub_{FIN} (f): Sub_{FIN} (G_{0}) → Sub_{FIN} (G_{1}) sending an object H to f(H). A morphism u: H → K given by c(g): H → K for some g ∈ G with gHg^{−1} ⊆ K is sent to the morphism given by c(f(g)): f(H) → f(K). There is an obvious transformation from the composite of the functor A(?): Sub_{FIN} (G_{1}) → RINGS with Sub_{FIN} (f) to the functor A_{inv}(?): Sub_{FIN} (G_{0}) → RINGS. It is given for an object H ∈ Sub_{FIN} (G_{0}) by the ring homomorphism A(f(H)) → A(H) induced by the group homomorphism f|_{H}: H → f(H). Thus we obtain a ring homomorphism A_{inv}(f): A_{inv}(G_{1}) → A_{inv}(G_{0}). So A_{inv} becomes a contravariant functor GROUPS→ RINGS.

Definition 3.1 reduces to the one for finite groups presented in Subsection 1 since for a finite group G the object G ∈ Sub_{FIN} (G) is a terminal object.

There is an obvious ring homomorphism, natural in G,

T^{G}: \overline{A}(G) → A_{inv}(G)  (3.2)

which is induced from the various ring homomorphisms \overline{A}(i_{H}): \overline{A}(G) → \overline{A}(H) = A(H) for the inclusions i_{H}: H → G for each finite subgroup H ⊆ G. The following examples show that it is neither injective nor surjective in general.

3.1 Some Computations of the Inverse-Limit-Version

Example 3.3 (A_{inv}(G) for torsionfree G). Suppose that G is torsionfree. Then Sub_{FIN} (G) is the trivial category with precisely one object and one morphisms. This implies that the projection pr: G → {1} induces a ring isomorphism

A_{inv}(pr): A_{inv}({1}) = ℤ ⟶^{≅} A_{inv}(G).

In particular we conclude from Example 2.6 that the canonical ring homomorphism

T^{ℤ}: \overline{A}(ℤ) → A_{inv}(ℤ)

of (3.2) is not injective.

Example 3.4 (Groups with appropriate maximal finite subgroups). Let G be a discrete group which is not torsionfree. Consider the following assertions concerning G:

(M) Every non-trivial finite subgroup of G is contained in a unique maximal finite subgroup;

(NM) If M ⊆ G is maximal finite, then N_{G}M = M.

The conditions (M) and (NM) imply the following: Let H be a non-trivial finite subgroup of G. Then there is a unique maximal finite subgroup M_{H} with H ⊆ M_{H} and the set of morphisms in Sub_{FIN} (G) from H to M_{H} consists of precisely one element which is represented by the inclusion H → M_{H}. Let {M_{i} | i ∈ I} be a complete set of representatives of the conjugacy classes of maximal finite subgroups of G. Denote by j_{i}: M_{i} → G, k_{i}: {1} → M_{i} and k: {1} → G the inclusions. Then we obtain a short exact sequence

0 → A_{inv}(G) \underrightarrow{A_{inv}(j_{\{1\}})×∏_{i∈I} A_{inv}(j_{i})} A_{inv}(\{1\}) × ∏_{i∈I} A_{inv}(M_{i})

\underrightarrow{Δ−∏_{i∈I} A_{inv}(k_{i})} ∏_{i∈I} A_{inv}(\{1\} → 0,

where Δ: A_{inv}({1} → ∏_{i∈I} A_{inv}({1} is the diagonal embedding. If we define Ã_{inv}(G) as the kernel of A_{inv}(G) → A_{inv}({1}), this gives an isomorphism

Ã_{inv}(G) \underrightarrow{∏_{i∈I} Ã_{inv}(j_{i})} ∏_{i∈I} Ã_{inv}(M_{i}).

Here are some examples of groups Q which satisfy conditions (M) and (NM):

• Extensions 1 → ℤ^{n} → G → F → 1 for finite F such that the conjugation action of F on ℤ^{n} is free outside 0 ∈ ℤ^{n}. The conditions (M) and (NM) are satisfied by [33, Lemma 6.3].

• Fuchsian groups F See for instance [33, Lemma 4.5]). In [33] the larger class of cocompact planar groups (sometimes also called cocompact NEC-groups) is treated.

• Finitely generated one-relator groups G Let G = ⟨(q_{i})_{i∈I} | r⟩ be a presentation with one relation. Let F be the free group with basis {q_{i} | i ∈ I}. Then r is an element in F. There exists an element s ∈ F and an integer m ≥ 1 such that r = s^{m}, the cyclic subgroup C generated by the class \overline{s} ∈ Q represented by s has order m, any finite subgroup of G is subconjugated to C and for any q ∈ Q the implication q^{−1}Cq ∩C ≠ 1 ⇒ q ∈ C holds. These claims follow from [34, Propositions 5.17, 5.18 and 5.19 in II.5 on pages 107 and 108]. Hence Q satisfies (M) and (NM) and the inclusion i: C → G induces an isomorphism

A_{inv}(i): A_{inv}(G) ⟶^{≅} A_{inv}(C).

Example 3.5 (Olshanskii’s group). There is for any prime number p > 10^{75} an infinite finitely generated group G all of whose proper subgroups are finite of order p [37]. Obviously G contains no subgroup of finite index. Hence the inclusion i: {1} → G induces an isomorphism

\overline{A}(i): \overline{A}(G) → A(\{1\}) = ℤ

(see Subsection 2.5). Let H be a finite non-trivial subgroup of G. Then H is isomorphic to ℤ/p and agrees with its normalizer. So the conditions appearing in Example 3.4 are satisfied. Hence we obtain an isomorphism

Ã_{inv}(G) ⟶^{≅} ∏_{\begin{matrix}(H)∈ccs_{f} (G) \\ H≠\{1\} \end{matrix}} Ã_{inv}(ℤ/p),

where ccs_{f}(G) is the set of conjugacy classes of finite subgroups. This implies that the natural map

T^{G}: \overline{A}(G) ⟶^{≅} A_{inv}(G)

of (3.2) is not surjective.

Example 3.6 (Extensions of ℤ^{n} with ℤ/p as quotient). Suppose that G can be written as an extension 1 → A → G → ℤ/p → 1 for some fixed prime number p and for A = ℤ^{n} for some integer n ≥ 0 and that G is not torsionfree. The conjugation action of G on the normal subgroup A yields the structure of a ℤ[ℤ/p]-module on A. Every non-trivial element g ∈ G of finite order G has order p and satisfies

N_{G}⟨g⟩ = C_{G}⟨g⟩ = A^{ℤ/p} × ⟨g⟩.

In particular the conjugation action of N_{G}⟨g⟩ on ⟨g⟩ is trivial. There is a bijection

μ: H^{1}(ℤ/p; A) ⟶^{≅} ccs_{f}(G),

where H^{1}(ℤ/p; A) is the first cohomology of ℤ/p with coefficients in the ℤ[ℤ/p]- module A. If we fix an element g ∈ G of order p and a generator s ∈ ℤ/p, the bijection μ sends [u] ∈ H^{1}(ℤ/p; A) to (⟨ug⟩) of the cyclic group ⟨ug⟩ of order p if [u] ∈ H^{1}(ℤ/p; A) is represented by the element u in the kernel of the second differential A → A, a ↦ ∑_{i=0}^{p−1} s^{i} · a. Hence we obtain an exact sequence

0 → A_{inv}(G) → A_{inv}({1}) × ∏_{H^{1}(ℤ/p;A)} A_{inv}(ℤ/p) → ∏_{H^{1}(ℤ/p;A)} A_{inv}({1}) → 0

This gives an isomorphism

Ã_{inv}(G) ⟶^{≅} ∏_{H^{1}(ℤ/p;A)} Ã(ℤ/p).

3.2 Character Theory and Burnside Congruences for the Inverse-Limit-Version

Next we define a character map for infinite groups G and determine its image generalizing Theorem 1.3.

Let ccs_{f}(G) be the set of conjugacy classes (H) of finite subgroups H ⊆ G. Given a group homomorphism f: G_{0} → G_{1}, let ccs_{f}(G_{0}) → ccs_{f}(G_{1}) be the map sending the G_{0}-conjugacy class of a finite subgroup H ⊆ G_{0} to the G_{1}- conjugacy class of f(H) ⊆ G_{1}. We obtain a covariant functor

ccs: Sub_{FIN} (G) → SETS, H ↦ ccs(H).

For each finite subgroup H ⊆ G the inclusion H → G induces a map ccs(H) → ccs_{f}(G) sending (K) to (K). These fit together to a bijection of sets

ϕ^{G}: colim_{H∈Sub _{FIN} (G)} ccs(H) ⟶^{≅} ccs_{f}(G).  (3.7)

One easily checks that ϕ^{G} is well-defined and surjective. Next we show injectivity. Consider two elements x_{0} and x_{1} in the source of ϕ^{G} with ϕ^{G}(x_{0}) = ϕ^{G}(x_{1}). For i = 0, 1 we can choose an object H_{i} ∈ Sub_{FIN} (G) and an element (K_{i}) ∈ ccs_{f}(H_{i}) such that the structure map of the colimit for the object (H_{i}) sends (K_{i}) to x_{i}. Then ϕ^{G}(x_{0}) = ϕ^{G}(x_{1}) means that the subgroups K_{0} and K_{1} of G are conjugated in G. Hence we can find g ∈ G with gK_{0}g^{−1} = K_{1}. The morphism K_{0} → H_{0} induced by the inclusion yields a map ccs(K_{0}) → ccs(H_{0}) sending (K_{0}) to (K_{0}). The morphism K_{0} → H_{1} induced by the conjugation homomorphism c(g): K_{0} → H_{1} yields a map ccs(K_{0}) → ccs(H_{1}) sending (K_{0}) to (K_{1}). This implies x_{0} = x_{1}.

By the universal property of the colimit we obtain an isomorphism of abelian groups

ψ^{G}: map (colim_{H∈Sub _{FIN} (G)} ccs(H), ℤ)

⟶^{≅} invlim_{(H)∈Sub _{FIN} (G)} map(ccs(H); ℤ). (3.8)

Define the character map

char_{inv}^{G}: A_{inv}(G) → ∏_{(H)∈ccs_{f} (G)} ℤ  (3.9)

to be the map for which the composition with the isomorphism

∏_{(H)∈ccs_{f} (G)} ℤ = map(ccs_{f}(G), ℤ) ⟶^{map(ϕ^{G},id)} map (colim_{H∈Sub _{FIN} (G)} ccs(H), ℤ)

⟶^{ψ^{G}} invlim_{H∈Sub _{FIN} (G)} map(ccs(H), ℤ)

is the map

A_{inv}(G) = invlim_{H∈Sub _{FIN} (G)} map(ccs(H), ℤ) ⟶^{invlim_{H∈Sub _{FIN} (G)} char^{H}}

invlim_{H∈Sub _{FIN} (G)} map(ccs(H); ℤ),

where char^{H}: A(H) → map(ccs(H), ℤ) is the map defined in (1.2).

Theorem 3.10 (Burnside ring congruences for A_{inv}(G)). Let x be an element in ∏_{(H)∈ccs_{f} (G)} ℤ. Then:

(i) The character map

char_{inv}^{G}: A_{inv}(G) → ∏_{(H)∈ccs_{f} (G)} ℤ

of (3.9) is injective;

(ii) The element x lies in the image of the character map

char_{inv}^{G}: A_{inv}(G) → ∏_{(H)∈ccs_{f} (G)} ℤ

of (3.9) if and only if for every finite subgroup K ⊆ G the following condition C(K) is satisfied: The image of x under the map induced by the inclusion i_{K}: K → G

(i_{K})^{∗} ∏_{(H)∈ccs_{f} (G)} ℤ = map(ccs(H), ℤ) → ∏_{(L)∈ccs_{f} (K)} ℤ = map(ccs(K), ℤ)

satisfies the Burnside ring congruences for the finite group K appearing in Theorem 1.3;

(iii) If K_{0} ⊆ K_{1} ⊆ G are two subgroups, then condition C(K_{1}) implies condition C(K_{0}).

Proof. This follows from Theorem 1.3 and the fact that the inverse limit is left exact.

Example 3.11. (Finitely many conjugacy classes of finite subgroups). Suppose that G has only finitely many conjugacy classes of finite subgroups. Then we conclude from Theorem 3.10 that the cokernel of the injective character map char_{inv}^{G}: A_{inv}(G) → ∏_{(H)∈ccs_{f} (G)} ℤ is finite. Hence A_{inv}(G) is a finitely generated free abelian group of rank | ccs_{f}(G)|.

Example 3.12 (A_{inv}(ℤ/p^{∞})). We have introduced in Example 2.10 the Prüfer group ℤ/p^{∞} as colim_{n→∞}ℤ/p^{n}. Each ℤ/p^{n} represents a finite subgroup and each finite subgroup arises in this way. Hence ccs(ℤ/p^{∞}) is on one-to-one-correspondence with ℤ^{≥0} = {n ∈ ℤ | n ≥ 0}. Thus x ∈ ∏_{(H)∈ccs(ℤ/p^{∞})} ℤ can be written as a sequence {x(n)} = {x(n) | n ∈ ℤ^{≥0}}, where x(n) corresponds to the value of x at ℤ/p^{n}.

Consider the finite subgroup ℤ/p^{m}. Its subgroups are given by ℤ/p^{k} for k = 0, 1, 2 … m. Then condition C(ℤ/p^{m}) reduces to the set of congruences for each k = 0, 1, 2, …, m − 1

∑_{C⊂(ℤ/p^{m})/(ℤ/p^{k})} Gen(C) · x(p_{k}^{−1}(C)) ≡ 0 mod p^{m−k},

where p_{k}: ℤ/p^{m} → (ℤ/p^{m})/(ℤ/p^{k}) is the projection. More explicitly, the condition C(ℤ/p^{m}) reduces to the set of congruences for each k = 0, 1, 2, …, m − 1

x(k) + ∑_{i=1}^{m−k} p^{i−1} · (p − 1) · x(k + i) ≡ 0 mod p^{m−k},

which can be rewritten as

∑_{i=0}^{m−k−1} p^{i} · (x(k + i) − x(k + i + 1)) ≡ 0 mod p^{m−k}.

One can see that C(ℤ/p^{m_{1}}) implies C(ℤ/p^{m_{0}}) for m_{0} ≤ m_{1} as predicted by Theorem 3.10 (iii).

Suppose that x satisfies C(ℤ/p^{m}) for m = 0, 1, 2, . . .. We want to show inductively for l = 0, 1, 2 … that x(j) ≡ x(j+1) mod p^{l} holds for j = 0, 1, 2, . . .. The induction begin l = 0 is trivial, the induction step from l − 1 to l ≥ 1 done as follows. The m-th equation appearing in condition C(l + m) yields

∑_{i=0}^{l−1} p^{i} · (x(m + i) − x(m + i + 1)) ≡ 0 mod p^{l}.

Since by induction hypothesis x(k + i) − x(k + i + 1) ≡ 0 mod p^{l−1} holds, this reduces to

x(m) − x(m + 1) ≡ 0 mod p^{l}.

This finishes the induction step.

Since x(j) ≡ x(j + 1) mod p^{l} holds for l = 0, 1, 2, …, we conclude x(j) = x(j + 1) for j = 0, 1, 2, . . .. On the other hand, if x(j) = x(j + 1) holds for j = 0, 1, 2, …, then x obviously satisfies the conditions C(ℤ/p^{m}) for m = 0, 1, 2, . . .. Theorem 3.10 (i) and (ii) shows that the character map

char_{inv}^{ℤ/p^{∞}}: A_{inv}(ℤ/p^{∞}) → ∏_{(H)∈ccs_{f} (ℤ/p^{∞})} ℤ

is injective and its image consists of the copy of the integers given by the constant series. This implies that the projection pr: ℤ/p^{∞} → {1} induces a ring isomorphism

A_{inv}(pr): A_{inv}({1}) = ℤ ⟶^{≅} A_{inv}(ℤ/p^{∞}).

In particular we conclude from Example 2.10 that the canonical ring homomorphism

T^{ℤ/p^{∞}}: \overline{A}(ℤ/p^{∞}) ⟶^{≅} A_{inv}(ℤ/p^{∞})

of (3.2) is bijective.

3.3 The-Inverse-Limit-Version of the Burnside Ring and Rational Representations

Analogously to A_{inv}(G) one defines R_{inv,F} (G) for a field F to be the commutative ring with unit given by the inverse limit of the contravariant functor

R_{inv,F} (?): Sub_{FIN} (G) → RINGS, H ↦ R_{F} (H).

This functor has been studied for F = ℂ for instance in [1], [2]. The system of maps P^{H}: A(H) → R_{ℚ}(H) for the finite subgroups H ⊆ G appearing in Theorem 1.20 defines a ring homomorphism

P_{inv}^{G}: A_{inv}(G) → R_{inv,ℚ}(G).  (3.13)

The system of the restriction maps for every finite subgroup H ⊆ G induces a homomorphism

S^{G,F} : Sw^{f}(G; F) → R_{inv,F} (G).  (3.14)

Although each of the maps P^{H} for the finite subgroups H ⊆ G are rationally surjective by Theorem 1.20, the map P_{inv} need not to be rationally surjective in general, since inverse limits do not respects surjectivity or rationally surjectivity in general.

Example 3.15 (R_{inv;ℚ}(ℤ/^{∞})). Since every finite subgroup of ℤ/p^{∞} is cyclic, we conclude from Theorem 1.20 that the map

P_{inv}^{ℤ/p^{∞}}: A_{inv}(ℤ/p^{∞}) ⟶^{≅} R_{inv,ℚ}(ℤ/p^{∞})

is bijective. We have already seen in Example 3.12 that p^{∗}: A_{inv}({1}) → A_{inv}(ℤ/p^{∞}) is bijective. We conclude from Example 2.10 that the following diagram is commutative and consists of isomorphisms

\overline{A}(ℤ/p^{∞}) ⟶_{≅}^{T^{ℤ/p^{∞}}} A_{inv}(ℤ/p^{∞})
\overline{P}^{ℤ/p^{∞}} ↓≅  _{P_{inv}^{ℤ/p^{∞}}} ↓≅
Sw^{f}(ℤ/p^{∞}; ℚ) ⟶_{≅}^{S_{inv}^{ℤ/p^{∞};ℚ}} R_{inv,ℚ}(ℤ/p^{∞})

and is isomorphic by the maps induced by the projection p: ℤ/p^{∞} → {1} to the following commutative diagram whose corners are all isomorphic to ℤ and whose arrows are all the identity under this identification.

\overline{A}(\{1\}) ⟶_{≅}^{T^{\{1\}}} A_{inv}(\{1\})
\overline{P}^{\{1\}}↓≅  _{P_{inv}^{\{1\}}}↓≅
Sw^{f}({1}; ℚ) ⟶_{≅}^{S_{inv}^{{1};ℚ}} R_{inv,ℚ}({1})

Example 3.16 (Sw^{f}(ℤ/p^{∞}; ℂ) and R_{inv;ℂ}(ℤ/^{∞})). On the other hand let us consider ℂ as coefficients. Consider the canonical map

S^{ℤ/p^{∞},ℂ}: Sw^{f}(ℤ/p^{∞}; ℂ) → R_{inv,ℂ}(ℤ/p^{∞})

which is induced by the restriction maps for all inclusions H → G of finite subgroups. If ϕ_{H}: R_{inv,ℂ}(ℤ/p^{∞}) → R_{ℂ}(H) is the structure map of the inverse limit defining R_{inv,ℂ}(ℤ/p^{∞}) for the finite subgroup H ⊆ ℤ/p^{∞}, then the composition

Sw^{f}(ℤ/p^{∞}; ℂ) ⟶^{S^{ℤ/p^{∞},ℂ}} R_{inv,ℂ}(ℤ/p^{∞}) ⟶^{ψ_{H}} R_{ℂ}(H)

is the map given by restriction with the inclusion of the finite subgroup H in ℤ/p^{∞}. We claim that this composition is surjective. Choose n with H = ℤ/p^{n}. We have to find for every 1-dimensional complex ℤ/p^{n}-representation V a 1dimensional complex ℤ/p^{∞}-representation W such that V is the restriction of V. If V is given by the homomorphism ℤ/p^{n} → S^{1}, \overline{k} ↦ exp(2πik/p^{n}), then the desired W is given by the homomorphism

ℤ/p^{∞} = ℤ[1/p]/ℤ → S^{1}, k ↦ exp(2πik).

This implies that both Sw^{f}(ℤ/p^{∞}; ℂ) and R_{inv,ℂ}(ℤ/p^{∞}) have infinite rank as abelian groups.

4 The Covariant Burnside Group

Next we give a third version for infinite groups which however will only be an abelian group, not necessarily a ring.

Definition 4.1 (Covariant Burnside group). Define the covariant Burnside group \underline{A}(G) of a group G to be the Grothendieck group which is associated to the abelian monoid under disjoint union of G-isomorphism classes of proper cofinite G-sets S, i.e., G-sets S for which the isotropy group of each element in S and the quotient G\S are finite.

The cartesian product of two proper cofinite G-sets with the diagonal action is proper but not cofinite unless G is finite. So for an infinite group G we do not get a ring structure on the Burnside group \underline{A}(G). If G is finite the underlying abelian group of the Burnside ring A(G) is just \underline{A}(G). Given a group homomorphism f: G_{0} → G_{1}, induction yields a homomorphism of abelian group \underline{A}(G_{0}) → \underline{A}(G_{1}) sending [S] to [G_{1} ×_{f} S]. Thus \underline{A} becomes a covariant functor from GROUPSto ℤ − MODULES.

In the sequel we denote by R[S] for a commutative ring R and a set S the free R-module with the set S as R-basis. We obtain an isomorphism of abelian groups

β^{G}: ℤ[ccs_{f}(G)] ⟶^{≅} \underline{A}(G), (H) ↦ [G/H].  (4.2)

The elementary proof of the following lemma is left to the reader.

Lemma 4.3. Let H and K be subgroups of G. Then

(i) G/H^{K} = {gH | g^{−1}Kg ⊂ H};

(ii) The map

ϕ: G/H^{K} → ccs(H), gH ↦ g^{−1}Kg

induces an injection

W_{G}K\(G/H^{K}) → ccs(H);

(iii) The W_{G}K-isotropy group of gH ∈ G/H^{K} is (gHg^{−1} ∩ N_{G}K)/K;

(iv) If H is finite, then G/H^{K} is a finite union of W_{G}K-orbits of the shape W_{G}K/L for finite subgroups L ⊂ W_{G}K.

The next definition makes sense because of Lemma 4.3 above.

Definition 4.4 (L^{2}-character map). Define for a finite subgroup K ⊂ G the L^{2}-character map at (K)

\underline{char}_{K}^{G}: \underline{A}(G) → ℚ, [S] ↦ ∑_{i=1}^{r} |L_{i}|^{−1}

if W_{G}K/L_{1}, W_{G}K/L_{2},…, W_{G}K/L_{r} are the W_{G}K-orbits of S^{K}. Define the global L^{2}-character map by

\underline{char}^{G}: \underline{A}(G) → ℚ[ccs_{f}(G)], [S] ↦ ∑_{(K)∈ccs_{f} (G)} \underline{char}_{K}^{G}([S]) · (K).

Notice that one gets from Lemma 4.3 the following explicit formula for the value of \underline{char}_{K}^{G}(G/H). Namely, define

L_{K}(H) := {(L) ∈ ccs(H) | L conjugate to K in G}.

For (L) ∈ L_{K}(H) choose L ∈ (L) and g ∈ G with g^{−1}Kg = L. Then

g(H ∩ N_{G}L)g^{−1} = gHg^{−1} ∩ N_{G}K;
|(gHg^{−1} ∩ N_{G}K)/K|^{−1} = \frac{|K|}{|H ∩ N_{G}L|}.

This implies

\underline{char}_{K}^{G}(G/H) = ∑_{(L)∈L_{K}(H)} \frac{|K|}{|H ∩ N_{G}L|}.  (4.5)

Remark 4.6 (Burnside integrality relations). Let T ⊆ ccs_{f}(G) be a finite subset closed under taking subgroups, i.e., if (H) ∈ T, then (K) ∈ T for every subgroup K ⊆ H. Since a finite subgroup contains only finitely many subgroups, one can write ccs_{f}(G) as the union of such subsets T. The union of two such subsets is again such a subset. So R[ccs_{f}(G)] is the colimit of the finitely generated free R-modules R[T] if T runs to the finite subsets of ccs_{f}(G) closed under taking subgroups.

Fix a subset T of ccs_{f}(G) closed under taking subgroups. One easily checks using Lemma 4.3 that the composition

ℤ[ccs_{f}(G)] ⟶^{β^{G}} \underline{A}(G) ⟶^{\underline{char}^{G}} ℚ[ccs_{f}(G)]

maps ℤ[T] to ℚ[T]. We numerate the elements in T by (H_{1}), (H_{2}), …, (H_{r}) such that H_{i} is subconjugated to (H_{j}) only if i ≤ j holds. Then the composition

ℚ[ccs_{f}(G)] ⟶^{β^{G}⊗_{ℤ}ℚ} \underline{A}(G) ⊗_{ℤ} ℚ ⟶^{\underline{char}^{G}} ℚ[ccs_{f}(G)]

induces a ℚ-homomorphism

ℚ[T] ⟶^{A_{T}} ℚ[T]

given with respect to the basis {(H_{i}) | i = 1, 2, … r} by a matrix A which is triangular and all whose diagonal entries are equal to 1. The explicit values of the entries in A_{T} are given by (4.5). The matrix A_{T} is invertible and one can actually write down an explicit formula for its inverse matrix B_{T} in terms of Möbius inversion [3, Chaper IV]. The matrix B_{T} yields an isomorphism

B_{T} : ℚ[T] ⟶^{≅} ℚ[T].

Given an element x ∈ ℚ[ccs_{f}(G)], we can find a finite subset T ⊆ ccs_{f}(G) closed under taking subgroups such that x lies already in ℚ[T]. Then x lies in the image of the injective L^{2}-character map

\underline{char}^{G}: \underline{A}(G) → ℚ[ccs_{f}(G)]

of Definition 4.4 if and only if

B_{T} : ℚ[T] ⟶^{≅} ℚ[T]

maps x to an element in ℤ[T]. This means that the following rational numbers

∑_{j=1}^{r} B_{T} (i, j) · x(j)

for i = 1, 2 …, r are integers, where B_{T} (i, j) and x(j) are the components of B_{T} and x belonging to (i, j) and j. We call the condition that these rational numbers are integral numbers the Burnside integrality relations.

Now suppose that G is finite. Then the global L^{2}-character of Definition 4.4 is related to the classical character map (1.2) by the factors |WK|^{−1}, i.e., we have for each subgroup K of G and any finite G-set S

ch_{K}^{G}(S) = |WK|^{−1} · |S^{K}|.  (4.7)

One easily checks that for finite G under the identification (4.7) the Burnside integrality relations can be reformulated as a set of congruences, which consists of one congruence modulo |W_{G}H| for every subgroup H ⊆ G (compare Subsection 1.1).

4.1 Relation to L^{2}-Euler characteristic and Universal Property of the Covariant Burnside Group

The Burnside group \underline{A}(G) can be characterized as the universal additive invariant for finite proper G-CW -complexes and the universal equivariant Euler characteristic of a finite proper G-CW -complex is mapped to the L^{2}-Euler characteristics of the W_{G}H-CW -complexes X^{H} by the character map at (H) for every finite subgroup H ⊆ G. In particular it is interesting to investigate the universal equivariant Euler characteristic of the classifying space for proper G-actions \underline{E}G provided that there is a finite G-CW -model for \underline{E}G. All this is explained in [26, Section 6.6.2].

The relation of the universal equivariant Euler characteristic to the equivariant Euler class which is by definition the class of the Euler operator on a cocompact proper smooth G-manifold with G-invariant Riemannian metric in equivariant K-homology defined by Kasparov is analyzed in [31]. Equivariant Lefschetz classes for G-maps of finite proper G-CW -complexes are studied in [32].

4.2 The Covariant Burnside Group and the Colimit-Version of the Burnside Ring Agree

Instead of the inverse-limit-version one may also consider the colimit-version

A_{cov}(G) := colim_{H∈Sub _{FIN} (G)}A(H)

where we consider A as a covariant functor from Sub_{FIN} (G) to the category of ℤ-modules by induction.

Theorem 4.8 (A_{cov}(G) and \underline{A}(G) agree). The obvious map induced by the various inclusions of a finite subgroup H ⊆ G

V^{G}: A_{cov}(G) ⟶^{≅} \underline{A}(G)

is a bijection of abelian groups.

Proof. Recall that \underline{A}(G) is the free abelian group with the set ccs_{f}(G) of conjugacy classes of finite subgroups as basis. Now the claim follows from the bijection (3.7).

The analogue for the representation ring is an open conjecture. Namely if we define for a field F of characteristic zero

R_{cov,F} (G) := colim_{H∈Sub _{FIN} (G)}R_{F} (H)

we can consider

Conjecture 4.9. The obvious map

W^{G,F} : R_{cov,F} (G) → K_{0}(F [G])

is a bijection of abelian groups.

This conjecture follows from the Farrell-Jones Conjecture for algebraic Ktheory for F [G] as explained in [30, Conjecture 3.3]. No counterexamples are known at the time of writing. For a status report about the Farrell-Jones Conjecture we refer for instance to [30, Section 5].

Let

P_{cov}^{G}: A_{cov}(G) → R_{cov,F} (G).  (4.10)

be the map induced by the maps P^{H}: A(H) → R_{F} (H), [S] ↦ [F [S]] for the various finite subgroups H ⊆ G.

4.3 The Covariant Burnside Group and the Projective Class Group

Given a finite proper G-set, the ℚ-module ℚ[S] with the set S as basis becomes a finitely generated projective ℚG-module by the G-action on S. Thus we obtain a homomorphism

\underline{P}^{G}: \underline{A}(G) → K_{0}(ℚG).  (4.11)

Conjecture 4.12. The map \underline{P}^{G}: \underline{A}(G) → K_{0}(ℚG) is rationally surjective.

This conjecture is motivated by the fact that it is implied by Theorem 1.20 and Theorem 4.8 together with Conjecture 4.9.

4.4 The Covariant Burnside Group as Module over the Finite-Set-Version

If S is a finite G-set and T is a cofinite proper G-set, then their cartesian product with the diagonal G-action is a cofinite proper G-set. Thus we obtain a pairing

μ_{A}^{G}: \overline{A}(G) × \underline{A}(G) → \underline{A}(G), ([S], [T]) ↦ [S × T]  (4.13)

Analogously one defines a pairing

μ_{K}^{G}: Sw^{f}(G; ℚ) × K_{0}(ℚG) → K_{0}(ℚG), ([M], [P]) ↦ [M ⊗_{ℚ} P] (4.14)

which turns K_{0}(ℚG) into a Sw^{f}(G; ℚ)-module. These two pairings are compatible in the obvious sense (see (0.1)).

4.5 A Pairing between the Inverse-Limit-Version and the Covariant Burnside Group

Given a finite group H, we obtain a homomorphism of abelian groups

ν^{H}: A(H) → hom_{ℤ}(A(H), ℤ), [S] ↦ ν^{H}(S),

where ν^{H}(S): A(H) → ℤ maps [T] to |G\(S × T)| for the diagonal G-operation on S × T. A group homomorphism f: H → K induces a homomorphism of abelian groups res_{ϕ}: A(K) → A(H) by restriction and a homomorphism of abelian groups ind_{ϕ}: A(H) → A(K) by induction. The latter induces a homomorphism of abelian groups (ind_{ϕ})^{∗}: A(K) → A(H). One easily checks that the collection of the homomorphisms ν^{H} for the subgroups H ⊆ G induces a natural transformation of the contravariant functors from Sub_{FIN} (G) to ℤ-modules given by A(?) and hom_{ℤ}(A(?), ℤ). Passing to the inverse limit, the canonical isomorphism of abelian groups

hom_{ℤ} (colim_{H∈Sub _{FIN} (G)}A(H), ℤ) ⟶^{≅} invlim_{H∈Sub _{FIN} (G)} hom_{ℤ}(A(H), ℤ)

and the isomorphism appearing in Theorem 4.8 yield a homomorphism of abelian groups

ν_{A}^{G}: A_{inv}(G) → hom_{ℤ}(\underline{A}(G); ℤ)

which we can also write a bilinear pairing

ν_{A}^{G}: A_{inv}(G) × \underline{A}(G) → ℤ.  (4.15)

For a field F of characteristic zero, there is an analogous pairing

ν_{R}^{G}: R_{inv,F} (G) × R_{cov,F} (G) → ℤ  (4.16)

which comes from the various homomorphisms of abelian groups for each finite subgroup H ⊆ G

R_{F} (H) → hom_{ℤ}(R_{F} (H); ℤ), [V] ↦ ([W] ↦ dim_{F} (F ⊗_{F G} (V ⊗_{F} W))) .

The pairings ν_{A}^{G} and ν_{R}^{G} are compatible in the obvious sense (see (0.1)).

The homomorphism of abelian groups \overline{ν_{R}^{G}}: R_{inv,F} (G) → hom_{ℤ}(R_{cov,F} (G), ℤ) associated to the pairing ν_{R}^{G} is injective. Its cokernel is finite if G has only finitely many conjugacy classes of finite subgroups. It is rationally surjective if there is an upper bound on the orders of finite subgroups of G.

Define the homomorphism Q_{A}^{G}: \underline{A}(G) → ℤ and Q_{K}^{G}: K_{0}(ℚG) → ℤ respectively by sending [S] to |G\S| and [P] to dim_{ℚ}(ℚ ⊗_{ℚG} P) respectively. Then the pairings μ_{A}^{G}, ν_{A}^{G}, μ_{K}^{G} and ν_{R}^{G} are compatible in the obvious sense (see (0.1)).

4.6 Some Computations of the Covariant Burnside Group

Example 4.17 (\underline{A}(G) for torsionfree G). Suppose that G is torsionfree. Then the inclusion i: {1} → G induces a ℤ-isomorphism

\underline{A}(i): \underline{A}(\{1\}) = ℤ ⟶^{≅} \underline{A}(G).

Example 4.18 (Extensions of ℤ^{n} with ℤ/p as quotient). Suppose that G can be written as an extension 1 → A → G → ℤ/p → 1 for some fixed prime number p and for A = ℤ^{n} for some integer n ≥ 0 and that G is not torsionfree. We use the notation of Example 3.6 in the sequel. We obtain an exact sequence

0 → ⨁_{H^{1}(ℤ/p;A)} \underline{A}(\{1\}) → \underline{A}(\{1\}) ⊕ ⨁_{H^{1}(ℤ/p;A)} \underline{A}(ℤ/p) → \underline{A}(G) → 0

If we define ˜ \underline{A}(G) as the kernel of \underline{A}(G) → \underline{A}(\{1\}), we obtain an isomorphism

⨁_{H^{1}(ℤ/p;A)} Ã(ℤ/p) ⟶^{≅} ˜ \underline{A}(G).

Let H_{0} be the trivial subgroup and H_{1}, H_{2}, …, H_{r} be a complete set of representatives of the conjugacy classes of finite subgroups. Then r = |H^{1}(ℤ/p; A)| and \underline{A}(G) is the free abelian group of rank r + 1 with \{[G/H_{0}], [G/H_{1}], … [G/H_{r}]\} as basis. Each H_{i} is isomorphic to ℤ/p. We compute using (4.5)

ch_{H_{0}}^{G}(G/H_{0}) = 1;
ch_{H_{0}}^{G}(G/H_{j}) = \frac{1}{p}  j = 1, 2, …, r;
ch_{H_{i}}^{G}(G/H_{j}) = 1  i = j, i, j = 1, 2, …, r;
ch_{H_{i}}^{G}(G/H_{j}) = 0  i ≠ j, i, j = 1, 2, …, r.

The Burnside integrality conditions (see Remark 4.6) become in this case for x = (x(i)) ∈ ⨁_{i=0}^{r} ℚ

x(0) − \frac{1}{p} · ∑_{i=1}^{r} x(i) ∈ ℤ;
x(i) ∈ ℤ i = 1, 2, …, r.

Example 4.19 (Groups with appropriate maximal finite subgroups). Consider the groups appearing in Example 3.4. In the notation of Example 3.4 we get an isomorphism of ℤ-modules

⨁_{i∈I} Ã(M_{i}) \underrightarrow{⨁_{i∈I} ˜ \underline{A}(j_{i})} ˜ \underline{A}(G).

5 Equivariant Cohomology Theories

In this section we recall the axioms of a (proper) equivariant cohomology theory of [27]. They are dual to the ones of a (proper) equivariant homology theory as described in [25, Section 1].

5.1 Axiomatic Description of a G-Cohomology Theory

Fix a group G and an commutative ring R. A G-CW -pair (X, A) is a pair of G-CW -complexes. Recall that a G-CW -complex X is proper if and only if all isotropy groups of X are finite, and is finite if X is obtained from A by attaching finitely many equivariant cells, or, equivalently, if G\X is compact. A G-cohomology theory H_{G}^{∗} with values in R-modules is a collection of covariant functors H_{G}^{n} from the category of G-CW -pairs to the category of R-modules indexed by n ∈ ℤ together with natural transformations δ_{G}^{n}(X, A): H_{G}^{n}(X, A) → H_{G}^{n+1}(A) := H_{G}^{n+1}(A, ∅) for n ∈ ℤ such that the following axioms are satisfied:

• G-homotopy invariance If f_{0} and f_{1} are G-homotopic maps (X, A) → (Y, B) of G-CW -pairs, then H_{G}^{n}(f_{0}) = H_{G}^{n}(f_{1}) for n ∈ ℤ;

• Long exact sequence of a pair Given a pair (X, A) of G-CW -complexes, there is a long exact sequence

⋯ ⟶^{δ_{G}^{n−1}} H_{G}^{n}(X, A) ⟶^{H_{G}^{n}(j)} H_{G}^{n}(X) ⟶^{H_{G}^{n}(i)} H_{G}^{n}(A) ⟶^{δ_{G}^{n}} ⋯,

where i: A → X and j: X → (X, A) are the inclusions;

• Excision Let (X, A) be a G-CW -pair and let f: A → B be a cellular G-map of G-CW -complexes. Equip (X ∪_{f} B, B) with the induced structure of a G-CW -pair. Then the canonical map (F, f): (X, A) → (X ∪_{f} B, B) induces an isomorphism

H_{G}^{n}(F, f): H_{G}^{n}(X, A) ⟶^{≅} H_{G}^{n}(X ∪_{f} B, B);

• Disjoint union axiom Let {X_{i} | i ∈ I} be a family of G-CW -complexes. Denote by j_{i}: X_{i} → ∐_{i∈I} X_{i} the canonical inclusion. Then the map

∏_{i∈I} H_{G}^{n}(j_{i}): H_{G}^{n} (∐_{i∈I} X_{i}) ⟶^{≅} ∏_{i∈I} H_{G}^{n}(X_{i})

is bijective.

If H_{G}^{∗} is defined or considered only for proper G-CW -pairs (X, A), we call it a proper G-cohomology theory H_{G}^{∗} with values in R-modules.

5.2 Axiomatic Description of an Equivariant Cohomology Theory

Let α: H → G be a group homomorphism. Given an H-space X, define the induction of X with α to be the G-space ind_{α} X which is the quotient of G × X by the right H-action (g, x) · h := (gα(h), h^{−1}x) for h ∈ H and (g, x) ∈ G × X. If α: H → G is an inclusion, we also write ind_{H}^{G} instead of ind_{α}.

A (proper) equivariant cohomology theory H_{?}^{∗} with values in R-modules consists of a collection of (proper) G-cohomology theories H_{G}^{∗} with values in Rmodules for each group G together with the following so called induction structure: given a group homomorphism α: H → G and a (proper) H-CW -pair (X, A) there are for each n ∈ ℤ natural homomorphisms

ind_{α}: H_{G}^{n}(ind_{α}(X, A)) → H_{H}^{n}(X, A)  (5.1)

satisfying

(i) Bijectivity If ker(α) acts freely on X, then ind_{α}: H_{G}^{n}(ind_{α}(X, A)) → H_{H}^{n}(X, A) is bijective for all n ∈ ℤ;

(ii) Compatibility with the boundary homomorphisms δ_{H}^{n} ∘ ind_{α} = ind_{α} ∘δ_{G}^{n};

(iii) Functoriality Let β: G → K be another group homomorphism. Then we have for n ∈ ℤ

ind_{β∘α} = ind_{α} ∘ ind_{β} ∘H_{K}^{n}(f_{1}): H_{H}^{n}(ind_{β∘α}(X, A)) → H_{K}^{n}(X, A),

where f_{1} : ind_{β} ind_{α}(X, A) ⟶^{≅} ind_{β∘α}(X, A), (k, g, x) ↦ (kβ(g), x) is the natural K-homeomorphism;

(iv) Compatibility with conjugation For n ∈ ℤ, g ∈ G and a (proper) G-CW -pair (X, A) the homomorphism ind_{c(g) : G→G}: H_{G}^{n}(ind_{c(g) : G→G}(X, A)) → H_{G}^{n}(X, A) agrees with H_{G}^{n}(f_{2}), where f_{2} is the G-homeomorphism f_{2} : (X, A) → ind_{c(g) : G→G}(X, A), x ↦ (1, g^{−1}x) and c(g)(g′) = gg′g^{−1}.

This induction structure links the various G-cohomology theories for different groups G.

Sometimes we will need the following lemma whose elementary proof is analogous to the one in [25, Lemma 1.2].

Lemma 5.2. Consider finite subgroups H, K ⊆ G and an element g ∈ G with gHg^{−1} ⊆ K. Let R_{g^{−1}} : G/H → G/K be the G-map sending g′H to g′g^{−1}K and c(g): H → K be the homomorphism sending h to ghg^{−1}. Let pr: (ind_{c(g) : H→K}{•}) → {•} be the projection to the one-point space {•}. Then the following diagram commutes

H_{G}^{n}(G/K) ⟶^{H_{G}^{n}(R_{g^{−1}} )} H_{G}^{n}(G/H)
ind_{K}^{G} ↓≅  ^{ind}_{H}^{G} ↓≅
H_{K}^{n}({•}) ⟶^{ind_{c(g)} ∘H_{K}^{n}(pr)} H_{H}^{n}({•})

5.3 Multiplicative Structures

Let H_{G}^{∗} be a (proper) G-cohomology theory. A multiplicative structure assigns to a (proper) G-CW -complex X with G-CW -subcomplexes A, B ⊆ X natural R-homomorphisms

∪: H_{G}^{m}(X, A) ⊗_{R} H_{G}^{n}(X, B) → H_{G}^{m+n}(X, A ∪ B).  (5.3)

This product is required to be compatible with the boundary homomorphism of the long exact sequence of a G-CW -pair, namely, for u ∈ H_{G}^{m}(A) and v ∈ H^{n}G(X) and i: A → X the inclusion we have δ(u ∪ v) = δ(u ∪ H^{n}(i)(v). Moreover, it is required to be graded commutative, to be associative and to have a unit 1 ∈ H_{G}^{0}(X) for every (proper) G-CW -complex X.

Let H_{?}^{∗} be a (proper) equivariant cohomology theory. A multiplicative structure on it assigns a multiplicative structure to the associated (proper) G-coho-mology theory H_{G}^{∗} for every group G such that for each group homomorphism α: H → G the maps given by the induction structure of (5.1)

ind_{α}: H_{G}^{n}(ind_{α}(X, A)) ⟶^{≅} H_{H}^{n}(X, A)

are in the obvious way compatible with the multiplicative structures on H_{G}^{∗} and H_{H}^{∗}.

Example 5.4. Equivariant cohomology theories coming from non-equi-variant ones). Let K^{∗} be a (non-equivariant) cohomology theory with multiplicative structure, for instance singular cohomology or topological K-theory. We can assign to it an equivariant cohomology theory with multiplicative structure H_{?}^{∗} in two ways. Namely, for a group G and a pair of G-CW -complexes (X, A) we define H_{G}^{n}(X, A) by K^{n}(G\(X, A)) or by K^{n}(EG ×_{G} (X, A)).

5.4 Equivariant Topological K-Theory

In [29] equivariant topological K-theory is defined for finite proper equivariant CW -complexes in terms of equivariant vector bundles. For finite G it reduces to the classical notion which appears for instance in [4]. Its relation to equivariant KK-theory is explained in [38]. This definition is extended to (not necessarily finite) proper equivariant CW -complexes in [29] in terms of equivariant spectra using Γ-spaces. This equivariant cohomology theory K_{?}^{∗} has the property that for any finite subgroup H of a group G we have

K_{G}^{n}(G/H) = K_{H}^{n}(\{•\}) = \{ \begin{matrix}R_{ℂ}(H) & n even; \\ \{0\} & n odd. \end{matrix}

6 Equivariant Stable Cohomotopy in Terms of Real Vector Bundles

In this section we give a construction of equivariant stable cohomotopy for finite proper G-CW -complexes for infinite groups G in terms of real vector bundles and maps between the associated sphere bundles. The result will be an equivariant cohomology theory with multiplicative structure for finite proper equivariant CW -complexes. It generalizes the well-known approach for finite groups in terms of representations. We will first give the construction, show why it reduces to the classical construction for a finite group and explain why we need to consider equivariant vector bundles and not only representations in the case of an infinite group.

6.1 Preliminaries about Equivariant Vector Bundles

We will need the following notation. Given a finite-dimensional (real) vector space V, we denote by S^{V} its one-point compactification. We will use the point at infinity as the base point of S^{V} in the sequel. Given two finite-dimensional vector spaces V and W, the obvious inclusion V ⊕ W → S^{V} ∧ S^{W} induces a natural homeomorphism

ϕ(V, W): S^{V ⊕W} ⟶^{≅} S^{V} ∧ S^{W} .  (6.1)

Let

∇: S^{ℝ} → S^{ℝ} ∨ S^{ℝ}.  (6.2)

be the pinching map, which sends x > 0 to ln(x) ∈ ℝ ⊆ S^{ℝ} in the first summand, x < 0 to − ln(−x) ∈ ℝ ⊆ S^{ℝ} in the second summand and 0 and ∞ to the base point in S^{ℝ} ∨S^{ℝ}. Under the identification S^{ℝ} = S^{1} this is the standard pinching map S^{1} → S^{1}/S^{0} ≅ S^{1} ∨ S^{1}, at least up to pointed homotopy.

We need some basics about G-vector bundles over proper G-CW -complexes. More details can be found for instance in [29, Section 1]. Recall that a G-CW - complex is proper if and only if all its isotropy groups are finite. A G-vector bundle ξ: E → X over X is a real vector bundle with a G-action on E such that ξ is G-equivariant and for each g ∈ G the map l_{g}: E → E is fiberwise a linear isomorphism. Such a G-vector bundle is automatically trivial in the equivariant sense that for each x ∈ X there is a G-neighborhood U, a G-map f: U → G/H and a H-representation V such that ξ|_{U} is isomorphic as G-vector bundle to the pullback of the G-vector bundle G ×_{H} V → G/H by f. Denote the fiber ξ^{−1}(x) over a point x ∈ X by E_{x}. This is a representation of the finite isotropy group G_{x} of x ∈ X. A map of G-vector bundles (\overline{f}, f) from ξ_{0} : E_{0} → X_{0} to ξ_{1} : E_{1} → X_{1} consists of G-maps \overline{f}: E_{0} → E_{1} and f: X_{0} → X_{1} with ξ_{1} ∘ \overline{f} = f ∘ ξ_{0} such that \overline{f} is fiberwise a (not necessarily injective or surjective) linear map.

Given a G-vector bundle ξ: E → X, let S^{ξ}: S^{E} → X be the locally trivial Gbundle whose fiber over x ∈ X is S^{E_{x}}. Consider two G-vector bundles ξ: E → X and μ: F → X. Let S^{ξ} ∧_{X} S^{μ}: S^{E} ∧_{X} S^{F} → X be the G-bundle whose fiber of x ∈ X is S^{E_{x}} ∧ S^{F_{x}}, in other word it is obtained from S^{ξ}: S^{E} → X and S^{μ}: S^{F} → X by the fiberwise smash product. Define S^{ξ} ∨_{X} S^{μ} analogously. From (6.1) we obtain a natural G-bundle isomorphism

ϕ(ξ, μ): S^{ξ⊕μ} ⟶^{≅} S^{ξ} ∧_{X} S^{μ}.  (6.3)

The next basic lemma is proved in [29, Lemma 3.7].

Lemma 6.4. Let f: X → Y be a G-map between finite proper G-CW -complexes and ξ a G-vector bundle over X. Then there is a G-vector bundle μ over Y such that ξ is a direct summand in f^{∗}μ.

6.2 The Definition of Equivariant Stable Cohomotopy Groups

Fix a proper G-CW -complex X. Let SPHB^{G}(X) be the following category. Objects are G-CW -vector bundles ξ: E → X over X. A morphism from ξ: E → X to μ: F → X is a bundle map u: S^{ξ} → S^{μ} covering the identity id: X → X which fiberwise preserve the base points. (We do not require that u is fiberwise a homotopy equivalence.)

A homotopy h between two morphisms u_{0}, u_{1} from ξ: E → X to μ: F → X is a bundle map h: S^{ξ} × [0, 1] → S^{μ} from the bundle S^{ξ} × id_{[0,1]} : S^{E} × [0, 1] → X×[0, 1] to S^{μ} which covers the projection X×[0, 1] → X and fiberwise preserve the base points such that its restriction to X × {i} is u_{i} for i = 0, 1.

Let \underline{ℝ^{k}} be the trivial vector bundle X × ℝ^{k} → X. We consider it as a Gvector bundle using the trivial G-action on ℝ^{k}. Fix an integer n ∈ ℤ. Given two objects ξ_{i}, two non-negative integers k_{i} with k_{i} + n ≥ 0 and two morphisms

u_{i}: S^{ξ_{i}⊕\underline{ℝ^{k_{i}}}} → S^{ξ_{i}⊕\underline{ℝ^{k_{i}+n}}}

for i = 0, 1, we call u_{0} and u_{1} equivalent, if there are objects μ_{i} in SPHB^{G}(X) for i = 0, 1 and an isomorphism of G-vector bundles v: μ_{0} ⊕ ξ_{0} ⟶^{≅} μ_{1} ⊕ ξ_{1} such that the following diagram in SPHB^{G}(X) commutes up to homotopy

S\underline{μ_{0}⊕ℝ^{k_{1}}} ∧_{X} S^{ξ_{0}⊕\underline{ℝ^{k_{0}}}} ⟶^{id ∧_{X}u_{0}} S\underline{μ_{0}⊕ℝ^{k_{1}}} ∧_{X} S^{ξ_{0}⊕\underline{ℝ^{k_{0}+n}}}
σ_{1}↓  ^{σ_{2}}↓
S^{μ_{0}⊕ξ_{0}⊕\underline{ℝ^{k_{0}+k_{1}}}}  S^{μ_{0}⊕ξ_{0}⊕\underline{ℝ^{k_{0}+k_{1}+n}}}
S^{v⊕id}↓  _{S^{v⊕id}}↓
S^{μ_{1}⊕ξ_{1}⊕\underline{ℝ^{k_{0}+k_{1}}}}  S^{μ_{1}⊕ξ_{1}⊕\underline{ℝ^{k_{0}+k_{1}+n}}}
σ_{3}↓  ^{σ_{4}}↓
S^{μ_{1}⊕\underline{ℝ^{k_{0}}}} ∧_{X} S^{ξ_{1}⊕\underline{ℝ^{k_{1}}}} ⟶^{id ∧_{X}u_{1}} S^{μ_{1}⊕\underline{ℝ^{k_{0}}}} ∧_{X} S^{ξ_{1}⊕\underline{ℝ^{k_{1}+n}}}

where σ_{i} stands for the obvious isomorphism coming from (6.3) and permutation.

We define π_{G}^{n}(X) to be the set of equivalence classes of such morphisms u: S^{ξ⊕\underline{ℝ^{k}}} → S^{ξ⊕\underline{ℝ^{k+n}}} under the equivalence relation mentioned above. It becomes an abelian group as follows.

The zero element is represented by the class of any morphism c: S^{ξ⊕\underline{ℝ^{k}}} → S^{ξ⊕\underline{ℝ^{k+n}}} which is fiberwise the constant map onto the base point.

Consider classes [u_{0}] and [u_{1}] represented by two morphisms of the shape u_{i}: S^{ξ_{i}⊕\underline{ℝ^{k_{i}}}} → S^{ξ_{i}⊕\underline{ℝ^{k_{i}+n}}} for i = 0, 1. Define their sum by the class represented by the morphism

S^{ξ_{0}⊕ξ_{1}⊕\underline{ℝ^{k_{0}+k_{1}+1}}} ⟶^{σ_{1}} S^{ξ_{0}⊕\underline{ℝ^{k_{0}}}} ∧_{X} S^{ξ_{1}⊕\underline{ℝ^{k_{1}}}} ∧_{X} S\underline{ℝ}

⟶^{id ∧_{X} id ∧_{X} \underline{∇}} S^{ξ_{0}⊕\underline{ℝ^{k_{0}}}} ∧_{X} S^{ξ_{1}⊕\underline{ℝ^{k_{1}}}} ∧_{X} (S\underline{ℝ} ∨_{X} S\underline{ℝ})

⟶^{τ} (S^{ξ_{0}⊕\underline{ℝ^{k_{0}}}} ∧_{X} S^{ξ_{1}⊕\underline{ℝ^{k_{1}}}} ∧_{X} S\underline{ℝ}) ∨_{X} (S^{ξ_{0}⊕\underline{ℝ^{k_{0}}}} ∧_{X} S^{ξ_{1}⊕\underline{ℝ^{k_{1}}}} ∧_{X} S\underline{ℝ})

(u_{0}∧_{X}id ∧_{X} id)∨_{X}(id ∧_{X}u_{1}∧_{X}id) ⟶

(S^{ξ_{0}⊕\underline{ℝ^{k_{0}+n}}} ∧_{X} S^{ξ_{1}⊕\underline{ℝ^{k_{1}}}} ∧_{X} S\underline{ℝ}) ∨_{X} (S^{ξ_{0}⊕\underline{ℝ^{k_{0}}}} ∧_{X} S^{ξ_{1}⊕\underline{ℝ^{k_{1}+n}}} ∧_{X} S\underline{ℝ})

⟶^{σ_{5}∨_{X}σ_{6}} (S^{ξ_{0}⊕ξ_{1}⊕\underline{ℝ^{k_{0}+k_{1}+1+n}}}) ∨_{X} (S^{ξ_{0}⊕ξ_{1}⊕\underline{ℝ^{k_{0}+k_{1}+1+n}}})

⟶^{id ∨_{X} id} S^{ξ_{0}⊕ξ_{1}⊕\underline{ℝ^{k_{0}+k_{1}+1+n}}}

where the isomorphisms σ_{i} are given by permutation and the isomorphisms (6.3) τ is given by the distributivity law for smash and wedge-products and \underline{∇} is defined fiberwise by the map ∇ of (6.2).

Consider a class [u] represented by the morphisms of the shape u: S^{ξ⊕\underline{ℝ^{k}}} → S^{ξ⊕\underline{ℝ^{k+n}}}. Define its inverse as the class represented by the composition

S^{ξ⊕\underline{ℝ^{k+1}}} ⟶^{σ_{1}} S^{ξ⊕\underline{ℝ^{k}}} ∧_{X} S\underline{ℝ} ⟶^{u∧_{X} \underline{− id}} S^{ξ⊕\underline{ℝ^{k+n}}} ∧_{X} S\underline{ℝ} ⟶^{σ_{2}} S^{ξ⊕\underline{ℝ^{k+1+n}}},

where \underline{− id} is fiberwise the map − id: ℝ → ℝ. The proof that this defines the structure of an abelian group is essentially the same as the one that the abelian group structure on the stable homotopy groups of a space is well-defined.

Next consider a pair (X, A) of proper G-CW -complexes. In order to define the abelian group π_{G}^{n}(X, A) we consider morphisms u: S^{ξ⊕\underline{ℝ^{k}}} → S^{ξ⊕\underline{ℝ^{k+n}}} with k + n ≥ 0 in SPHB^{G}(X) such that u is trivial over A, i.e., for every point a ∈ A the map u_{a}: S^{ξ_{a}⊕ℝ^{k}} → S^{ξ_{a}⊕ℝ^{k+n}} is the constant map onto the base point. In the definition of the equivalence relation for such pairs we require that the homotopies of two morphisms are stationary over A. Then define π_{G}^{n}(X, A) as the set of equivalence classes of morphism u: S^{ξ⊕\underline{ℝ^{k}}} → S^{ξ⊕\underline{ℝ^{k+n}}} in SPHB^{G}(X) with k + n ≥ 0 which are trivial over A using this equivalence relation. The definition of the abelian group structure goes through word by word.

Notice that in the definition of π_{G}^{n}(X, A) we cannot use as in the classical settings cones or suspensions since these contain G-fixed points and hence are not proper unless G is finite. The properness is needed to ensure that certain basic facts about bundles carry over to the equivariant setting.

6.3 The Proof of the Axioms of an Equivariant Cohomology Theory with Multiplicative Structure

In this subsection we want to prove

Theorem 6.5 (Equivariant stable cohomotopy in terms of equivariant vector bundles). Equivariant stable cohomotopy π_{?}^{∗} defines an equivariant cohomology theory with multiplicative structure for finite proper equivariant CW - complexes. For every finite subgroup H of the group G the abelian groups π_{G}^{n}(G/H) and π_{H}^{n} are isomorphic for every n ∈ ℤ and the rings π_{G}^{0}(G/H) and π_{H}^{0} = A(H) are isomorphic.

Consider a G-map f: (X, A) → (Y, B) of pairs of proper G-CW -complexes. Using the pullback construction one defines a homomorphism of abelian groups

π_{G}^{n}(f): π_{G}^{n}(Y, B) → π_{n}^{G}(X, A).

Thus π_{G}^{n} becomes a contravariant functor from the category of proper G-CW - pairs to the category of abelian groups.

Lemma 6.6. Let f_{0}, f_{1} : (X, A) → (Y, B) be two G-maps of pairs of proper G-CW -complexes. If they are G-homotopic, then π_{G}^{n}(f_{0}) = π_{G}^{n}(f_{1}).

Proof. By the naturality of π_{G}^{n} it suffices to prove that π_{G}^{n}(h) = id holds for the G-map

h: (X, A) × [0, 1] → (X, A) × [0, 1], (x, t) ↦ (x, 0).

Let the element [u] ∈ π_{G}^{n}((X, A) × [0, 1]) be given by the morphism u: S^{ξ⊕\underline{ℝ^{k}}} → S^{ξ⊕\underline{ℝ^{k+n}}} in SPHB^{G}(X × [0, 1]) with k + n ≥ 0 which is trivial over X × \{0, 1\} ∪ A × [0, 1]. By [29, Theorem 1.2] there is an isomorphism of G-vector bundles v: ξ ⟶^{≅} h^{∗}ξ covering the identity id: X ×[0, 1] → X ×[0, 1] such that v restricted to X×{0} is the identity id: ξ|_{X×{0}} → ξ|_{X×{0}}. The composition of morphisms in SPHB^{G}(X × [0, 1])

u′: S^{h^{∗}ξ⊕\underline{ℝ^{k}}} ⟶^{S^{v^{−1}⊕id}} S^{ξ⊕\underline{ℝ^{k}}} ⟶^{u} S^{ξ⊕\underline{ℝ^{k+n}}} ⟶^{S^{v}⊕id} S^{h^{∗}ξ⊕\underline{ℝ^{k+n}}}

has the property that its restriction to X × {0} agrees with the restriction of h^{∗}u to X × {0}. Hence this composite u′ is homotopic to the morphism h^{∗}u itself. Namely, if we write h^{∗}ξ = i_{0}^{∗}ξ × [0, 1] for i_{0} : X → X × [0, 1], x ↦ (x, 0), then the homotopy is given at time s ∈ [0, 1] by the morphism

S^{i_{0}^{∗}ξ⊕\underline{ℝ^{k}}} × [0, 1] → S^{i_{0}^{∗}ξ⊕\underline{ℝ^{k+n}}} × [0, 1], (z, t) ↦ (pr ∘u′(z, st), t)

for pr: S^{i_{0}^{∗}ξ⊕\underline{ℝ^{k+n}}} × [0, 1] → S^{i_{0}^{∗}ξ⊕\underline{ℝ^{k+n}}} the projection. Obviously this homotopy is stationary over A. We conclude from the equivalence relation appearing in the definition of π_{G}^{n} and the definition of π_{G}^{n}(h) that π_{G}^{n}(h)([u]) = [u′] = [u] holds.

Next we define the suspension homomorphism

σ_{G}^{n}(X, A): π_{G}^{n}(X, A) → π_{G}^{n+1}((X, A) × ([0, 1], {0, 1})).  (6.7)

For a G-vector bundle ξ over X let ξ × [0, 1] be the obvious G-vector bundle over X × [0, 1], which is the same as the pullback of ξ for the projection X × [0, 1] → X. Consider a morphism u: S^{ξ} → S^{μ} in SPHB^{G}(X) which is trivial over A. Let

σ(u): S^{ξ×[0,1]} = S^{ξ} × [0, 1] → S^{(μ⊕\underline{ℝ})×[0,1]} = (S^{μ} ∧_{X} S\underline{ℝ}) × [0, 1]

be the morphism in SPHB^{G}(X × [0, 1]) given by

(z, t) ∈ S^{ξ} × [0, 1] ↦ ((u(z) ∧ e(t)) , t) ∈ (S^{μ} ∧_{X} S\underline{ℝ}) × [0, 1],

where e: [0, 1] → S\underline{ℝ} comes from the homeomorphism (0, 1) → ℝ, t ↦ ln(x) − ln(1 − x). The morphism σ(u) is trivial over X × {0, 1} ∪ A × [0, 1]. We define the in (X, A) natural homomorphism of abelian groups σ_{G}^{n}(X, A) by sending the class of u to the class of σ(u).

Lemma 6.8. The homomorphism σ_{G}^{n}(X, A) of (6.7) is bijective for all pairs of proper G-CW -complexes (X, A).

Proof. We want to construct an inverse

τ_{G}^{n+1} : π_{G}^{n+1}((X, A) × ([0, 1], {0, 1})) → π_{G}^{n}(X, A)

of σ_{G}^{n}(X, A). Consider two G-vector bundles ξ and μ over X and a morphism v: S^{ξ×[0,1]} → S^{μ×[0,1]} in SPHB^{G}(X × [0, 1]) which is trivial over A × {0, 1}. Define a morphism in SPHB^{G}(X) which is trivial over A

τ(v): S^{ξ⊕\underline{ℝ}} = S^{ξ} ∧_{X} S\underline{ℝ} → S^{μ}

by sending (z, (e(t)) ∈ S^{ξ} ∧_{X} S\underline{ℝ} to pr ∘v(z, t) for pr: S^{μ×[0,1]} = S^{μ} ×[0, 1] → S^{μ} the projection, e: [0, 1] → S\underline{ℝ} the map defined above and t ∈ [0, 1]. Next consider an element [u] ∈ π_{G}^{n+1}(X, A) represented by a morphism u: S^{ξ⊕\underline{ℝ^{k}}} → S^{ξ⊕\underline{ℝ^{k+n+1}}} in SPHB^{G}(X × [0, 1]) for k + n ≥ 0 which is trivial over X × \{0, 1\} ∪ A × [0, 1]. Choose an isomorphism of G-vector bundles v: ξ_{0} × [0, 1] ⟶^{≅} ξ which covers the identity on X × [0, 1] and is the identity over X × {0}, where ξ_{0} is the restriction of ξ to X = X×\{0\} (see [29, Theorem 1.2]). Let u′: S^{(ξ_{0} ⨁ \underline{ℝ^{k}})×[0,1]} → S^{(ξ_{0} ⨁ \underline{ℝ^{n}})×[0,1]} be the composition

S^{(ξ_{0} ⨁ \underline{ℝ^{k}})×[0,1]} ⟶^{S^{v⊕id}} S^{(ξ ⨁ \underline{ℝ^{k}})} ⟶^{u} S^{ξ⊕\underline{ℝ^{k+n+1}}} ⟶^{S^{v^{−1}⊕id}} S^{(ξ_{0} ⨁ \underline{ℝ^{k+n+1}})×[0,1]}.

Notice that [u] = [u′] holds in π_{G}^{n+1}((X, A) × ([0, 1], {0, 1})). Define τ_{G}^{n+1}([u]) by the class of τ(u′): S^{ξ_{0} ⨁ \underline{ℝ^{k+1}}} → S^{ξ_{0} ⨁ \underline{ℝ^{k+n+1}}}.

Consider [u] ∈ π_{G}^{n}(X, A) represented by the morphism u: S^{ξ⊕\underline{ℝ^{k}}} → S^{ξ⊕\underline{ℝ^{k+n}}} in SPHB^{G}(X) which is trivial over A. Then τ_{G}^{n+1} ∘ σ_{G}^{n}([u]) is represented by the morphism τ ∘ σ(u) which can be identified with

S^{ξ⊕\underline{ℝ^{k+1}}} ⟶^{σ_{1}} S^{ξ⊕\underline{ℝ^{k}}} ∧_{X} S\underline{ℝ} ⟶^{u∧_{X}id} S^{ξ ⨁ \underline{ℝ^{k+n}}} ∧_{X} S\underline{ℝ} ⟶^{σ_{2}} S^{ξ⊕\underline{ℝ^{k+1+n}}⊕\underline{ℝ}}

But the latter morphism represents the same class as u. This shows τ_{G}^{n+1} ∘ σ_{G}^{n}([u]) = [u] and hence τ_{G}^{n+1} ∘ σ_{G}^{n} = id. The proof of σ_{G}^{n}([u]) ∘ τ_{G}^{n+1} = id is analogous.

So far we have only assumed that the G-CW -complex X is proper. In the sequel we will need additionally that it is finite since this condition appears in Lemma 6.4.

Lemma 6.9. Let (X_{1}, X_{0}) be a pair of finite proper G-CW -complexes and f: X_{0} → X_{2} be a cellular G-map of finite proper G-CW -complexes. Define the pair of finite proper G-CW -complexes (X, X_{2}) by the cellular G-pushout

X_{0} ⟶^{f} X_{2}
↓  ↓
X_{1} ⟶_{F} X

Then the homomorphism

π_{G}^{n}(F, f): π_{G}^{n}(X, X_{2}) ⟶^{≅} π_{G}^{n}(X_{1}, X_{0})

is bijective for all n ∈ ℤ.

Proof. We begin with surjectivity. Consider an element a ∈ π_{G}^{n}(X_{1}, X_{0}) represented by a morphism

u: S^{ξ⊕\underline{ℝ^{k}}} → S^{ξ⊕\underline{ℝ^{k+n}}}

in SPHB^{G}(X_{1}) which is trivial over X_{0}. By Lemma 6.4 there is a G-vector bundle μ over X, a G-vector bundle ξ′ over X_{1} and an isomorphism of G-vector bundles v: ξ ⊕ ξ′ ⟶^{≅} F^{∗}μ. Consider the morphism u′ in SPHB^{G}(X_{1}) which is given by the composition

S^{F^{∗}μ⊕\underline{ℝ^{k}}} ⟶^{S^{v^{−1}⊕id}} S^{ξ⊕ξ′⊕\underline{ℝ^{k}}} ⟶^{σ_{1}} S^{ξ′} ∧_{X_{1}} S^{ξ⊕\underline{ℝ^{k}}} ⟶^{id ∧_{X_{1}} u} S^{ξ′} ∧_{X_{1}} S^{ξ⊕\underline{ℝ^{k+n}}}

⟶^{σ_{2}} S^{ξ⊕ξ′⊕\underline{ℝ^{k+n}}} ⟶^{S^{v⊕id}} S^{F^{∗}μ⊕\underline{ℝ^{k+n}}}.

By definition it represents the same element in π_{G}^{n}(X_{1}, X_{0}) as u. Hence we can assume without loss of generality for the representative u of a that the bundle ξ is of the form F^{∗}μ for some G-vector bundle μ over X. Since the morphism u is trivial over X_{0}, we can find a morphism SPHB^{G}(X)

\overline{u}: S^{μ⊕\underline{ℝ^{k}}} → S^{μ⊕\underline{ℝ^{k+n}}}

which is trivial over X_{2} and satisfies F^{∗}(\overline{u}) = u. Hence the morphism \overline{u} defines an element in π_{G}^{n}(X, X_{1}) such that π_{G}^{n}(F, f)([\overline{u}]) = [u] = a holds.

It remains to prove injectivity of π_{G}^{n}(F, f). Consider an element b in the kernel of π_{G}^{n}(F, f). Choose a morphism

u: S^{ξ⊕\underline{ℝ^{k}}} → S^{ξ⊕\underline{ℝ^{k+n}}}

in SPHB^{G}(X) which is trivial over X_{2} and represents b. Then F^{∗}u: S^{F^{∗}ξ⊕\underline{ℝ^{k}}} → S^{F^{∗}ξ⊕\underline{ℝ^{k+n}}} represents zero in π_{G}^{n}(X_{1}, X_{0}). Hence we can find a bundle μ over X_{1} such that the composition

S^{μ⊕F^{∗}ξ⊕\underline{ℝ^{k}}} ⟶^{σ_{1}} S^{μ}∧_{X_{1}}S^{F^{∗}ξ⊕\underline{ℝ^{k}}} ⟶^{id ∧_{X_{1}} u} S^{μ}∧_{X_{1}}S^{F^{∗}ξ⊕\underline{ℝ^{k+n}}} ⟶^{σ_{2}} S^{μ⊕F^{∗}ξ⊕\underline{ℝ^{k+n}}}

is homotopic relative X_{0} to the trivial morphism. As in the proof of the surjectivity we can arrange using Lemma 6.4 that μ is of the shape F^{∗}ξ′ for some G-vector bundle ξ′ over X. By replacing u by id_{S^{ξ′}} ∧_{X}u we can achieve that b = [u] still holds and additionally the morphism in SPHB^{G}(X_{1})

F^{∗}u: S^{F^{∗}(ξ⊕\underline{ℝ^{k}})} → S^{F^{∗}(ξ⊕\underline{ℝ^{k+n}})}

is homotopic relative X_{0} to the trivial map. Since u is trivial over X_{2}, we can extend this homotopy trivially from X_{1} to X to show that u itself is homotopic relative X_{2} to the trivial map. But this means b = [u] = 0 in π_{G}^{n}(X, X_{2}). This finishes the proof of Lemma 6.9.

Lemma 6.10. Let A ⊆ Y ⊂ X be inclusions of finite proper G-CW -complexes. Then the sequence

π_{G}^{n}(X, Y) ⟶^{π_{G}^{n}(j)} π_{G}^{n}(X, A) ⟶^{π_{G}^{n}(i)} π_{G}^{n}(Y, A)

is exact at π_{G}^{n}(X, A) for all n ∈ ℤ, where i and j denote the obvious inclusions.

Proof. The inclusion j ∘ i: (Y, A) → (X, Y) induces the zero map π_{G}^{n}(X, Y) → π_{n}^{G}(Y, A) since an element in a ∈ π_{G}^{n}(X, Y) is represented by a morphism u: S^{ξ⊕\underline{ℝ^{k}}} → S^{ξ⊕\underline{ℝ^{k+n}}} in SPHB^{G}(X) which is trivial over Y and π_{G}^{n}(j ∘ i)(a) is represented by the restriction of u to Y.

Consider an element a ∈ π_{G}^{n}(X, A) which is mapped to zero under π_{G}^{n}(i). Choose a morphism u: S^{ξ⊕\underline{ℝ^{k}}} → S^{ξ⊕\underline{ℝ^{k+n}}} in SPHB^{G}(X) which is trivial over A and represents a. Hence we can find a G-vector bundle ξ′ over Y such that the morphism in SPHB^{G}(Y) given by the composition

S^{ξ′⊕i^{∗}ξ⊕\underline{ℝ^{k}}} ⟶^{σ_{1}} S^{ξ′} ∧_{Y} S^{i^{∗}ξ⊕\underline{ℝ^{k}}} ⟶^{id ∧_{Y} i^{∗}u} S^{ξ′} ∧_{Y} S^{i^{∗}ξ⊕\underline{ℝ^{k+n}}} ⟶^{σ_{2}} S^{ξ′⊕i^{∗}ξ⊕\underline{ℝ^{k+n}}}

is homotopic to the trivial map relative A.

As in the proof of Lemma 6.9 we can arrange using Lemma 6.4 that ξ′ is the of the shape i^{∗}μ for some G-vector bundle μ over X. Hence we can achieve by replacing u by id_{S^{μ}} ∧_{X}u that a = [u] still holds in π_{G}^{n}(X, A) and additionally the morphisms in SPHB^{G}(Y)

i^{∗}u: S^{j^{∗}ξ⊕\underline{ℝ^{k}}} → S^{i^{∗}ξ⊕\underline{ℝ^{k+n}}}

is homotopic relative A to the trivial map. One proves inductively over the number of equivariant cells in X − Y and [29, Theorem 1.2] that this homotopy can be extended to a homotopy relative A of the morphism u to another morphism v: S^{ξ} → S^{ξ⊕\underline{ℝ^{n}}} in SPHB^{G}(X) which is trivial over Y. But this implies that the element [v] ∈ π_{G}^{n}(X, Y) represented by v is mapped to a = [u] under π_{G}^{n}(i).

In order to define a G-cohomology theory we must construct a connecting homomorphism for pairs. In the sequel maps denoted by i_{k} are the obvious inclusions and maps denoted by pr_{k} are the obvious projections. Consider a pair of finite proper G-CW -complexes (X, A) and n ∈ ℤ, n ≥ 0. We define the homomorphism of abelian groups

δ_{G}^{n}(X, A): π_{G}^{n}(A) → π_{G}^{n+1}(X, A)  (6.11)

to be the composition

π_{G}^{n}(A) ⟶^{σ_{G}^{n}(A)} π_{G}^{n+1}(A × [0, 1], A × {0, 1})

⟶^{(π_{G}^{n+1}(i_{1}))^{−1}} π_{G}^{n+1}(X ∪_{A×{0}} A × [0, 1], X ⨿ A × {1})

⟶^{π_{G}^{n+1}(i_{2})} π_{G}^{n+1}(X ∪_{A×{0}} A × [0, 1], A × {1})

⟶^{(π_{G}^{n+1}(pr_{1}))^{−1}} π_{G}^{n+1}(X, A),

where σ_{G}^{n}(A) is the suspension isomorphism (see Lemma 6.8), the map π_{G}^{n+1}(i_{1}) is bijective by excision (see Lemma 6.9) and π_{G}^{n+1}(pr_{1}) is bijective by homotopy invariance (see Lemma 6.6) since pr_{1} is a G-homotopy equivalence of pairs.

Lemma 6.12. Let (X, A) be a pair for proper finite G-CW -complexes. Let i: A → X and j: X → (X, A) be the inclusions. Then the following long sequence is exact and natural in (X, A):

⋯ ⟶^{δ_{G}^{n−1}} π_{G}^{n}(X, A) ⟶^{π_{G}^{n}(j)} π_{G}^{n}(X) ⟶^{π_{G}^{n}(i)} π_{G}^{n}(A)

⟶^{δ_{G}^{n}} π_{G}^{n+1}(X, A) ⟶^{π_{G}^{n+1}(j)} π_{G}^{n+1}(X) ⟶^{π_{G}^{n+1}(i)} π_{G}^{n+1}(A) ⟶^{δ_{G}^{n−1}} · · · .

Proof. It is obviously natural. It remains to prove exactness.

Exactness at π_{G}^{n}(X) follows from Lemma 6.10.

Exactness at π_{G}^{n}(X, A) follows from the following commutative diagram

π_{G}^{n+1} (X ∪_{A×{0}} A × [0, 1], X ⨿ A × {1}) ⟶_{≅}^{(σ_{G}^{n}(A))^{−1}∘π_{G}^{n+1}(i_{1})} π_{G}^{n}(A)
π_{G}^{n+1}(i_{2})↓  ^{δ_{G}^{n+1}(X,A)}↓
π_{G}^{n+1} (X ∪_{A×{0}} A × [0, 1], A × {1})  ⟶_{≅}^{(π_{G}^{n+1}(pr_{1}))^{−1}} π_{G}^{n+1}(X, A)
π_{G}^{n+1}(i_{3})↓  ^{π_{G}^{n+1}(i)}↓
π_{G}^{n+1} (X ⨿ A × {1}, A × {1})  ⟶_{≅}^{π_{G}^{n+1}(i_{4})}  π_{G}^{n+1}(X)

whose left column is exact at π_{G}^{n+1} (X ∪_{A×{0}} A × [0, 1], A × {1}) by Lemma 6.10.

Exactness at π_{G}^{n}(A) is proved analogously by applying Lemma 6.10 to the inclusions

(X × {0, 1}, X × {0} ⨿ A × {1})

⊆ (X × [0, 1], X × {0} ⨿ A × {1})

⊆ (X × [0, 1], X × {0, 1}) .

We conclude from Lemma 6.6, Lemma 6.9 and Lemma 6.12 that π_{G}^{∗} defines a G-cohomology theory on the category of finite proper G-CW -complexes.

Consider a finite proper G-CW -complex X with two subcomplexes A, B ⊆ X. We want to define a multiplicative structure, i.e., a cup-product

∪: π_{G}^{m}(X, A) × π_{G}^{n}(X, B) → π_{G}^{m+n}(X, A ∪ B).  (6.13)

Given elements a ∈ π_{G}^{m}(X, A) and b ∈ π_{G}^{n}(X, B), choose appropriate morphisms u: S^{ξ⊕\underline{ℝ^{k}}} → S^{ξ⊕\underline{ℝ^{m+k}}} and v: S^{η⊕\underline{ℝ^{l}}} → S^{η⊕\underline{ℝ^{n+l}}} in SPHB^{G}(X) representing a and b. Define a ∪ b by the class of the composition of morphisms in SPHB^{G}(X) which are trivial over A ∪ B.

S^{ξ⊕η⊕\underline{ℝ^{k+l}}} ⟶^{σ_{1}} S^{ξ⊕\underline{ℝ^{k}}} ∧_{X} S^{η⊕\underline{ℝ^{l}}}

⟶^{u∧_{X}v} S^{ξ⊕\underline{ℝ^{k+m}}} ∧_{X} S^{η⊕\underline{ℝ^{l+n}}} ⟶^{σ_{2}} S^{ξ⊕η⊕\underline{ℝ^{k+l+m+n}}}.

Next we deal with the induction structure. Consider a group homomorphism α: H → G. The pullback construction for the α: H → G-equivariant map X → ind_{α} X = G ×_{α} X, x ↦ 1G ×_{α} x defines a functor SPHB^{G}(ind_{α} X) → SPHB^{H}(X) which yields a homomorphism of abelian groups

ind_{α}: π_{G}^{n} (ind_{α}(X, A)) → π_{H}^{n}(X).  (6.14)

Now suppose that the kernel of H acts trivially on X. Let ξ: E → X be a Hvector bundle. Then G ×_{α} X is a proper H-CW -complex. Since H acts freely on X, we obtain a well-defined G-vector bundle G ×_{α} ξ: G ×_{α} E → G ×_{α} X. Thus we obtain a functor ind_{α}: SPHB^{H}(X) → SPHB^{G}(G ×_{α} X). It defines a homomorphism of abelian groups

ind_{α}: π_{H}^{n}(X) → π_{G}^{n} (ind_{α}(X, A)) .  (6.15)

which turns out to be an inverse of the induction homomorphism (6.14).

Now we have all ingredients of an equivariant cohomology theory with a multiplicative structure. We leave it to the reader to verify all the axioms. This finishes the proof of Theorem 6.5.

6.4 Comparison with the Classical Construction for Finite Groups

Next we want to show that for a finite group G our construction reduces to the classical one. We first explain why the finite group case is easier.

Remark 6.16 (Advantages in the case of finite groups). The finite group case is easier because for finite groups the following facts are true. The first fact is that every G-CW -complex is proper. Hence one can view pointed G-CW - complexes, where the base point is fixed under the G-action and one can carry out constructions like mapping cones without losing the property proper. We need proper to ensure that certain basic facts about G-vector bundles are true (see [29, Section 1]). The second fact is that every G-vector bundle over a finite G-CW -complex ξ is a direct summand in a trivial G-vector bundle, i.e., a Gvector bundle given by the projection V × X → X for some G-representation V. This makes for instance Lemma 6.4 superfluous whose proof is non-trivial in the infinite group case (see [29, Lemma 3.7]).

Next we identify π_{G}^{n}(X) defined in Subsection 6.2 with the classical definitions which we have explained in Subsection 1.4 provided that G is a finite group.

Consider an element a ∈ π_{G}^{n}(X) with respect to the definition given in Subsection 1.4. Obviously we can find a positive integer k ∈ ℤ with k + n ≥ 0 such that a is represented for some complex G-representation V by a G-map f: S^{V} ∧S^{k} ∧X_{+} → S^{V} ∧S^{k+n}. Let ξ be the trivial G-vector bundle V ×X → X. Define a map

\overline{f}: S^{ξ⊕\underline{ℝ^{k}}} ⟶^{σ_{1}} (S^{V} ∧ S^{k}) × X ⟶^{f×pr_{X}} (S^{V} ∧ S^{k+n}) × X ⟶^{σ_{2}} S^{ξ⊕\underline{ℝ^{k+n}}}.

It is a morphism in SPHB^{G}(X) and hence defines an element a′ ∈ π_{G}^{n}(X) with respect to the definition of Subsection 6.2. Thus we get a homomorphism of abelian groups a ↦ a′ from the definition of Subsection 1.4 to the one of Subsection 6.2.

Consider a morphism u: S^{ξ⊕\underline{ℝ^{k}}} → S^{ξ⊕\underline{ℝ^{k+n}}} in SPHB^{G}(X) representing an element in b = [u] in π_{G}^{n}(X) as defined in Subsection 6.2. Choose a G-vector bundle μ, a complex G-representation V and an isomorphism of (real) G-vector bundles ϕ: μ ⊕ ξ ⟶^{≅} V × X. Then the morphism

v: S^{V ⊕ℝ^{k}} × X ⟶^{(ϕ⊕id)^{−1}} S^{μ⊕ξ⊕\underline{ℝ^{k}}} ⟶^{σ_{1}} S^{μ} ∧_{X} S^{ξ⊕\underline{ℝ^{k}}}

⟶^{id ∧u} S^{μ} ∧_{X} S^{ξ⊕\underline{ℝ^{k+n}}} ⟶^{σ_{2}} S^{μ⊕ξ⊕\underline{ℝ^{k+n}}} ⟶^{ϕ⊕id} S^{V ⊕ℝ^{k+n}} × X

is equivalent to u and hence b = [v]. Since v covers the identity and is fiberwise a pointed map, its composition with the projection S^{V ⊕ℝ^{k+n}} × X → S^{V ⊕ℝ^{k+n}} yields a map

\overline{v}: S^{V ⊕ℝ^{k}} ∧ X_{+} → S^{V ⊕ℝ^{k+n}}.

It defines an element b′ := [\overline{v}] in π_{G}^{n}(X) with respect to the definition given in Subsection 1.4. The map b ↦ b′ is the inverse of the map a → a′ before.

Remark 6.17 (Why consider G-vector bundles?). One may ask why we consider G-vector bundles ξ in Section 6. It would be much easier if we would only consider trivial G-vector bundles V × X for G-representations V. Then we would not need Lemma 6.4. The proof that π_{G}^{∗} is a G-cohomology theory with a multiplicative structure would go through and for finite groups we would get the classical notion. The problem is that the induction structure does not exists anymore as the following example shows.

Consider a finitely generated group such that G_{mrf} is trivial. Then any Grepresentation V is trivial (see Subsection 2.5). This implies that a morphism u: S^{ξ⊕\underline{ℝ^{k}}} → S^{ξ⊕\underline{ℝ^{k+n}}} in SPHB^{G}(X) for ξ = V × X is the same as a (nonequivariant) map

S^{V ⊕ℝ^{k}} ∧ (G\X_{+}) → S^{V ⊕ℝ^{k+n}}.

This yields an identification of π_{G}^{n}(X) with respect to the definition, where all G-vector bundles are of the shape X×V, with the (non-equivariant) stable cohomotopy group π_{s}^{n}(G\X). If G contains a non-trivial subgroup H ⊆ G, then the existence of an induction structure would predict for X = G/H that π_{G}^{n}(G/H) is isomorphic to π_{H}^{n}, which is in general different from π_{s}^{n}(G\(G/H)) = π_{{1}}^{n}.

So we need to consider G-vector bundles in order to get induction structures and hence an equivariant cohomology theory. In particular our definition guarantees π_{G}^{n}(G/H) = π_{H}^{n} for every group G with a finite subgroup H ⊆ G.

Remark 6.18 (The coefficients of equivariant stable cohomotopy). It is important to have information about the values π_{G}^{n}(G/H) for a finite subgroup H ⊆ G of a group G. By the induction structure and the identification above π_{G}^{n}(G/H) agrees with the abelian groups π_{H}^{n} = π_{−n}^{H} defined in Subsection 1.4.

The equivariant homotopy groups π_{−n}^{H} are computed in terms of the splitting due to Segal and tom Dieck (see [49, Theorem 7.7 in Chapter II on page 154], [41, Proposition 2]) by

π_{G}^{n}(G/H) = π_{−n}^{H} = ⨁_{(K)∈ccs(H)} π_{−n}^{s}(BW_{H}K).

The abelian group π_{q}^{s} is finite for q ≥ 1 by a result of Serre [42] (see also [18]), is ℤ for q = 0 and is trivial for q ≤ −1. Since W_{H}K is finite, H̃_{p}(BW_{H}K; ℤ) is finite for all p ∈ ℤ. We conclude from the Atiyah-Hirzebruch spectral sequence that π_{−n}^{s}(BW_{H}K) is finite for n ≤ −1. This implies |π_{G}^{n}(G/H)| < ∞ for n ≤ −1 and that π_{G}^{n}(G/H) = 0 for n ≥ 1. We know already π_{H}^{0} = A(H) from Theorem 1.13. Thus we get

|π_{G}^{n}(G/H)| < ∞  n ≤ −1;
π_{G}^{0}(G/H) = A(H);
π_{G}^{n}(G/H) = {0}  n ≥ 1.

Remark 6.19. (Equivariant stable Cohomotopy for arbitrary G-CW - complexes). In order to construct an equivariant cohomology theory or an (equivariant homology theory) for arbitrary G-CW -complexes it suffices to construct a contravariant (covariant) functor from the category of small groupoids to the category of spectra (see [39], [30, Proposition 6.8]). In a different paper we will carry out such a construction yielding equivariant cohomotopy and homotopy for arbitrary equivariant CW -complexes and will identify the result with the one presented here for finite proper G-CW -complexes.

6.5 Rational Computation of Equivariant Cohomotopy

The cohomotopy theoretic Hurewicz homomorphism yields a transformation of cohomology theories

π_{s}^{∗}(X) → H^{∗}(X; ℤ)

from the (non-equivariant) stable cohomotopy to singular cohomology with ℤcoefficients. It is rationally an isomorphism provided that X is a finite CW - complex. It is compatible with the multiplicative structures. The analogue for equivariant cohomotopy is described next.

Let G be a group and H ⊆ G be a finite subgroup. Consider a pair of finite proper G-CW -complexes (X, A). Lemma 4.3 implies that (X^{H}, A^{H}) is a pair of finite proper W_{G}H-CW -complexes and W_{G}H\(X^{H}, A^{H}) is a pair of finite CW -complexes. Taking the H-fixed point set yields a homomorphism

α_{(H)}^{n}(X, A): π_{G}^{n}(X, A) → π_{W_{G}H}^{n}(X^{H}, A^{H}).

This map is natural and compatible with long exact sequences of pairs and Mayer-Vietoris sequences.

The induction structure with respect to the homomorphism W_{G}H → {1} yields a homomorphism

β_{W_{G}H}^{n}(X^{H}, A^{H}): π_{s}^{n} (W_{G}H\(X^{H}, A^{H})) → π_{W_{G}H}^{n}(X^{H}, A^{H}).

We claim that β_{W_{G}H}^{n}(Z, B) is a rational isomorphism for any pair of finite proper W_{G}H-CW -complexes (Z, B). Since β_{W_{G}H}^{∗} is natural and compatible with the long exact sequences of pairs and Mayer-Vietoris sequences, it suffices to prove the claim for Z = W_{G}H/L and B = ∅ for any finite subgroup L ⊂ W_{G}H. But then β_{W_{G}H}^{n} reduces to the obvious map π^{n}({•}) → π_{L}^{n} which is a rational isomorphism by Remark 6.18.

Let

h^{n}(X^{H}, A^{H}): π_{s}^{n} (W_{G}H\(X^{H}, A^{H})) → H^{n} (W_{G}H\(X^{H}, A^{H}); ℤ)

be the cohomotopy theoretic Hurewicz homomorphism. Let

γ^{n} (W_{G}H\(X^{H}, A^{H})) : H^{n} (W_{G}H\(X^{H}, A^{H}); ℤ) ⊗_{ℤ} ℚ

→ H^{n} (W_{G}H\(X^{H}, A^{H}); ℚ)

be the natural map. Define a ℚ-homomorphism by the composition

ζ_{G}^{n}(X, A)_{(H)} : π_{G}^{n}(X, A) ⊗_{ℤ} ℚ ⟶^{α_{(H)}^{n}(X,A)⊗_{ℤ}id_{ℚ}} π_{W_{G}H}^{n}(X^{H}, A^{H}) ⊗_{ℤ} ℚ

⟶^{(β_{W_{G}H}^{n}(X^{H},A^{H})⊗_{ℤ}id_{ℚ})^{−1}} π_{s}^{n} (W_{G}H\(X^{H}, A^{H})) ⊗_{ℤ} ℚ

⟶^{h^{n}(X^{H},A^{H})⊗_{ℤ}id_{ℚ}} H^{n} (W_{G}H\(X^{H}, A^{H}); ℤ) ⊗_{ℤ} ℚ

⟶^{γ^{n}(W_{G}H\(X^{H},A^{H}))} H^{n} (W_{G}H\(X^{H}, A^{H}); ℚ) .

Define

ζ_{G}^{n}(X, A) = ∏_{(H)∈ccs(G)} ζ_{G}^{n}(X, A)_{(H)} : π_{G}^{n}(X, A) ⊗_{ℤ} ℚ

→ ∏_{(H)∈ccs(G)} H^{n} (W_{G}H\(X^{H}, A^{H}); ℚ) . (6.20)

Theorem 6.21. The maps

ζ_{G}^{n}(X, A): π_{G}^{n}(X, A) ⊗_{ℤ} ℚ ⟶^{≅} ∏_{(H)∈ccs(G)} H^{n} (W_{G}H\(X^{H}, A^{H}); ℚ)

are bijective for all n ∈ ℤ and all pairs of finite proper G-CW -complexes (X, A). They are compatible with the obvious multiplicative structures.

Proof. One easily checks that ζ_{G}^{∗} defines a transformation of G-homology theories, i.e., is natural in (X, A) and compatible with long exact sequences of pairs and Mayer-Vietoris sequences. Hence it suffices to show that ζ_{G}^{n}(G/K) is bijective for all n ∈ ℤ and finite subgroups K ⊂ G. The source and target of ζ_{G}^{n}(G/K) are trivial for n ≠ 0 (see Remark 6.18). The map ζ_{G}^{0}(G/K) can be identified using Lemma 4.3 (ii) with the rationalization of the character map

char^{H}: A(H) → ∏_{(H)∈ccs(G)} ℤ

defined in (1.2) which is bijective by Theorem 1.3.

6.6 Relating Equivariant Stable Cohomotopy and Equivariant Topological K-Theory

We have introduced two equivariant cohomology theories with multiplicative structure, namely equivariant cohomotopy (see Theorem 6.5) and equivariant topological K-theory (see Subsection 5.4).

Let X be a finite proper G-CW -complex with G-CW -subcomplexes A and B and let a ∈ π_{G}^{m}(X, A) be an element. We want to assign to it for every m ∈ ℤ a homomorphism of abelian groups

ϕ_{G}^{m,n}(X, A)(a): K_{G}^{n}(X, B) → K_{G}^{m+n}(X, A ∪ B).  (6.22)

Choose an integer k ∈ ℤ with k ≥ 0, k + m ≥ 0 and a morphism u: S^{ξ⊕\underline{ℝ^{k}}} → S^{ξ⊕\underline{ℝ^{k+m}}} in SPHB^{G}(X) which is trivial over A and represents a. Let v be the morphism in SPHB^{G}(X) which is given by the composite

v: S^{ξ⊕ξ⊕\underline{ℝ^{k}}} ⟶^{σ} S^{ξ} ∧_{X} S^{ξ⊕\underline{ℝ^{k}}} ⟶^{id ∧_{X}u} S^{ξ} ∧_{X} S^{ξ⊕\underline{ℝ^{k+m}}} ⟶^{σ^{−1}} S^{ξ⊕ξ⊕\underline{ℝ^{k+m}}}.

Then v is another representative of a. The bundle ξ ⊕ ξ carries a canonical structure of a complex vector bundle and we denote this complex vector bundle by ξ_{ℂ}.

Let σ^{k}(X, A ∪ B): K_{G}^{m+n}(X, A ∪ B) ⟶^{≅} K_{G}^{m+n+k} ((X, A ∪ B) × (D^{k}, S^{k−1}))

be the suspension isomorphism. Let pr_{k}: X × D^{k} → X be the projection and pr_{k}^{∗} ξ_{ℂ} be the complex vector bundle obtained from ξ_{ℂ} by the pull back construction. Associated to it is a Thom isomorphism

T_{pr_{k}^{∗} ξ_{ℂ}}^{m+n+k}: K_{G}^{m+n+k} ((X, A ∪ B) × (D^{k}, S^{k−1}))

⟶^{≅} K_{G}^{m+n+k+2·dim(ξ)} (S^{pr_{k}^{∗} ξ_{ℂ}}, S^{pr_{k}^{∗} ξ_{ℂ}|_{X×S^{k−1}∪(A∪B)×D^{k}}} ∪ (X × D^{k})_{∞}) ,

where (X × D^{k})_{∞} is the copy of X × D^{k} given by the various points at infinity in the fibers S^{pr_{k}^{∗} ξ_{ℂ}} and pr_{k}^{∗} ξ_{ℂ}|_{X×S^{k−1}∪(A∪B)×D^{k}} is the restriction of pr_{k}^{∗} ξ_{ℂ} to X × S^{k−1} ∪ (A ∪ B) × D^{k} (see [29, Theorem 3.14]). Let

p_{k}: (S^{pr_{k}^{∗} ξ_{ℂ}}, S^{pr_{k}^{∗} ξ_{ℂ}|_{X×S^{k−1}∪(A∪B)×D^{k}}} ∪ (X × D^{k})_{∞})

(S^{ξ⊕ξ⊕\underline{ℝ^{k}}}, S^{ξ⊕ξ⊕\underline{ℝ^{k}}|_{A∪B}} ∪ X_{∞})

be the obvious projection which induces by excision an isomorphism on K_{G}^{∗}. Define an isomorphism

μ^{m+n,m+n+k+2·dim(ξ)} : K_{G}^{m+n}(X, A ∪ B)

→ K_{G}^{m+n+k+2·dim(ξ)} (S^{ξ⊕ξ⊕\underline{ℝ^{k}}}, S^{ξ⊕ξ⊕\underline{ℝ^{k}}|_{A∪B}} ∪ X_{∞})

by the composite K_{G}^{m+n+k+2·dim(ξ)}(p_{k})^{−1} ∘ T_{pr_{k}^{∗} ξ_{ℂ}}^{n+m+k} ∘ σ^{k}(X, A ∪ B). Define

μ^{n,m+n+k+2·dim(ξ)} : K_{G}^{n}(X, B)

→ K_{G}^{m+n+k+2·dim(ξ)} (S^{ξ⊕ξ⊕\underline{ℝ^{k+m}}}, S^{ξ⊕ξ⊕\underline{ℝ^{k+m}}|_{B}} ∪ X_{∞})

analogously. Let the desired map ϕ_{G}^{m,n}(X, A)(a) be the composite

ϕ_{G}^{m,n}(X; A, B)(a): K_{G}^{n}(X, B)

⟶^{μ^{n,m+n+k+2·dim(ξ)}} K_{G}^{m+n+k+2·dim(ξ)} (S^{ξ⊕ξ⊕\underline{ℝ^{k+m}}}, S^{ξ⊕ξ⊕\underline{ℝ^{k+m}}|_{B}} ∪ X_{∞})

⟶^{K_{G}^{m+n+k+2·dim(ξ)}(v)} K_{G}^{m+n+k+2·dim(ξ)} (S^{ξ⊕ξ⊕\underline{ℝ^{k}}}, S^{ξ⊕ξ⊕\underline{ℝ^{k}}|_{A∪B}} ∪ X_{∞})

⟶^{(μ^{m+n,m+n+k+2·dim(ξ)})^{−1}} K_{G}^{m+n}(X, A ∪ B).

We leave it to the reader to check that the definition of ϕ_{G}^{m,n}(X; A, B)(a) is independent of the choices of k and u. The maps ϕ_{G}^{m,n}(X; A, B)(a) for the various elements a ∈ π_{G}^{m}(X, A) define pairings

ϕ_{G}^{m,n}(X; A, B): π_{G}^{m}(X, A) × K_{G}^{n}(X, B) → K_{G}^{m+n}(X, A ∪ B). (6.23)

The verification of the next theorem is left to the reader.

Theorem 6.24. (Equivariant topological K-theory as graded algebra over equivariant stable cohomotopy).

(i) Naturality The pairings ϕ_{G}^{m,n}(X; A.B) are natural in (X; A, B);

(ii) Algebra structure The collection of the pairings ϕ_{G}^{m,n}(X; ∅, A) defines on K_{G}^{∗}(X, A) the structure of a graded algebra over the graded ring π_{G}^{∗}(X) ;

(iii) Compatibility with induction Let ϕ: H → G be a group homomorphism and (X, A) be a pair of proper finite G-CW -complexes. Then the following diagram commutes

π_{G}^{m} (ind_{α}(X, A)) × K_{G}^{n} (ind_{α}(X, B)) ⟶^{ϕ_{G}^{m,n}(ind_{α}(X;A,B))} K_{G}^{m+n} (ind_{α}(X, A ∪ B))
ind_{α} × ind_{α}↓  ^{ind}_{α}↓
π_{H}^{m}(X, A) × K_{H}^{n}(X, B)  ⟶^{ϕ_{H}^{m,n}(X;A,B)}  K_{H}^{m+n}(X, A ∪ B)

(iv) For a ∈ π_{G}^{m−1}(A) and b ∈ K_{G}^{n}(X) we have

ϕ_{G}^{m,n}(X; ∅, ∅)(δ(a), b) = δ (ϕ_{G}^{m−1,n}(A; ∅, ∅)(a, K_{G}^{n}(j)(b))) ,

where δ: π^{m−1}(A) → π_{G}^{m}(X) and δ: K_{G}^{m+n−1}(A) → K_{G}^{m+n}(X) are boundary operators for the pair (X, A) and j: A → X is the inclusion.

For every pair (X, A) of finite proper G-CW -complexes define a homomorphism

ψ_{G}^{n}(X, A): π_{G}^{n}(X, A) → K_{G}^{n}(X, A),  a ↦ ϕ_{G}^{n,0}(X, A, ∅)(a, 1_{X}), (6.25)

where 1_{X} ∈ K_{G}^{0}(X) is the unit element. Then Theorem 6.24 implies

Theorem 6.26 (Transformation from equivariant stable cohomotopy to equivariant topological K-theory). We obtain a natural transformation of equivariant cohomology theories with multiplicative structure for pairs of equivariant proper finite CW -complexes by the maps

ψ_{?}^{∗}: π_{?}^{∗} → K_{?}^{∗}.

If H ⊆ G is a finite subgroup of the group G, then the map

ψ_{G}^{n}(G/H): π_{G}^{n}(G/H) → K_{G}^{0}(G/H)

is trivial for n ≥ 1 and agrees for n = 0 under the identifications π_{G}^{0}(G/H) = π_{H}^{0} = A(H) and K_{G}^{0}(G/H) = K_{H}^{0}({•}) = R_{ℂ}(H) with the ring homomorphism

A(H) → R_{ℂ}(H), [S] ↦ [ℂ[S]]

which assigns to a finite H-set the associated complex permutation representation.

7 The Homotopy Theoretic Burnside Ring

In this section we introduce another version of the Burnside ring which is of homotopy theoretic nature and probably the most sophisticated and interesting one.

7.1 Classifying Space for Proper G-Actions

We need the following notion due to tom Dieck [44].

Definition 7.1 (Classifying space for proper G-actions). A model for the classifying space for proper G-actions is a proper G-CW -complex \underline{E}G such that \underline{E}G^{H} is contractible for every finite subgroup H ⊆ G.

Recall that a G-CW -complex is proper if and only if all its isotropy groups are finite. If \underline{E}G is a model for the classifying space for proper G-actions, then for every proper G-CW -complex X there is up to G-homotopy precisely one G-map X → \underline{E}G. In particular two models are G-homotopy equivalent and the G-homotopy equivalence between two models is unique up to G-homotopy. If G is finite, a model for \underline{E}G is G/G. If G is torsionfree, \underline{E}G is the same as EG which is by definition the total space of the universal principal G-bundle G → EG → BG.

Here is a list of groups G together with specific models for \underline{E}G with the property that the model is a finite G-CW -complex.

G  \underline{E}G

word hyperbolic groups

Rips complex

discrete cocompact subgroup G ⊆ L of a Lie group L with finite π_{0}(L)

L/K for a maximal compact subgroup K ⊆ L

G acts by isometries properly and cocompactly on a CAT(0)-space X, for instance on a tree or a simply-connected complete Riemannian manifold with non-positive sectional curvature

X

arithmetic groups Borel-Serre completion mapping class groups Teichmüller space outer automorphisms of finitely generated free groups

outer space

More information and more references about \underline{E}G can be found for instance in [6] and [28].

7.2 The Definition of the Homotopy Theoretic Burnside Ring

We have introduced the equivariant cohomology theory with multiplicative structure for proper finite equivariant CW -complexes π_{?}^{∗} in Section 6 and the classifying space \underline{E}G for proper G-actions in Subsection 7.1.

Definition 7.2 (Homotopy theoretic Burnside ring). Let G be a (discrete) group such that there exists a finite model \underline{E}G for the universal space for proper G-actions. Define the homotopy theoretic Burnside ring to be

A_{ho}(G) := π_{G}^{0}(\underline{E}G).

If G is finite, π_{G}^{0}(\underline{E}G) agrees with π_{G}^{0} which is isomorphic to the Burnside ring A(G) by Theorem 1.13. So the homotopy theoretic definition A_{ho}(G) reflects this aspect of the Burnside ring which has not been addressed by the other definitions before.

After the program described in Remark 6.19 has been carried out, the assumption in Definition 7.2 that there exists a finite model for \underline{E}G can be dropped and thus the Homotopy Theoretic Burnside ring A_{ho}(G) can be defined by π_{G}^{0}(\underline{E}G) and analyzed for all discrete groups G.

If G is torsionfree, A_{ho}(G) agrees with π_{s}^{0}(BG).

Theorem 6.26 implies that the map (see (6.25))

ψ_{G}^{0}(\underline{E}G): A_{ho}(G) = π_{G}^{0}(\underline{E}G) → K_{G}^{0}(\underline{E}G)

is a ring homomorphism. It reduces for finite G to the ring homomorphism A(G) → R_{ℂ}(G) sending the class of a finite G-set to the class of the associated complex permutation representation.

7.3 Relation between the Homotopy Theoretic and the Inverse-Limit-Version

Suppose there is a finite model for \underline{E}G. Then there is an equivariant Atiyah-Hirzebruch spectral sequence which converges to π_{G}^{n}(\underline{E}G) and whose E^{2}-term is given in terms of Bredon cohomomology

E_{2}^{p,q} = H_{ℤSub _{FIN} (G)}^{p}(\underline{E}G; π_{?}^{q}).

Here π_{?}^{q} is the contravariant functor

π_{?}^{q}: Sub_{FIN} (G) → ℤ − MODULES, H ↦ π_{H}^{q}

and naturality comes from restriction with a group homomorphism H → K representing a morphism in Sub_{FIN} (G). Usually the Bredon cohomology is defined over the orbit category, but in our case we can pass to the category Sub_{FIN} (G) because of Lemma 5.2. Details of the construction of H_{ℤSub _{FIN} (G)}^{p}(\underline{E}G; π_{q}^{?}) can be found for instance in [27, Section 3]. We will only need the following elementary facts. There is a canonical identification

H_{ℤSub _{FIN} (G)}^{0}(\underline{E}G; π_{?}^{q}) ≅ invlim_{H∈Sub _{FIN} (G)}π_{H}^{q}.  (7.3)

If we combine (7.3) with Theorem 1.13 we get an identification

H_{ℤSub _{FIN} (G)}^{0}(\underline{E}G; π_{?}^{0}) ≅ A_{inv}(G).  (7.4)

The assumption that \underline{E}G is finite implies together with Remark 6.18

|H_{ℤSub _{FIN} (G)}^{p}(\underline{E}G; π_{?}^{q})| < ∞  if q ≤ −1;  (7.5)
H_{ℤSub _{FIN} (G)}^{p}(\underline{E}G; π_{?}^{q}) = \{0\}  if p > dim(\underline{E}G) or p ≤ −1 or q ≥ 1. (7.6)

The equivariant Atiyah-Hirzebruch spectral sequence together with (7.4), (7.5) and (7.6) implies

Theorem 7.7 (Rationally A_{ho}(G) and A_{inv}(G) agree). Suppose that there is a finite model for \underline{E}G. Then the edge homomorphism

edge^{G}: A_{ho}(G) = π_{G}^{0}(\underline{E}G) → A_{inv}(G)

is a ring homomorphism whose kernel and the cokernel are finite.

The edge homomorphism appearing in Theorem 7.7 can be made explicit. Consider a morphism u: S^{ξ⊕\underline{ℝ^{k}}} → S^{ξ⊕\underline{ℝ^{k}}} in SPHB^{G}(\underline{E}G) representing the element a ∈ π_{G}^{0}(\underline{E}G). In order to specify edge^{G}(a) we must define for every finite subgroup H ⊆ G an element edge^{G}(a)_{H} ∈ A(H). Choose a point x ∈ \underline{E}G^{H}. Then u induces a pointed H-map S^{ξ_{x}⊕ℝ^{k}} → S^{ξ_{x}⊕ℝ^{k}}. It defines an element in π_{H}^{0}. Let edge^{G}(a)_{H} be the image of this element under the ring isomorphism deg^{H}: π_{H}^{0} ⟶^{≅} A(H) appearing in Theorem 1.13. One easily checks that the collection of these elements edge_{H}^{G}(a) does define an element in the inverse limit A_{inv}(G). So essentially edge^{G} is the map which remembers just the system of the maps of the various fibers.

Remark 7.8. (Rank of the abelian group A_{ho}(G)). A kind of character map for the homotopy theoretic version would be the composition of edge and the character map char_{inv}^{G} of (3.9). Since we assume that \underline{E}G has a finite model, there are only finitely many conjugacy classes of finite subgroups and the Burnside ring congruences appearing in Theorem 3.10 becomes easier to handle. In particular we conclude from Example 3.11 and Theorem 7.7 that A_{ho}(G) is a finitely generated abelian group whose rank is the number | ccs_{f}(G)| of conjugacy classes of finite subgroups of G.

7.4 Some Computations of the Homotopy Theoretic Burnside Ring

Example 7.9 (Groups with appropriate maximal finite subgroups). Suppose that the group G satisfies the conditions appearing in Example 3.4 and admits a finite model for \underline{E}G. In the sequel we use the notation introduced in Example 3.4. Then one can construct a G-pushout (see [28, Section 4.11])

\begin{matrix}∐_{i∈I} G ×_{M_{i}} EM_{i} & ⟶^{i} & EG \\ ↓ & & ↓ \\ ∐_{i∈I} G/M_{i} & ⟶ & \underline{E}G \end{matrix}  (7.10)

Taking the G-quotient, yields a non-equivariant pushout. There are long exact Mayer-Vietoris sequence associated to (7.10) and to the G-quotient. (We ignore the problem that G ×_{E} M_{i} and EG may not be finite. It does not really matter since both are free or because we will in a different paper extend the definition of equivariant stable cohomotopy to all proper equivariant CW -complexes). These are linked by the induction maps with respect to the projections G → {1}. Splicing these two long exact sequences together, yields the long exact sequence

⋯ → ∏_{i∈I} ker (res_{M_{i}}^{\{1\}}: π_{M_{i}}^{−1} → π_{s}^{−1}) → π_{s}^{0}(G\\underline{E}G) → A_{ho}(G)

→ ∏_{i∈I} Ã(M_{i}) → π_{s}^{1}(G\\underline{E}G) → ⋯

Example 7.11 (Extensions of ℤ^{n} with ℤ/p as quotient). Suppose that G satisfies the assumptions appearing in Example 3.6. Then G admits a finite model for \underline{E}G. In the sequel we use the notation introduced in Example 3.6. Then variation of the argument above yields a long exact sequence

⋯ → ∏_{H^{1}(ℤ/p;A)} ker (res_{ℤ/p}^{\{1\}}: π_{ℤ/p}^{−1}(BA^{ℤ/p}) → π_{s}^{−1}(BA^{ℤ/p})) → π_{s}^{0}(G\\underline{E}G)

→ A_{ho}(G) → ∏_{H^{1}(ℤ/p;A)} ker (res_{ℤ/p}^{\{1\}}: π_{ℤ/p}^{0}(BA^{ℤ/p}) → π_{s}^{0}(BA^{ℤ/p})) → π_{s}^{1}(G\\underline{E}G) → ⋯

where ℤ/p acts trivially on BA^{ℤ/p}. If r is the rank of the finitely generated free abelian group A^{ℤ/p}, then

ker (res_{ℤ/p}^{\{1\}}: π_{ℤ/p}^{n}(BA^{ℤ/p}) → π_{s}^{n}(BA^{ℤ/p})) = ⨁_{k=0}^{r} ker (res_{ℤ/p}^{\{1\}}: π_{ℤ/p}^{n−k} → π_{s}^{n−k})^{\binom{r}{k}} .

8 The Segal Conjecture for Infinite Groups

We can now formulate a version of the Segal Conjecture for infinite groups. Let ϵ^{G}: A_{ho}(G) → ℤ be the ring homomorphism which sends an element represented by a morphism u: S^{ξ⊕\underline{ℝ^{k}}} → S^{ξ⊕\underline{ℝ^{k}}} in SPHB^{G}(\underline{E}G) to the mapping degree of the map induced on the fiber u_{x}: S^{ξ_{x}⊕ℝ^{k}} → S^{ξ_{x}⊕ℝ^{k}} for some x ∈ \underline{E}G. This is the same as the composition

A_{ho}(G) ⟶^{edge^{G}} A_{inv}(G) ⟶^{char_{inv}^{G}} ∏_{(H)∈ccs_{f} (G)} ℤ ⟶^{pr_{{1}}} ℤ,

where char_{inv}^{G} is the ring homomorphism defined in (3.9) and pr_{{1}} the projection onto the factor belonging to the trivial group. We define the augmentation ideal I_{G} of A_{ho}(G) to be the kernel of the ring homomorphism ϵ^{G}. Recall that for a finite proper G-CW -complex X the abelian group π_{G}^{n}(X) is a π_{G}^{0}(X)- module. The classifying map f: X → \underline{E}G is unique up to G-homotopy. Suppose that \underline{E}G is finite. Then f induces a uniquely defined ring homomorphism π_{G}^{0}(f): A_{ho}(G) = π_{G}^{0}(\underline{E}G) → π_{G}^{0}(X) and we can consider π_{G}^{n}(X) is a A_{ho}(G)- module.

Conjecture 8.1 (Segal Conjecture for infinite groups). Let G be a group such that there is a finite model for the classifying space of proper G-actions \underline{E}G. Then for every finite proper G-CW -complex there is an isomorphism

π_{s}^{n}(EG ×_{G} X) ⟶^{≅} π_{G}^{n}(X)_{I_{G}},

where π_{G}^{n}(X)_{I_{G}} is the I_{G}-adic completion of the A_{ho}(G)-module π_{G}^{n}(X). In particular we get for all n ∈ ℤ an isomorphism

π_{s}^{n}(BG) ⟶^{≅} π_{G}^{n}(\underline{E}G)_{I_{G}}

and especially for n = 0

π_{s}^{0}(BG) ⟶^{≅} A_{ho}(G)_{I_{G}}.

If G is finite, Conjecture 8.1 reduces to the classical Segal Conjecture (see Theorem 1.14).

The classical Segal Conjecture (see Theorem 1.14) for a finite group does not say much if the finite group is torsionfree, i.e. is trivial. The same remark holds for the Segal Conjecture 8.1 for infinite groups if the group under consideration is torsionfree as explained below.

Since G is torsionfree, G acts freely on X and \underline{E}G = EG. Thus we obtain an identification

π_{s}^{n}(EG ×_{G} X) = π_{s}^{n}(G\X).

The Burnside ring A_{ho}(G) becomes π_{s}^{0}(BG) and I_{G} corresponds to the kernel of the ring homomorphism π_{s}^{0}(BG) → π_{s}^{0}({•}). By assumption BG is a finite CW -complex. It is not hard to check that (I_{G})^{dim(BG)+1} = {0}. Hence we obtain an identification

π_{G}^{n}(X)_{I_{G}} ≅ π_{G}^{n}(X) ≅ π_{s}^{n}(G\X).

Under these two identifications the map appearing in Conjecture 8.1 is the identity and hence an isomorphism.

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