Equivariant Cohomological Chern Characters

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

August 30, 2004

Abstract We construct for an equivariant cohomology theory for proper equivariant CW -complexes an equivariant Chern character, provided that certain conditions about the coefficients are satisfied. These conditions are fulfilled if the coefficients of the equivariant cohomology theory possess a Mackey structure. Such a structure is present in many interesting examples. Key words: equivariant cohomology theory, equivariant Chern character. Mathematics Subject Classification 2000: 55N91.

0. Introduction

The purpose of this paper is to construct an equivariant Chern character for a proper equivariant cohomology theory H_{?}^{∗} with values in R-modules for a commutative associative ring R with unit which satisfies ℚ ⊆ R. It is a natural transformation of equivariant cohomology theories

ch_{?}^{∗}: H_{?}^{∗} → BH_{?}^{∗}

for a given equivariant cohomology theory H_{?}^{∗}. Here BH_{?}^{∗} is the associated equivariant cohomology theory which is defined by the Bredon cohomology with coefficients coming from the coefficients of H_{?}^{∗}. The notion of an equivariant cohomology theory and examples for it are presented in Section 1 and the associated Bredon cohomology is explained in Section 3. The point is that BH_{?}^{∗} is much simpler and easier to compute than H_{?}^{∗}. If H_{?}^{∗} satisfies the disjoint union axiom, then

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

ch_{G}^{n}(X, A): H_{G}^{n}(X, A) ⟶^{≅} BH_{G}^{n}(X, A)

is bijective for every discrete group G, proper G-CW -pair (X, A) and n ∈ ℤ.

The Chern character ch_{?}^{∗} is only defined if the coefficients of H_{?}^{∗} satisfy a certain injectivity condition (see Theorem 4.6). This condition is fulfilled if the coefficients of H_{?}^{∗} come with a Mackey structure (see Theorem 5.5) what is the case in many interesting examples.

The equivariant cohomological Chern character is a generalization to the equivariant setting of the classical non-equivariant Chern character for a (nonequivariant) cohomology theory H^{∗} (see Example 4.1)

ch^{n}(X, A): H^{n}(X, A) ⟶^{≅} ∏_{p+q=n} H^{p}(X, A, H^{q}(∗)).

The equivariant cohomological Chern character has already been constructed in the special case, where H_{?}^{∗} is equivariant topological K-theory K_{?}^{∗}, in [12]. Its homological version has already been treated in [8] and plays an important role in the computation of the source of the assembly maps appearing in the Farrell-Jones Conjecture for K_{n}(RG) and L_{n}^{⟨−∞⟩}(RG) and the Baum-Connes Conjecture for K_{n}(C_{r}^{∗}(G)) (see also [9]).

The detailed formulation of the main result of this paper is presented in Theorem 5.5.

The equivariant Chern character will play a key role in the proof of the following result which will be presented in [11]. Theorem 0.1 (Rational computation of the topological K-theory of BG). Let G be a discrete group. Suppose that there is a finite G-CW -model for the classifying space \underline{E}G for proper G-actions. Then there is a ℚ-isomorphism

\overline{ch}_{G}^{n}: K^{n}(BG) ⊗_{ℤ} ℚ

⟶^{≅} ∏_{i∈ℤ} H^{2i+n}(BG; ℚ) × ∏_{p prime} ∏_{(g)∈con_{p}(G)} H^{2i+n}(BC_{G}⟨g⟩; ℚ_{p}).

Here con_{p}(G) is the set of conjugacy classes (g) of elements g ∈ G of order p^{m} for some integer m ≥ 1 and C_{G}⟨g⟩ is the centralizer of the cyclic subgroup generated by g in G.

The assumption in Theorem 0.1 that there is a finite G-CW -model for the classifying space \underline{E}G for proper G-actions is satisfied for instance, if G is wordhyperbolic in the sense of Gromov, if G is a cocompact subgroup of a Lie group with finitely many path components, if G is a finitely generated one-relator group, if G is an arithmetic group, a mapping class group of a compact surface or the group of outer automorphisms of a finitely generated free group. For more information about \underline{E}G we refer for instance to [1] and [10].

A group G is always understood to be discrete and a ring R is always understood to be associative with unit throughout this paper.

The paper is organized as follows:

1. Equivariant Cohomology Theories
2. Modules over a Category
3. The Associated Bredon Cohomology Theory
4. The Construction of the Equivariant Cohomological Chern Character
5. Mackey Functors
6. Multiplicative Structures

References

1. Equivariant Cohomology Theories

In this section we describe the axioms of a (proper) equivariant cohomology theory. They are dual to the ones of a (proper) equivariant homology theory as described in [8, Section 1].

Fix a group G and an commutative ring R. A G-CW -pair (X, A) is a pair of G-CW -complexes. It is proper if all isotropy groups of X are finite. It is relative finite if X is obtained from A by attaching finitely many equivariant cells, or, equivalently, if G\(X/A) is compact. Basic information about G-CW -pairs can be found for instance in [7, Section 1 and 2]. 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).

Sometimes also the following axiom is required.

• 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.

The role of the disjoint union axiom is explained by the following result. Its proof for non-equivariant cohomology theories (see for instance [16, 7.66 and 7.67]) carries over directly to G-cohomology theories.

Lemma 1.1. Let H_{G}^{∗} and K_{G}^{∗} be (proper) G-cohomology theories. Then

(a) Suppose that H_{G}^{∗} satisfies the disjoint union axiom. Then there exists for every (proper) G-CW -pair (X, A) with an exhaustion by subcomplexes A = X_{−1} ⊆ X_{0} ⊆ X_{1} ⊆ … ⊆ ⋃_{n≥−1} X_{n} = X a natural short exact sequence

0 → lim_{n→∞}^{1}H_{G}^{p−1}(X_{n}∪A, A) → H^{p}(X, A) → lim_{n→∞} H_{G}^{p}(X_{n}∪A, A) → 0;

(b) Let T^{∗}: H_{G}^{∗} → K_{G}^{∗} be a transformation of (proper) G-cohomology theories, i.e. a collection of natural transformations T^{n}: H_{G}^{n} → K_{G}^{n} of contravariant functors from the category of (proper) G-CW -pairs to the category of R-modules indexed by n ∈ ℤ which is compatible with the boundary operator associated to (proper) G-CW -pairs. Suppose that T^{n}(G/H) is bijective for every (proper) homogeneous space G/H and n ∈ ℤ.

Then T^{n}(X, A): H_{G}^{∗}(X, A) → K_{G}^{∗}(X, A) is bijective for all n ∈ ℤ provided that (X, A) is relative finite or that both H^{∗} and K^{∗} satisfy the disjoint union axiom.

Remark 1.2 (The disjoint union axiom is not compatible with −⊗_{ℤ} ℚ). Let H_{G}^{∗} be a G-cohomology theory with values in ℤ-modules. Then H_{G}^{∗} ⊗_{ℤ} ℚ is a G-cohomology theory with values in ℚ-modules since ℚ is flat as ℤ-module. However, even if H^{∗} satisfies the disjoint union axiom, H_{G}^{∗} ⊗_{ℤ} ℚ does not satisfy the disjoint union axiom since − ⊗_{ℤ} ℚ is not compatible with products over arbitrary index sets.

Example 1.3 (Rationalizing topological K-theory). Consider for instance the (non-equivariant) cohomology theory with values in ℤ-modules satisfying the disjoint union axiom given by topological K-theory K^{∗}. Let K^{∗}(−; ℚ) be the cohomology theory associated to the rationalization of the K-theory spectrum. This is a (non-equivariant) cohomology theory with values in ℚmodules satisfying the disjoint union axiom. There is a natural transformation

T^{∗}(X): K^{∗}(X) ⊗_{ℤ} ℚ → K^{∗}(X; ℚ)

of (non-equivariant) cohomology theory with values in ℚ-modules. The ℚ-map T^{n}({pt.}) is bijective for all n ∈ ℤ. Hence T^{n}(X) is bijective for all finite CW -complexes by Lemma 1.1 (b). Notice that K^{∗}(X) ⊗_{ℤ} ℚ does not satisfy the disjoint union axiom in contrast to K^{∗}(X; ℚ). Hence we cannot expect T^{n}(X) to be bijective for all CW -complexes. Consider the case X = BG for a finite group G. Since H^{p}(BG; ℚ) ≅ H^{p}({pt.}; ℚ) for all p ∈ ℤ, one obtains K^{p}(BG; ℚ) ≅ K^{p}({pt.}; ℚ) for all p ∈ ℤ. By the Atiyah-Segal Completion Theorem K^{p}(BG) ⊗_{ℤ} ℚ ≅ K^{p}({pt.}) ⊗_{ℤ} ℚ is only true if and only if the finite group G is trivial.

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 theory 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) such that ker(α) acts freely on X, there are for each n ∈ ℤ natural isomorphisms

ind_{α}: H_{G}^{n}(ind_{α}(X, A)) ⟶^{≅} H_{H}^{n}(X, A)  (1.4)

satisfying

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

(b) Functoriality Let β: G → K be another group homomorphism such that ker(β ∘ α) acts freely on X. 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;

(c) 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. It will play a key role in the construction of the equivariant Chern character even if we want to carry it out only for a fixed group G. In all of the relevant examples the induction homomorphism ind_{α} of (1.4) exists for every group homomorphism α: H → G, the condition that ker(α) acts freely on X is only needed to ensure that ind_{α} is bijective. If α is an inclusion, we sometimes write ind_{H}^{G} instead of ind_{α}.

We say that H_{?}^{∗} satisfies the disjoint union axiom if for every group G the G-cohomology theory H_{G}^{∗} satisfies the disjoint union axiom.

We will later need the following lemma whose elementary proof is analogous to the one in [8, Lemma 1.2].

Lemma 1.5. 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}{pt.}) → {pt.} be the projection. 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}({pt.}) ⟶^{ind_{c(g)} ∘H_{K}^{n}(pr)} H_{H}^{n}({pt.})

Example 1.6 (Borel cohomology). Let K^{∗} be a cohomology theory for (nonequivariant) CW -pairs with values in R-modules. Examples are singular cohomology and topological K-theory. Then we obtain two equivariant cohomology theories with values in R-modules by the following constructions

H_{G}^{n}(X, A) = K^{n}(G\X, G\A);
H_{G}^{n}(X, A) = K^{n}(EG ×_{G} (X, A)).

The second one is called the equivariant Borel cohomology associated to K. In both cases H_{G}^{∗} inherits the structure of a G-cohomology theory from the cohomology structure on K^{∗}.

The induction homomorphism associated to a group homomorphism α: H → G is defined as follows. Let a: H\X ⟶^{≅} G\(G ×_{α} X) be the homeomorphism sending Hx to G(1, x). Define b: EH ×_{H} X → EG×_{G} G×_{α} X by sending (e, x) to (Eα(e), 1, x) for e ∈ EH, x ∈ X and Eα: EH → EG the α-equivariant map induced by α. The desired induction map ind_{α} is given by K^{∗}(a) and K^{∗}(b). If the kernel ker(α) acts freely on X, the map b is a homotopy equivalence and hence in both cases ind_{α} is bijective.

If K^{∗} satisfies the disjoint union axiom, the same is true for the two equivariant cohomology theories constructed above.

Example 1.7 (Equivariant K-theory). In [12] G-equivariant topological (complex) K-theory K_{G}^{∗}(X, A) is constructed for any proper G-CW -pair (X, A) and shown that K_{G}^{∗} defines a proper G-cohomology theory satisfying the disjoint union axiom. Given a group homomorphism α: H → G, it induces an injective group homomorphism \overline{α}: H/ ker(α) → G. Let

Infl_{H/ ker(α)}^{H}: K_{H/ ker(α)}^{∗}(ker(α)\X) → K_{H}^{∗}(X)

be the inflation homomorphism of [12, Proposition 3.3] and

ind\overline{α}: K_{H/ ker(α)}^{∗}(ker(α)\X) ⟶^{≅} K_{G}^{∗}(ind\overline{α}(ker(α)\X))

be the induction isomorphism of [12, Proposition 3.2 (b)]. Define the induction homomorphism

ind_{α}: K_{G}^{∗}(ind_{α} X) → K_{H}^{∗}(X)

by Infl_{H/ ker(α)}^{H} ∘(ind\overline{α})^{−1}, where we identify ind_{α} X = ind\overline{α}(ker(α)\X). On the level of complex finite-dimensional vector bundles the induction homomorphism ind_{α} corresponds to considering for a G-vector bundle E over G ×_{α} X the Hvector bundle obtained from E by the pullback construction associated to the α-equivariant map X → G ×_{α} X, x ↦ (1, x).

Thus we obtain a proper equivariant cohomology theory K_{?}^{∗} with values in ℤ-modules which satisfies the disjoint union axiom. There is also a real version KO_{?}^{∗}.

Example 1.8 (Equivariant cohomology theories and spectra). Denote by GROUPOIDS the category of small groupoids. Let Ω-SPECTRAbe the category of Ω-spectra, where a morphism f : E → F is a sequence of maps f_{n}: E_{n} → F_{n} compatible with the structure maps and we work in the category of compactly generated spaces (see for instance [3, Section 1]). A contravariant GROUPOIDS-Ω-spectrum is a contravariant functor E: GROUPOIDS → Ω-SPECTRA .

Next we explain how we can associate to it an equivariant cohomology theory H_{?}^{∗}(−; E) satisfying the disjoint union axiom, provided that E respects equivalences, i.e. it sends an equivalence of groupoids to a weak equivalence of spectra. This construction is dual to the construction of an equivariant homology theory associated to a covariant GROUPOIDS-spectrum as explained in [13, Section 6.2], [14, Theorem 2.10 on page 21].

Fix a group G. We have to specify a G-cohomology theory H_{G}^{∗}(−; E). Let Or(G) be the orbit category whose set of objects consists of homogeneous Gspaces G/H and whose morphisms are G-maps. For a G-set S we denote by G^{G}(S) its associated transport groupoid. Its objects are the elements of S. The set of morphisms from s_{0} to s_{1} consists of those elements g ∈ G which satisfy gs_{0} = s_{1}. Composition in G^{G}(S) comes from the multiplication in G. Thus we obtain for a group G a covariant functor

G^{G}: Or(G) → GROUPOIDS,  G/H ↦ G^{G}(G/H),  (1.9)

and a contravariant Or(G)-Ω-spectrum E ∘ G^{G}. Given a G-CW -pair (X, A), we obtain a contravariant pair of Or(G)-CW -complexes (X^{?}, A^{?}) by sending G/H to (map_{G}(G/H, X), map_{G}(G/H, A)) = (X^{H}, A^{H}). The contravariant Or(G)- spectrum E ∘ G^{G} defines a cohomology theory on the category of contravariant Or(G)-CW -complexes as explained in [3, Section 4]. It value at (X^{?}, A^{?}) is defined to be H_{G}^{∗}(X, A; E). Explicitly, H_{G}^{n}(X, A; E) is the (−n)-th homotopy group of the spectrum map_{Or (G)} (X_{+}^{?} ∪_{A_{+}^{?}} cone(A_{+}^{?}), E ∘ G^{G}). We need Ω-spectra in order to ensure that the disjoint union axiom holds.

We briefly explain for a group homomorphism α: H → G the definition of the induction homomorphism ind_{α}: H_{G}^{n}(ind_{α} X; E) → H_{H}^{n}(X; E) in the special case A = ∅. The functor induced by α on the orbit categories is denoted in the same way

α: Or(H) → Or(G), H/L ↦ ind_{α}(H/L) = G/α(L).

There is an obvious natural transformation of functors Or(H) → GROUPOIDS

T: G^{H} → G^{G} ∘ α.

Its evaluation at H/L is the functor of groupoids G^{H}(H/L) → G^{G}(G/α(L)) which sends an object hL to the object α(h)α(L) and a morphism given by h ∈ H to the morphism α(h) ∈ G. Notice that T (H/L) is an equivalence if ker(α) acts freely on H/L. The desired isomorphism

ind_{α}: H_{G}^{n}(ind_{α} X; E) → H_{H}^{n}(X; E)

is induced by the following map of spectra

map_{Or (G)} (map_{G}(−, ind_{α} X_{+}), E ∘ G^{G})
⟶^{≅} map_{Or (G)} (α_{∗}(map_{H}(−, X_{+})), E ∘ G^{G})
⟶^{≅} map_{Or (H)} (map_{H}(−, X_{+}), E ∘ G^{G} ∘ α)

⟶^{map_{Or (H)}(id,E(T))} map_{Or (H)} (map_{H}(−, X_{+}), E ∘ G^{H}) .

Here α_{∗} map_{H}(−, X_{+}) is the pointed Or(G)-space which is obtained from the pointed Or(H)-space map_{H}(−, X_{+}) by induction, i.e. by taking the balanced product over Or(H) with the Or(H)-Or(G) bimodule mor_{Or (G)}(??, α(?)) [3, Definition 1.8]. The second map is given by the adjunction homeomorphism of induction α_{∗} and restriction α^{∗} (see [3, Lemma 1.9]). The first map comes from the homeomorphism of Or(G)-spaces

α_{∗} map_{H}(−, X_{+}) → map_{G}(−, ind_{α} X_{+})

which is the adjoint of the obvious map of Or(H)-spaces map_{H}(−, X_{+}) → α^{∗} map_{G}(−, ind_{α} X_{+}) whose evaluation at H/L is given by ind_{α}.

2. Modules over a Category

In this section we give a brief summary about modules over a small category C as far as needed for this paper. They will appear in the definition of the equivariant Chern character.

Let C be a small category and let R be a commutative ring. A contravariant RC-module is a contravariant functor from C to the category R - MOD of Rmodules. Morphisms of contravariant RC-modules are natural transformations. Given a group G, let Ĝ be the category with one object whose set of morphisms is given by G. Then a contravariant RĜ-module is the same as a right RGmodule. Therefore we can identify the abelian category MOD -RĜ with the abelian category of right RG-modules MOD-RG in the sequel. Many of the constructions, which we will introduce for RC-modules below, reduce in the special case C = Ĝ to their classical versions for RG-modules. The reader should have this example in mind. There is also a covariant version. In the sequel RCmodule means contravariant RC-module unless stated explicitly differently.

The category MOD -RC of RC-modules inherits the structure of an abelian category from R - MOD in the obvious way, namely objectwise. For instance a sequence 0 → M → N → P → 0 of contravariant RC-modules is called exact if its evaluation at each object in C is an exact sequence in R - MOD. The notion of an injective and of a projective RC-module is now clear. For a set S denote by RS the free R-module with S as basis. An RC-module is free if it is isomorphic to RC-module of the shape ⨁_{i∈I} R mor_{C}(?, c_{i}) for some index set I and objects c_{i} ∈ C. Notice that by the Yoneda-Lemma there is for every RC-module N and every object c a bijection of sets

hom_{RC}(R mor_{C}(?, c), N) ⟶^{≅} N(c), ϕ ↦ ϕ(id_{x}).

This implies that every free RC-module is projective and a RC-module is projective if and only if it is a direct summand in a free RC-module. The category of RC-modules has enough projectives and injectives (see [7, Lemma 17.1] and [17, Example 2.3.13]).

Given a contravariant RC-module M and a covariant RC-module N, their tensor product over RC is defined to be the following R-module M ⊗_{RC} N. It is given by

M ⊗_{RC} N = ⨁_{c∈Ob(C)} M(c) ⊗_{R} N(c)/ ∼,

where ∼ is the typical tensor relation mf ⊗n = m⊗fn, i.e. for every morphism f: c → d in C, m ∈ M(d) and n ∈ N(c) we introduce the relation M(f)(m) ⊗ n − m ⊗ N(f)(n) = 0. The main property of this construction is that it is adjoint to the hom_{R}-functor in the sense that for any R-module L there are natural isomorphisms of R-modules

hom_{R}(M ⊗_{RC} N, L) ⟶^{≅} hom_{RC}(M, hom_{R}(N, L));  (2.1)
hom_{R}(M ⊗_{RC} N, L) ⟶^{≅} hom_{RC}(N, hom_{R}(M, L)).  (2.2)

Consider a functor F: C → D. Given a RD-module M, define its restriction with F to be F^{∗}M := M ∘F. Given a contravariant RC-module M, its induction with F is the contravariant RD-module F_{∗}M given by

(F_{∗}M)(??) := M(?) ⊗_{RC} R mor_{D}(??, F (?)),  (2.3)

and coinduction with F is the contravariant RD-module F_{!}M given by

(F_{!}M)(??) := hom_{RC}(R mor_{D}(F (?),??), M(?)).  (2.4)

Restriction with F can be written as F^{∗}N(?) = hom_{RD}(R mor_{D}(??, F (?)), N(??)), the natural isomorphisms sends n ∈ N(F (?)) to the map

R mor_{D}(??, F (?)) → N(??), ϕ: ?? → F (?) ↦ N(ϕ)(n).

Restriction with F can also be written as F^{∗}N(?) = R mor_{D}(F (?),??) ⊗_{RD} N(??), the natural isomorphisms sends ϕ ⊗_{RD} n to N(ϕ)(n). We conclude from (2.2) that (F_{∗}, F^{∗}) and (F^{∗}, F_{!}) form adjoint pairs, i.e. for a RC-module M and a RD-module N there are natural isomorphisms of R-modules

hom_{RD}(F_{∗}M, N) ⟶^{≅} hom_{RC}(M, F^{∗}N);  (2.5)
hom_{RD}(F^{∗}N, M) ⟶^{≅} hom_{RC}(N, F_{!}M).  (2.6)

Consider an object c in C. Let aut(c) be the group of automorphism of c. We can think of { \atop aut(c)} as a subcategory of C in the obvious way. Denote by

i(c): { \atop aut(c)} → C

the inclusion of categories and abbreviate the group ring R[aut(c)] by R[c] in the sequel. Thus we obtain functors

i(c)^{∗}: MOD -RC → MOD -R[c];  (2.7)
i(c)_{∗}: MOD -R[c] → MOD -RC;  (2.8)
i(c)_{!} : MOD -R[c] → MOD -RC.  (2.9)

The projective splitting functor

S_{c}: MOD -RC → MOD -R[c]  (2.10)

sends M to the cokernel of the map

⨁_{f: c→d f not an isomorphism} M(f): ⨁_{f: c→d f not an isomorphism} M(d) → M(c).

The injective splitting functor

T_{c}: MOD -RC → MOD -R[c]  (2.11)

sends M to the kernel of the map

∏_{f: d→c f not an isomorphism} M(f): M(c) → ∏_{f: d→c f not an isomorphism} M(d).

From now on suppose that C is an EI-category, i.e. a small category such that endomorphisms are isomorphisms. Then we can define the inclusion functor

I_{c}: MOD -R[c] → MOD -RC  (2.12)

by I_{c}(M)(?) = M ⊗_{R[c]} R mor(?, c) if c ≅ ? in C and by I_{c}(M)(?) = 0 otherwise. Let B be the RC-R[c]-bimodule, covariant over C and a right module over R[c], given by

B(c,?) = \begin{matrix}R mor_{C}(c,?) & if c ≅ ?; \\ 0 & if c ≅ ?. \end{matrix}

Let C be the R[c]-RC-bimodule, contravariant over C and a left module over R[c], given by

C(?, c) = \begin{matrix}R mor_{C}(?, c) & if c ≅ ?; \\ 0 & if c ≅ ?. \end{matrix}

One easily checks that there are natural isomorphisms

S_{c}M ≅ M ⊗_{RC} B;
I_{c}N ≅ hom_{R[c]}(B, N);
T_{c}M ≅ hom_{RC}(C, M);
I_{c}N ≅ N ⊗_{R[c]} C.

Lemma 2.13. Let C be an EI-category and c, d objects in C.

(a) We obtain adjoint pairs (i(c)_{∗}, i(c)^{∗}), (i(c)^{∗}, i(c)_{!}), (S_{c}, I_{c}) and (I_{c}, T_{c});

(b) There are natural equivalences of functors S_{c} ∘i(c)_{∗} ⟶^{≅} id and T_{c} ∘i(c)_{!} ⟶^{≅} id of functors MOD -R[c] → MOD -R[c]. If c ≅ d̸, then S_{c} ∘ i(d)_{∗} = T_{c} ∘ i(d)_{!} = 0;

(c) The functors S_{c} and i(c)_{∗} send projective modules to projective modules. The functors I_{c} and i(c)_{!} send injective modules to injective modules.

Proof. (a) follows from (2.5), (2.6) and (2.1).

(b) This follows in the case T_{c} ∘ i(d)_{!} from the following chain of canonical isomorphisms

T_{c} ∘ i(d)_{!}(M) = hom_{RC}(C(?, c), hom_{R[d]}(R mor_{C}(d,?), M))

⟶^{≅} hom_{R[d]}(C(?, c) ⊗_{RC} R mor_{C}(d,?), M) ⟶^{≅} hom_{R[c]}(C(c, d), M),

and analogously for S_{d} ∘ i(c)_{∗}.

(c) The functors S_{c} and i(c)_{∗} are left adjoint to an exact functor and hence respect projective. The functors T_{c} and i(c)_{!} are right adjoint to an exact functor and hence respect injective.

The length l(c) ∈ ℕ ∪ {∞} of an object c is the supremum over all natural numbers l for which there exists a sequence of morphisms c_{0} ⟶^{f_{1}} c_{1} ⟶^{f_{2}} c_{2} ⟶^{f_{3}} … ⟶^{f_{l}} c_{l} such that no f_{i} is an isomorphism and c_{l} = c. The colength col(c) ∈ ℕ ∪ {∞} of an object c is the supremum over all natural numbers l for which there exists a sequence of morphisms c_{0} ⟶^{f_{1}} c_{1} ⟶^{f_{2}} c_{2} ⟶^{f_{3}} … ⟶^{f_{l}} c_{l} such that no f_{i} is an isomorphism and c_{0} = c. If each object c has length l(c) < ∞, we say that C has finite length. If each object c has colength col(c) < ∞, we say that C has finite colength.

Theorem 2.14. (Structure theorem for projective and injective RCmodules). Let C be an EI-category. Then

(a) Suppose that C has finite colength. Let M be a contravariant RC-module such that the R aut(c)-module S_{c}M is projective for all objects c in C. Let σ_{c}: S_{c}M → M(c) be an R aut(c)-section of the canonical projection M(c) → S_{c}M. Consider the map of RC-modules

μ(M): ⨁_{(c)∈Is(C)} i(c)_{∗}S_{c}M \underrightarrow{⨁_{(c)∈Is(C)} i(c)_{∗}σ_{c}} ⨁_{(c)∈Is(C)} i(c)_{∗}M(c)

\underrightarrow{⨁_{(c)∈Is(C)} α(c)} M,

where α(c): i(c)_{∗}M(c) = i(c)_{∗}i(c)^{∗}M → M is the adjoint of the identity i(c)^{∗}M → i(c)^{∗}M under the adjunction (2.5). The map μ(M) is always surjective. It is bijective if and only if M is a projective RC-module;

(b) Suppose that C has finite length. Let M be a contravariant RC-module such that the R aut(c)-module T_{c}M is injective for all objects c in C. Let ρ_{c}: M(c) → T_{c}M be an R aut(c)-retraction of the canonical injection T_{c}M → M(c). Consider the map of RC-modules

ν(M): M \underrightarrow{∏_{(c)∈Is(C)} β(c)} ∏_{(c)∈Is(C)} i(c)_{!}M(c)

\underrightarrow{∏_{(c)∈Is(C)} i(c)_{!}ρ_{c}} ∏_{(c)∈Is(C)} i(c)_{∗}T_{c}M

where β(c): M → i(c)_{!}i(c)^{∗}M = i(c)_{!}M(c) is the adjoint of the identity i(c)^{∗}M → i(c)^{∗}M under the adjunction (2.6). The map ν(M) is always injective. It is bijective if and only if M is an injective RC-module.

Proof. (a) A contravariant RC-module is the same as covariant RC^{op}-module, where C^{op} is the opposite category of C, just invert the direction of every morphisms. The corresponding covariant version of assertion (a) is proved in [8, Theorem 2.11].

(b) is the dual statement of assertion (a). We first show that ν(M) is always injective. We show by induction over the length l(x) of an object x ∈ C that ν(M)(x) is injective. Let u be an element in the kernel of ν(M)(x). Consider a morphism f: y → x which is not an isomorphism. Then l(y) < l(x) and by induction hypothesis ν(M)(y) is injective. Since the composite ν(M)(y) ∘ M(f) factorizes through ν(M)(x), we have u ∈ ker(M(f)). This implies u ∈ T_{x}M. Consider the composite

T_{x}M ⟶^{i} M(x) ⟶^{ν(M)(x)} ∏_{(c)∈Is(C)} i(c)_{!}T_{c}M(x) ⟶^{pr_{x}} i(x)_{!}T_{x}M(x) ⟶^{j} T_{x}M,

where i is the inclusion, pr_{x} is the projection onto the factor belonging to the isomorphism class of x and j is the isomorphism hom_{R[x]}(R mor_{C}(x, x), T_{x}M) ⟶^{≅} T_{x}M sending ϕ to ϕ(id_{x}). Since this composite is the identity on T_{x}M and u lies in the kernel of ν(M)(x), we conclude u = 0.

In particular we see that an injective RC-module M is trivial if and only if i(d)_{!}T_{d}M(x) is trivial for all objects d ∈ C.

If ν(M) is bijective and each T_{c}M is an injective R[c]-module, then M is an injective RC-module, since i(c)_{!} sends injective R[c]-modules to injective RC-modules by Lemma 2.13 (c) and the product of injective modules is again injective.

Now suppose that M is injective. Let N be the cokernel of ν(M). We have the exact sequence

0 → M ⟶^{ν(M)} ∏_{(c)∈Is(C)} i(c)_{∗}T_{c}M ⟶^{pr} N → 0.  (2.15)

Since M is injective, this is a split exact sequence of injective RC-modules. Fix an object d. The functors i(d)_{!} and T_{d} send split exact sequences to split exact sequences. Therefore we obtain a split exact sequence if we apply i(d)_{!}T_{d} to (2.15). Using Lemma 2.13 (b) the resulting exact sequence is isomorphic to the exact sequence

0 → i(d)_{!}T_{d}M ⟶^{id} i(d)_{!}T_{d}M → i(d)_{!}T_{d}N → 0.

Hence i(d)_{!}T_{d}N vanishes for all objects d. This implies that N is trivial and because of (2.15) that ν(M) is bijective.

For more details about modules over a category we refer to [7, Section 9A].

3. The Associated Bredon Cohomology Theory

Given a proper equivariant cohomology theory with values in R-modules, we can associate to it another proper equivariant cohomology theory with values in R-modules satisfying the disjoint union axiom called Bredon cohomology, which is much simpler. The equivariant Chern character will identify this simpler proper equivariant cohomology theory with the given one.

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 f is injective and 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.

Denote by Or(G, F) ⊆ Or(G) and Sub(G, F) ⊆ Sub(G) the full subcategories, whose objects G/H and H are given by finite subgroups H ⊆ G. Both Or(G, F) and Sub(G, F) are EI-categories of finite length.

Given a proper G-cohomology theory H_{G}^{∗} with values in R-modules we obtain for n ∈ ℤ a contravariant ROr(G, F)-module

H_{G}^{n}(G/?): Or(G, F) → R - MOD, G/H ↦ H_{G}^{n}(G/H).  (3.1)

Let (X, A) be a pair of proper G-CW -complexes. Then there is a canonical identification X^{H} = map(G/H, X)^{G}. Thus we obtain contravariant functors

Or(G, F) → CW -PAIRS,  G/H ↦ (X^{H}, A^{H});
Sub(G, F) → CW -PAIRS,  G/H ↦ C_{G}H\(X^{H}, A^{H}),

where CW -PAIRS is the category of pairs of CW -complexes. If we compose them with the covariant functor CW -PAIRS → ℤ-CHCOM sending (Z, B) to its cellular ℤ-chain complex, then we obtain the contravariant ℤOr(G, F)- chain complex C_{∗}^{Or (G,F)}(X, A) and the contravariant ℤSub(G, F)-chain complex C_{∗}^{Sub (G,F)}(X, A). Both chain complexes are free in the sense that each chain module is a free ℤOr(G, F)-module resp. ℤSub(G, F)-module. Namely, if X_{n} is obtained from X_{n−1} ∪ A_{n} by attaching the equivariant cells G/H_{i} × D^{n} for i ∈ I_{n}, then

C_{n}^{Or (G,F)}(X, A) ≅ ⨁_{i∈I_{n}} ℤ mor_{Or (G,F)}(G/?, G/H_{i});  (3.2)
C_{n}^{Sub (G,F)}(X, A) ≅ ⨁_{i∈I_{n}} ℤ mor_{Sub (G,F)}(?, H_{i}).  (3.3)

Given a contravariant ROr(G, F)-module M, the equivariant Bredon cohomology (see [2]) of a pair of proper G-CW -complexes (X, A) with coefficients in M is defined by

H_{Or (G,F)}^{n}(X, A; M) := H^{n} (hom_{ℤOr (G,F)}(C_{∗}^{Or (G,F)}(X, A), M)) . (3.4)

This is indeed a proper G-cohomology theory satisfying the disjoint union axiom. Hence we can assign to a proper G-homology theory H_{G}^{∗} another proper Gcohomology theory which we call the associated Bredon cohomology

BH_{G}^{n}(X, A) := ∏_{p+q=n} H_{Or (G,F)}^{p}(X, A; H_{G}^{q}(G/?)).  (3.5)

There is an obvious ℤSub(G; F)-chain map

pr_{∗} C_{∗}^{Or (G,F)}(X, A) ⟶^{≅} C_{∗}^{Sub (G,F)}(X, A)

which is bijective because of (3.2), (3.3) and the canonical identification

pr_{∗} ℤ mor_{Or (G,F)}(G/?, G/H_{i}) = ℤ mor_{Sub (G,F)}(?, H_{i}).

Given a covariant ℤSub(G, F)-module M, we get from the adjunction (pr_{∗}, pr^{∗}) (see Lemma 2.13 (a)) natural isomorphisms

H_{ROr (G,F)}^{n}(X, A; res_{pr} M)

⟶^{≅} H^{n} (hom_{ℤSub (G,F)} (C_{∗}^{Sub (G,F)}(X, A), M)) . (3.6)

This will allow us to work with modules over the category Sub(G; F) which is smaller than the orbit category and has nicer properties from the homological algebra point of view. The main advantage of Sub(G; F) is that the automorphism groups of every object is finite.

Suppose, we are given a proper equivariant cohomology theory H_{?}^{∗} with values in R-modules. We get from (3.1) for each group G and n ∈ ℤ a covariant RSub(G, F)-module

H_{G}^{n}(G/?): Sub(G, F) → R - MOD, H ↦ H_{G}^{n}(G/H).  (3.7)

We have to show that for g ∈ C_{G}H the G-map R_{g^{−1}} : G/H → G/H, g′H → g′g^{−1}H induces the identity on H_{G}^{n}(G/H). This follows from Lemma 1.5. We will denote the covariant ROr(G, F)-module obtained by restriction with pr: Or(G, F) → Sub(G, F) from the RSub(G, F)-module H_{G}^{n}(G/?) of (3.7) again by H_{G}^{n}(G/?) as introduced already in (3.1).

It remains to show that the collection of G-cohomology theories BH_{G}^{∗}(X, A) defined in (3.4) inherits the structure of a proper equivariant cohomology theory, i.e. we have to specify the induction structure. We leave it to the reader to carry out the obvious dualization of the construction for homology in [8, Section 3] and to check the disjoint union axiom.

4. The Construction of the Equivariant Cohomological Chern Character

We begin with explaining the cohomological version of the homological Chern character due to Dold [4].

Example 4.1 (The non-equivariant Chern character). Consider a (nonequivariant) cohomology theory H^{∗} with values in R-modules. Suppose that ℚ ⊆ R. For a space X let X_{+} be the pointed space obtained from X by adding a disjoint base point ∗. Since the stable homotopy groups π_{p}^{s}(S^{0}) are finite for p ≥ 1 by results of Serre [15], the condition ℚ ⊆ R imply that the Hurewicz homomorphism induces isomorphisms

hur_{R}: π_{p}^{s}(X_{+}) ⊗_{ℤ} R ⟶^{hur ⊗_{ℤ} id_{R}} H_{p}(X) ⊗_{ℤ} R ⟶^{≅} H_{p}(X; R)

and that the canonical map

α: H^{p}(X; H^{q}({pt.})) ⟶^{≅} hom_{ℚ}(H_{p}(X; ℚ), H^{q}({pt.})) ⟶^{≅} hom_{R}(H_{p}(X; R), H^{q}({pt.}))

is bijective. Define a map

D^{p,q}: H^{p+q}(X) → hom_{R}(π_{p}^{s}(X_{+}) ⊗_{ℤ} R, H^{q}({pt.}))  (4.2)

as follows. Denote in the sequel by σ^{k} the k-fold suspension isomorphism. Given a ∈ H^{p+q}(X) and an element in π_{p}^{s}(X_{+}, ∗) represented by a map f: S^{p+k} → S^{k}∧ X_{+}, we define D^{p,q}(a)([f]) ∈ H^{q}({pt.}) as the image of a under the composite

H^{p+q}(X) ⟶^{≅} H̃^{p+q}(X_{+}) ⟶^{σ^{k}} H̃^{p+q+k}(S^{k} ∧ X_{+}) ⟶^{H̃^{p+q+k}(f)} H̃^{p+q+k}(S^{p+k})

⟶^{(σ^{p+k})^{−1}} H̃^{q}(S^{0}) ⟶^{≅} H^{q}({pt.}).

Then the (non-equivariant) Chern character for a CW -complex X is given by the following composite

ch^{n}(X): H^{n}(X) \underrightarrow{∏_{p+q=n} D^{p,q}} ∏_{p+q=n} hom_{R} (π_{p}^{s}(X_{+}, ∗) ⊗_{ℤ} R, H^{q}(∗))

\underrightarrow{∏_{p+q=n} hom_{R}(hur_{R}^{−1},id)} ∏_{p+q=n} hom_{R}(H_{p}(X; R), H^{q}(∗))

\underrightarrow{∏_{p+q=n} α^{−1}} ∏_{p+q=n} H^{p}(X, H^{q}(∗)).

There is an obvious version for a pair of CW -complexes

ch^{n}(X, A): H^{n}(X, A) ⟶^{≅} ∏_{p+q=n} H^{p}(X, A, H^{q}(∗)).

We get a natural transformation ch^{∗} of cohomology theories with values in Rmodules. One easily checks that it is an isomorphism in the case X = {pt.}. Hence ch^{n}(X, A) is bijective for all relative finite CW -pairs (X, A) and n ∈ ℤ by Lemma 1.1 (b). If H^{∗} satisfies the disjoint union axiom, then ch^{n}(X, A) is bijective for all CW -pairs (X, A) and n ∈ ℤ by Lemma 1.1 (b).

Let R be a commutative ring with ℚ ⊆ R. Consider an equivariant cohomology theory H_{?}^{∗} with values in R-modules. Let G be a group and let (X, A) be a proper G-CW -pair. We want to construct an R-homomorphism for a finite subgroup H ⊆ G

\underline{ch}_{G}^{p,q}(X, A)(H): H_{G}^{p+q}(X, A)

→ hom_{R} (H_{p}(C_{G}H\(X^{H}, A^{H}); R), H_{G}^{q}(G/H)) . (4.3)

We define it only in the case A = ∅, the general case is completely analogous.

H_{G}^{p+q}(X)
H_{G}^{p+q}(v_{H})↓
H_{G}^{p+q}(ind_{m_{H}} X^{H})
H_{G}^{p+q}(ind_{m_{H}} pr_{2})↓
H_{G}^{p+q}(ind_{m_{H}} EG × X^{H})
ind^{m_{H}} ↓≅
H_{C_{G}H×H}^{p+q}(EG × X^{H})
(^{ind}_{pr : C_{G}H×H→H})^{−1}↓≅
H_{H}^{p+q}(EG ×_{C_{G}H} X^{H})
D_{H}^{p,q}(EG×_{C_{G}H}X^{H})↓
hom_{R} (π_{p}^{s}((EG ×_{C_{G}H} X^{H})_{+}) ⊗_{ℤ} R, H_{H}^{q}({pt.}))
hom_{R}(hur_{R}(EG×_{C_{G}H}X^{H}),id)^{−1}↓
hom_{R} (H_{p}(EG ×_{C_{G}H} X^{H}; R), H_{H}^{q}({pt.}))
hom_{R}^{(H_{p}(pr_{1};R),id)}^{−1}↓
hom_{R} (H_{p}(C_{G}H\X^{H}; R), H_{H}^{q}({pt.}))
hom_{R}(id;(ind_{H}^{G})^{−1})↓
hom_{R} (H_{p}(C_{G}H\X^{H}; R), H_{G}^{q}(G/H))

Here are some explanations, more details can be found in [8, Section 4].

We have a left free C_{G}H-action on EG × X^{H} by g(e, x) = (eg^{−1}, gx) for g ∈ C_{G}H, e ∈ EG and x ∈ X^{H}. The map pr_{1} : EG ×_{C_{G}H} X^{H} → C_{G}H\X^{H} is the canonical projection. Since the projection BL → {pt.} induces isomorphisms H_{p}(BL; R) ⟶^{≅} H_{p}({pt.}; R) for all p ∈ ℤ and finite groups L because of ℚ ⊆ R, we obtain for every p ∈ ℤ an isomorphism

H_{p}(pr_{1}; R): H_{p}(EG ×_{C_{G}H} X^{H}; R) ⟶^{≅} H_{p}(C_{G}H\X^{H}; R).

The group homomorphism pr: C_{G}H ×H → H is the obvious projection and the group homomorphism m_{H}: C_{G}H × H → G sends (g, h) to gh. The C_{G}H × Haction on EG × X^{H} comes from the obvious C_{G}H-action and the trivial Haction. In particular we equip EG ×_{C_{G}H} X^{H} with the trivial H-action. The kernels of the two group homomorphisms pr and m_{H} act freely on EG × X^{H}. We denote by pr_{2} : EG × X^{H} → X^{H} the canonical projection. The G-map v_{H}: ind_{m_{H}} X^{H} = G ×_{m_{H}} X^{H} → X sends (g, x) to gx.

Since H is a finite group, a CW -complex Z equipped with the trivial Haction is a proper H-CW -complex. Hence we can think of H_{H}^{∗} as an (nonequivariant) homology theory if we apply it to a CW -pair Z with respect to the trivial H-action. Define the map

D_{H}^{p,q}(Z): H_{H}^{p+q}(Z) → hom_{R}(π_{p}^{s}(Z_{+}) ⊗_{ℤ} R, H_{H}^{q}({pt.}))

for a CW -complex Z by the map D^{p,q} of (4.2).

A calculation similar to the one in [8, Lemma 4.3] shows that the system of maps \underline{ch}_{G}^{p,q}(X, A)(H) (4.3) fit together to an in X natural R-homomorphism

\underline{\underline{ch}}_{G}^{p,q}(X, A): H_{G}^{p+q}(X, A)

→ hom_{Sub (G;F)} (H_{p}(C_{G}?\X^{?}; R), H_{G}^{q}(G/?)) . (4.4)

For any contravariant RSub(G; F)-module M and p ∈ ℤ there is an in (X, A) natural R-homomorphism

α_{G}^{p}(X, A; M): H_{RSub (G;F)}^{p}(X, A; M) → hom_{ℚSub (G;F)}(H_{p}(C_{G}?\X^{?}; ℚ), M)

⟶^{≅} hom_{RSub (G;F)}(H_{p}(C_{G}?\X^{?}; R), M) (4.5)

which is bijective if M is injective as ℚSub(G; F)-module.

Theorem 4.6 (The equivariant Chern character). Let R be a commutative ring R with ℚ ⊆ R. Let H_{?}^{∗} be a proper equivariant cohomology theory with values in R-modules. Suppose that the RSub(G; F)-module H_{G}^{q}(G/?) of (3.7), which sends G/H to H_{G}^{q}(G/H), is injective as ℚSub(G; F)-module for every group G and every q ∈ ℤ.

Then we obtain a transformation of proper equivariant cohomology theories with values in R-modules

ch_{?}^{∗}: H_{?}^{∗} ⟶^{≅} BH_{?}^{∗},

if we define for a group G and a proper G-CW -pair (X, A)

ch_{G}^{n}(X, A): H_{G}^{n}(X, A) → BH_{G}^{n}(X, A) := ∏_{p+q=n} H_{RSub (G;F)}^{p}(X, A; H_{G}^{q}(G/?))

by the composite

H_{G}^{n}(X, A) \underrightarrow{∏_{p+q=n} \underline{\underline{ch}}_{G}^{p,q}(X,A)} ∏_{p+q=n} hom_{RSub (G;F)} (H_{p}(C_{G}?\X^{?}; R), H_{G}^{q}(G/?))

\underrightarrow{∏_{p+q=n} α_{G}^{p}(X,A;H_{G}^{q}(G/?))^{−1}} ∏_{p+q=n} H_{RSub (G;F)}^{p}(X, A; H_{G}^{q}(G/?))

of the maps defined in (4.4) and (4.5).

The R-map ch_{G}^{n}(X, A) is bijective for all proper relative finite G-CW -pairs (X, A) and n ∈ ℤ. If H_{?}^{∗} satisfies the disjoint union axiom, then the R-map ch_{G}^{n}(X, A) is bijective for all proper G-CW -pairs (X, A) and n ∈ ℤ.

Proof. First one checks that ch_{G}^{∗} defines a natural transformation of proper Gcohomology theories. One checks for each finite subgroup H ⊆ G and n ∈ ℤ that ch_{G}^{n}(G/H) is the identity if we identify for any RSub(G; F)-module M

H_{RSub (G;F)}^{p}(G/H; M) = H^{p} (hom_{RSub (G;F)}(C_{∗}^{RSub (G;F)}(G/H), M)

= \{ \begin{matrix}hom_{RSub (G;F)} (R mor_{Sub (G;F)}(?, G/H), M) = M(G/H) & if p = 0; \\ 0 & if p ≠ 0. \end{matrix}

Finally apply Lemma 1.1 (b).

Remark 4.7 (The Atiyah-Hirzebruch spectral sequence for equivariant cohomology). There exists a Atiyah-Hirzebruch spectral sequence for equivariant cohomology (see [3, Theroem 4.7 (2)]). It converges to H_{G}^{p+q}(X, A) and has as E_{2}-term the Bredon cohomology groups H_{RSub (G;F)}^{p}(X, A; H_{G}^{q}(G/?)). The conclusion of Theorem 4.6 is that the spectral sequences collapses.

Example 4.8 (Equivariant Chern character for K^{∗}(G\(X, A))). Let K^{∗} be a (non-equivariant) cohomology theory with values in R-modules for a commutative ring with ℚ ⊆ R. In Example 1.6 we have assigned to it an equivariant cohomology theory by

H_{G}^{n}(X, A) = K^{n}(G\(X, A)).

We claim that the assumptions appearing in Theorem 4.6 are satisfied We have to show that the constant functor

\underline{K^{q}(\{pt.\})}: Sub(G; F) → ℚ - MOD, H ↦ K^{q}(\{pt.\})

is injective. Let i: Sub({1}) → Sub(G; F) the obvious inclusion of categories. Since the object {1} is an initial object in Sub(G; F), the ℚSub(G; F)-modules i_{!}(K^{q}(\{pt.\})) and \underline{K^{q}(\{pt.\})} are isomorphic. Since i_{!} sends an injective ℚ-module to an injective RSub(G; F)-module by Lemma2.13 (c) and H^{q}({pt.}) is injective as ℚ-module, \underline{K^{q}(\{pt.\})} is injective as ℚSub(G; F)-module. From Theorem 4.6 we get a transformation of equivariant cohomology theories

ch_{G}^{n}(X, A): K^{n}(G\(X, A)) ⟶^{≅} ∏_{p+q=n} H_{RSub (G;F)}^{p}(X, A; \underline{H^{q}(\{pt.\})})

= ∏_{p+q=n} H^{p}(G\(X, A); H^{q}({pt.})).

One easily checks that this is precisely the Chern character of Example 4.1 applied to K^{∗} and the CW -pair G\(X, A).

5. Mackey Functors

In Theorem 4.6 the assumption appears that the contravariant RSub(G; F)- module H_{G}^{q}(G/?) is injective for each q ∈ ℤ. We want to give a criterion which ensures that this assumption is satisfies and which turns out to apply to all cases of interest.

Let R be a commutative ring. Let FGINJ be the category of finite groups with injective group homomorphisms as morphisms. Let M: FGINJ→ R - MOD be a bifunctor, i.e. a pair (M_{∗}, M^{∗}) consisting of a covariant functor M_{∗} and a contravariant functor M^{∗} from FGINJ to R - MOD 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

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

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

(c) 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 G be a group. In the sequel we denote for a subgroup H ⊆ G by N_{G}H the normalizer and by C_{G}H the centralizer of H in G and by W_{G}H the quotient N_{G}H/H · C_{G}H. Notice that W_{G}H is finite if H is finite. Let R be a commutative ring. Let M be a Mackey functor with values in R-modules. It induces a contravariant RSub(G, F)-module denoted in the same way

M: Sub(G, F) → R - MOD, (f: H → K) ↦ (M^{∗}(f): M(H) → M(K)) .

We want to use Theorem 2.14 (b) to show that M is injective and analyse its structure. The R[W_{G}H]-module T_{H}M introduced in (2.11) is the same as the kernel of

∏_{K⊊H} M(i_{K}): M(H) → ∏_{K⊊H} M(K),

where for each subgroup K ⊊ H different from H we denote by i_{K} the inclusion. Suppose that R[W_{G}H]-module T_{H}M is injective for every finite subgroup H ⊆ G. For every finite subgroup H ⊆ G choose a retraction ρ_{H}: M(H) → T_{H}M of the inclusion T_{H}M → M(H). Denote by I = Is(Sub(G, F)) the set of isomorphism classes of objects in Sub(G; F) which is the same as the set of conjugacy classes (H) of finite subgroups H of G. Let

ν = ν(M): M → ∏_{(K)∈I} i(K)_{!} ∘ T_{K}(M)  (5.1)

be the homomorphism of RSub(G, F)-modules uniquely determined by the property that for any (K) ∈ I its composition with the projection onto the factor indexed by (K) is the adjoint of ρ_{K}: M(K) → T_{K}M for the adjoint pair (i(K)^{∗}, i(K)_{!}).

Theorem 5.2 (Injectivity and Mackey functors). Let G be a group and let R be a commutative ring such that the order of every finite subgroup of G is invertible in R. Suppose that the R[W_{G}H]-module T_{H}M is injective for each finite subgroup H ⊆ G. Then M is injective as RSub(G, F)-module and the map ν of (5.1) is bijective.

Proof. The map ν of (5.1) is the map ν(M) appearing in Theorem 2.14 (b). Because of Theorem 2.14 (b) it suffices to show for each finite subgroup H ⊆ G that ν(M)(H) is surjective.

Fix for any (K) ∈ I a representative K. Then choose for any W_{G}K · f ∈ W_{G}K\ mor(K, H) an element f ∈ conhom(K, H) which represents a morphism f: K → H in Sub(G; F) which belongs to W_{G}K·f ∈ W_{G}K\ mor(K, H). Notice that W_{G}K is the automorphism group of the object K in Sub(G; F) and W_{G}K, mor(K, H) and W_{G}K\ mor(K, H) are finite. With these choices we get for every object H in Sub(G; F) an identification

i(K)_{!}T_{K}M(H) = hom_{RW_{G}K}(R mor(K, H), T_{K}M) = ∏_{\begin{matrix}W_{G}K·f∈ \\ W_{G}K\ mor(K,H) \end{matrix}} T_{K}M^{W_{G}K_{f}}

where W_{G}K_{f} ⊆ W_{G}K is the isotropy group of f under the W_{G}K-action on mor(K, H). Under this identification ν(H) becomes the map

ν(H): M(H) → ∏_{(K)∈I} ∏_{\begin{matrix}W_{G}K·f∈ \\ W_{G}K\ mor(K,H) \end{matrix}} T_{K}M^{W_{G}K_{f}}

for which the component of ν(H)(m), which belongs (K) ∈ I and W_{G}K · f ∈ W_{G}K\ mor(K, H), is ρ_{K} ∘ res_{f}(m) for m ∈ M(H). Notice that the image of res_{f} always is contained in M(K)^{W_{G}K_{f}}. Next we define a map

μ(H): ⨁_{\begin{matrix}(K)∈I, \\ (K)≤(H) \end{matrix}} ⨁_{\begin{matrix}W_{G}K·f∈ \\ W_{G}K\ mor(K,H) \end{matrix}} T_{K}M^{W_{G}K_{f}} → M(H)

by requiring that its restriction to the summand, which belongs to (K) ∈ I and W_{G}K · f ∈ W_{G}K\ mor(K, H), is the composite of the inclusion T_{K}M^{W_{G}K_{f}} → M(K) with ind_{f}: M(K) → M(H). We want to show that the composite

ν(H) ∘ μ(H): ⨁_{\begin{matrix}(K)∈I, \\ (K)≤(H) \end{matrix}} ⨁_{\begin{matrix}W_{G}K·f∈ \\ W_{G}K\ mor(K,H) \end{matrix}} T_{K}M^{W_{G}K_{f}}

→ ∏_{(K)∈I} ∏_{\begin{matrix}W_{G}K·f∈ \\ W_{G}K\ mor(K,H) \end{matrix}} T_{K}M^{W_{G}K_{f}} = ⨁_{\begin{matrix}(K)∈I, \\ (K)≤(H) \end{matrix}} ⨁_{\begin{matrix}W_{G}K·f∈ \\ W_{G}K\ mor(K,H) \end{matrix}} T_{K}M^{W_{G}K_{f}}

is bijective. If K is subconjugated to H, we write (K) ≤ (H). Fix (K), (L) ∈ I with (K) ≤ (H) and (L) ≤ (H) and W_{G}K · f ∈ mor(K, H) and W_{G}L · g ∈ mor(L, H). Then the homomorphism T_{K}M^{W_{G}K^{f}} → T_{L}M^{W_{G}L^{g}} given by ν(H) ∘ μ(H) and the summands corresponding to (K, f) and (L, g) is induced by the composite

α_{(K,f),(L,g)} : T_{K}M^{W_{G}K^{f}} ⟶^{i} M(K) ⟶^{ind_{f : K→im(f)}} M(im(f)) ⟶^{ind_{im(f)}^{H}} M(H)

⟶^{res_{H}^{im(g)}} M(im(g)) ⟶^{res_{g: L→im(g)}} M(L) ⟶^{ρ_{L}} T_{L}M, (5.3)

where i is the inclusion. The double coset formula implies

res_{H}^{im(g)} ∘ ind_{im(f)}^{H}

= ∑_{im(g)h im(f)∈ im(g)\H/ im(f)} ind_{c(h) : im(f)∩h^{−1} im(g)h→im(g)} ∘ res_{im(f)}^{im(f)∩h^{−1} im(g)h} . (5.4)

The composite

T_{K}M^{W_{G}K^{f}} ⟶^{i} M(K) ⟶^{ind_{f : K→im(f)}} M(im(f))

⟶^{res_{im(f)}^{im(f)∩h^{−1} im(g)h}} M(im(f) ∩ h^{−1} im(g)h)

is trivial by the definition of T_{K}M if im(f) ∩ h^{−1} im(g)h ≠ im(f) holds. Hence α_{(K,f),(L,g)} ≠ 0 is only possible if im(f) ∩ h^{−1} im(g)h = im(f) for some h ∈ H and hence (K) ≤ (L) hold.

Suppose that (K) = (L). Then K = L by our choice of representatives. Suppose that α_{(K,f),(K,g)} ≠ 0. We have already seen that this implies im(f) ∩ h^{−1} im(g)h = im(f) for some h ∈ H. Since | im(f)| = |K| = | im(g)| we conclude h^{−1} im(g)h = im(f) and therefore W_{G}K · f = W_{G}K · g in W_{G}K\ mor(K, H). This implies already f = g as group homomorphism K → H by our choice of representatives. The double coset formula (5.4) implies that α_{(K,f),(K,f)} is |H ∩ N_{G} im(f)| · id_{T_{K}M^{^{W_{G}K}^{f}}} since for all h ∈ N_{G} im(f) ∩ H the composite

T_{K}M^{W_{G}K^{f}} ⟶^{i} M(K) ⟶^{ind_{f : K→im(f)}} M(im(f))

⟶^{ind_{c(h) : im(f)→im(f)}} M(im(f))

agrees with T_{K}M^{W_{G}K^{f}} ⟶^{i} M(K) ⟶^{ind_{f : K→im(f)}} M(im(f)). Since the order of |H ∩ N_{G} im(f)| is invertible in R by assumption, α_{(K,f),(K,f)} is bijective.

We conclude that ν(H)∘μ(H) can be written as a matrix of maps which has upper triangular form and isomorphisms on the diagonal. Therefore ν(H)∘μ(H) is surjective. This shows that ν(H) is surjective. This finishes the proof of Theorem 5.2.

Theorem 5.5 (The equivariant Chern character and Mackey structures). Let R be a commutative ring with ℚ ⊆ R. Let H_{?}^{∗} be a proper equivariant cohomology theory. Define a contravariant functor

H_{?}^{q}({pt.}): FGINJ→ R - MOD

by sending a homomorphism α: H → K to the composite

H_{K}^{q}({pt.}) ⟶^{H^{q}(pr)} H_{K}^{q}(K/H) ⟶^{ind_{α}} H_{H}^{q}({pt.})

where pr: H/K = ind_{α}({pt.}) → {pt.} is the projection and ind_{α} comes from the induction structure of H_{?}^{∗}. Suppose that it extends to a Mackey functor for every q ∈ ℤ. Then

(a) For every group G the RSub(G; F)-module H_{G}^{q}(G/?) of (3.7) is injective as RSub(G; F)-module, provided that R is semisimple;

(b) We obtain a natural transformation of proper equivariant cohomology theories with values in R-modules

ch_{?}^{∗}(X, A): H_{?}^{∗} → BH_{?}^{∗}.

In particular we get for every proper G-CW -pair (X, A) and every n ∈ ℤ a natural R-homomorphism

ch_{G}^{n}(X, A): H_{G}^{n}(X, A)

→ BH_{G}^{n}(X, A) := ∏_{p+q=n} H_{RSub (G;F)}^{p}(X, A; H_{G}^{q}(G/?)).

It is bijective for all proper relative finite G-CW -pairs (X, A) and n ∈ ℤ. If H_{?}^{∗} satisfies the disjoint union axiom, it is bijective for all proper G-CW -pairs (X, A) and n ∈ ℤ; (c) Define for finite subgroup H ⊆ G the R[W_{G}H]-module T_{H}(H_{H}^{q}({pt.})) by

ker ( ∏_{L⊊H} ind_{L}^{K} ∘H^{q}(pr: H/L → {pt.}): H_{H}^{q}({pt.}) → ∏_{L⊊H} H_{L}^{q}({pt.})) .

Then the Bredon cohomology BH_{G}^{n}(X, A) of a proper G-CW -pair (X, A) is naturally R-isomorphic to

∏_{p+q=n}  ∏_{(H),H⊆G finite} H_{W_{G}H}^{p}(C_{G}H\(X^{H}, A^{H}); T_{H}(H_{H}^{q}({pt.}))

provided that R is semisimple.

Proof. (a) This follows from Theorem 5.2 since for every finite subgroup H ⊆ G the group W_{G}H is finite and hence the ring R[W_{G}H] is semisimple and every R[W_{G}H]-module is injective.

(b) This follows from assertion (a) applied in the case R = ℚ together with Theorem 4.6.

(c) Since R is semisimple and contains ℚ and W_{G}H is finite, the ring R[W_{G}H] is semisimple and every R[W_{G}H]-module is injective for every finite subgroup H. Because the map ν of (5.1) is an isomorphism by Theorem 5.2, it remains to show for a C_{G}H-module N

H_{RSub (G;F)}^{p}(X, A; i(H)_{!}N) ≅ H_{W_{G}H}^{p}(C_{G}H\(X^{H}, A^{H}); N)

This follows from the adjunction (i(H)^{∗}, i(H)_{!}) of Lemma 2.13 (a).

Example 5.6 (Mackey structures for Borel cohomology). Let K^{∗} be a cohomology theory for (non-equivariant) CW -pairs with values in R-modules for a commutative ring R such that ℚ ⊆ R and R is semisimple. In Example 1.6 we have assigned to it an equivariant cohomology theory called equivariant Borel cohomology by

H_{G}^{n}(X, A) = K^{n}(EG ×_{G} (X, A)).

We claim that the assumptions appearing in Theorem 5.5 are satisfied. Namely, the contravariant functor

FGINJ→ R - MOD, H ↦ K^{n}(BH)

extends to a Mackey functor, the necessary covariant functor comes from the Becker-Gottlieb transfer (see for instance [5] and [6, Corollary 6.4 on page 206]). Hence we get from Theorem 5.5 for every group G and every proper G-CW -pair (X, A) natural R-maps

ch_{G}^{n}(X, A): K^{n}(EG ×_{G} (X, A)) ⟶^{≅} ∏_{p+q=n} H_{RSub (G;F)}^{p}(X, A; K^{q}(B?))

≅ ∏_{p+q=n} ∏_{(H),H⊆G finite} hom_{RW_{G}H}(H_{p}(C_{G}H\X^{H}; R), T_{H}(K^{q}(BH)),

if we define

T_{H}(K^{q}(BH)) := ker ( ∏_{K⊊H} K^{q}(BK → BH): K^{q}(BH) → ∏_{K⊊H} K^{q}(BK)) .

If (X, A) is relative finite or if K^{∗} satisfies the disjoint union axiom, then these maps ch_{G}^{n}(X, A) are bijective.

Remark 5.7. We mention that this does not prove Theorem 0.1 since we cannot apply it to K^{∗} := K^{∗} ⊗_{ℤ} ℚ. The problem is that K^{∗} ⊗_{ℤ} ℚ defines all axioms of a cohomology theory but not the disjoint union axiom. But this is needed if we want to deal with classifying spaces BG of groups which are not finite CW -complexes, for instance of groups containing torsion (see also Remark 1.2 and Example 1.3).

A proof of Theorem 0.1 will be given in [11].

Example 5.8 (Equivariant K-theory and Mackey structures). In Example 1.7 we have introduced the equivariant cohomology theory K_{?}^{∗} given by topological K-theory. Recall that it takes values in R-modules for R = ℤ. Notice that for a finite group H we get an identification of K_{H}^{0}({pt.}) with the complex representation ring R(H) and the associated contravariant functor

K_{?}^{q}: FGINJ→ R - MOD, H ↦ K_{H}^{q}({pt.}) = R(?)

sends an injective group homomorphism α: H → G of finite groups to the homomorphism of abelian groups R(G) → R(H) given by restriction with α. Induction with α induces a covariant functor H ↦ R(H) and it turns out that this defines a Mackey structure on K_{?}^{q}.

For rationalized equivariant topological K-theory K_{?}^{∗} ⊗_{ℤ} ℚ the equivariant Chern character of Theorem 5.5 can be identified with the one constructed in [12] for proper relative finite G-CW -pairs (X, A).

6. Multiplicative Structures

Next we want to introduce a multiplicative structure on a proper equivariant cohomology theory H_{?}^{∗} and show that it induces one on the associated Bredon cohomology BH_{?}^{∗} such that the equivariant Chern character is compatible with it.

We begin with the non-equivariant case. Let H^{∗} be a (non-equivariant) cohomology theory with values in R-modules. A multiplicative structure assigns to a CW -complex X with CW -subcomplexes A, B ⊆ X natural R-homomorphisms

∪: H^{n}(X, A) ⊗_{R} H^{n′}(X, B) → H^{n+n′}(X, A ∪ B).  (6.1)

This product is required to be compatible with the boundary homomorphism of the long exact sequence of a pair, to be graded commutative, to be associative and to have a unit 1 ∈ H^{0}({pt.}).

Given a multiplicative structure on H^{∗}, we obtain for every p, q ∈ ℤ a pairing

∪: H^{q}({pt.}) ⊗_{R} H^{q′}({pt.}) → H^{q+q′}({pt.}).

It yields on singular (or equivalently cellular) cohomology a product

H^{p}(X, A; H^{q}({pt.}))⊗_{R}H^{p′}(X, B; H^{q′}({pt.})) → H^{p+p′}(X, A∪B; H^{q+q′}({pt.})).

The collection of these pairings induce a multiplicative structure on the cohomology theory given by ∏_{p+q=n} H^{p}(X, A; H^{q}({pt.})). The straightforward proof of the next lemma is left to the reader. Lemma 6.2. Let R be a commutative ring with ℚ ⊆ R. Let H^{∗} be a (nonequivariant) cohomology theory satisfying the disjoint union axiom which comes with a multiplicative structure.

Then the (non-equivariant) Chern character of Example 4.1

ch^{n}(X, A): H^{n}(X, A) ⟶^{≅} ∏_{p+q=n} H^{p}(X, A, H^{q}(∗))

is compatible with the given multiplicative structure on H^{∗} and the induced multiplicative structure on the target.

Next we deal with the equivariant version. We only deal with the proper case, the definitions below make also sense without this condition.

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 Rhomomorphisms

∪: H_{G}^{n}(X, A) ⊗_{R} H_{G}^{n′}(X, B) → H_{G}^{n+n′}(X, A ∪ B).  (6.3)

This product is required to be compatible with the boundary homomorphism of the long exact sequence of a G-CW -pair, 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 (1.4)

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}^{∗}.

Next we explain how a given multiplicative structure on H_{?}^{∗} induces one on BH_{?}^{∗}. We have to specify for every group G a multiplicative structure on the G-cohomology theory BH_{G}^{∗}. Consider a G-CW -complex X with G-CW - subcomplexes A, B ⊆ X. For two contravariant ROr(G; F)-chain complexes C_{∗} and D_{∗} define the contravariant ROr(G; F)-chain complexes C_{∗} ⊗_{R} D_{∗} by sending G/H to the tensor product of R-chain complexes C_{∗}(G/H) ⊗_{R} D_{∗}(G/H). Let

a_{∗}: C_{∗}^{ROr (G;F)}(X, A) ⊗_{R} C_{∗}^{ROr (G;F)}(X, B) ⟶^{≅} C_{∗}^{ROr (G;F)}((X, A) × (X, B))

be the isomorphism of ROr(G; F)-chain complexes which is given for an object G/H by the natural isomorphism of cellular R-chain complexes

C_{∗}(X^{H}, A^{H}) ⊗_{R} C_{∗}(X^{H}, B^{H}) ⟶^{≅} C_{∗}((X^{H}, A^{H}) × (X^{H}, B^{H})).

The multiplicative structure on H_{G}^{∗} yields a map of contravariant ROr(G; F)- modules

c: H_{G}^{q}(G/?) ⊗_{R} H_{G}^{q}(G/?) → H_{G}^{q+q}(G/?).

Let

Δ: (X; A ∪ B) → (X, A) × (X, B), x ↦ (x, x)

be the diagonal embedding. Define a R-cochain map by the composite

b^{∗}: hom_{ROr (G;F)} (C_{∗}^{ROr (G;F)}(X, A), H_{G}^{q}(G/?))

⊗_{R} hom_{ROr (G;F)} (C_{∗}^{ROr (G;F)}(X, B), H_{G}^{q′}(G/?)) ⟶^{⊗_{R}}

hom_{ROr (G;F)} (C_{∗}^{ROr (G;F)}(X, A) ⊗_{R} C_{∗}^{ROr (G;F)}(X, B), H_{G}^{q}(G/?) ⊗_{R} H_{G}^{q′}(G/?))
⟶^{hom_{ROr (G;F)}((a_{∗})^{−1},c)} hom_{ROr (G;F)} (C_{∗}^{ROr (G;F)}((X, A) × (X, B)), H_{G}^{q+q′}(G/?))

⟶^{hom_{ROr (G;F)}(C_{∗}^{ROr (G;F)}(Δ),id)} hom_{ROr (G;F)} (C_{∗}^{ROr (G;F)}(X, A ∪ B), H_{G}^{q+q′}(G/?)) .

There is a canonical R-map

H^{∗}(C^{∗} ⊗_{R} D^{∗}) → H^{∗}(C^{∗} ⊗_{R} D^{∗})

for two R-cochain complexes C^{∗} and D^{∗}. This map together with the map induced by b^{∗} on cohomology yields an R-homomorphism

H_{ROr (G;F)}^{p} (X, A; H_{G}^{q}(G/?)) ⊗_{R} H_{ROr (G;F)}^{p′}(X, B; H_{G}^{q′}(G/?))

→ H_{ROr (G;F)}^{p+p′} (X, A ∪ B; H_{G}^{q}(G/?)) .

The collection of these R-homomorphisms yields the desired multiplicative structure on BH_{G}^{∗}. We leave it to the reader to check that the axioms of a multiplicative structure on BH_{G}^{∗} are satisfied and that all these are compatible with the induction structure so that we obtain a multiplicative structure on the equivariant cohomology theory BH_{?}^{∗}.

We also omit the lengthy but straightforward proof of the following result which is based on Theorem 4.6, Lemma 6.2 and the compatibility of the multiplicative structure with the induction structure.

Theorem 6.4 (The equivariant Chern character and multiplicative structures). Let R be a commutative ring such that ℚ ⊆ R. Suppose that H_{?}^{∗} is a proper cohomology theory with values in R-modules which comes with a multiplicative structure. Suppose that the RSub(G; F)-module H_{G}^{q}(G/?) of (3.7), which sends G/H to H_{G}^{q}(G/H), is injective for each q ∈ ℤ.

Then the natural transformation of equivariant cohomology theories appearing in Theorem 4.6

ch_{?}^{∗}: H_{?}^{∗} → BH_{?}^{∗}

is compatible with the given multiplicative structure on H_{?}^{∗} and the induced multiplicative structure on BH_{?}^{∗}.

Remark 6.5 (External products and restriction structures). One can also define an external product for a proper equivariant cohomology theory H_{?}^{∗} with values in R-modules. It assigns to every two groups G and H, a proper G-CW -pair (X, A), a proper H-CW -pair(Y, B) and p, q ∈ ℤ an R-homomorphism

×: H_{G}^{p}(X, A) ⊗_{R} H_{H}^{q}(Y, B) → H_{G×H}^{p+q}((X, A) × (Y, B)).

One requires graded commutativity, associativity, the existence of a unit 1 ∈ H_{{1}}^{0}({pt.}) and compatibility with the induction structure and the boundary homomorphism associated to a pair. One can show that BH_{?}^{∗} inherits an external product and prove the analogon of Theorem 6.4 for external products.

One can also introduce the notion of a restriction structure on H_{?}^{∗}. It yields for every injective group homomorphism α: H → G, every proper G-CW -pair (X, A) and p ∈ ℤ an R-homomorphism

res_{α}: H_{G}^{p}(X, A)) → H_{H}^{p}(res_{α}(X, A)).

Again certain axioms are required such as compatibility with the boundary homomorphism associated to pair, compatibility with induction for group isomorphisms α: H ⟶^{≅} G, compatibility with conjugation, the double coset formula and compatibility for projections onto quotients under free actions. One can show that BH_{?}^{∗} inherits a restriction structure and prove the analogon of Theorem 6.4 for restriction structures.

An external product together with a restriction structure yields a multiplicative structure as follows. Consider G-CW -pairs (X, A) and (X, B). Let d: G → G × G and D: (X, A ∪ B) → (X, A) × (X, B) be the diagonal maps. Define

∪: H_{G}^{m}(X, A) ⊗_{R} H_{G}^{n}(X, A) → H_{G}^{m+n}(X, A ∪ B)  (6.6)

to be the composite

H_{G}^{m}(X, A) ⊗_{R} H_{G}^{n}(X, A) ⟶^{×} H_{G×G}^{m+n}((X, A) × (X, B))

⟶^{res_{d}} H_{G}^{m+n}((X, A) × (X, B)) ⟶^{H_{G}^{m+n}(D)} H_{G}^{m+n}(X, A ∪ B).

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