Equivariant Cohomological Chern Characters

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much simpler and easier to compute than H? ∗ . If H∗ ?satisfies the disjoint unionaxiom, thench n G(X, A): HG(X, n A) −→ ∼= BHG(X, n 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 acertain injectivity condition (see Theorem 4.6). This condition is fulfilled if thecoefficients of H? ∗ come with a Mackey structure (see Theorem 5.5) what is thecase in many interesting examples.The equivariant cohomological Chern character is a generalization to theequivariant 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=nH p (X, A, H q (∗)).The equivariant cohomological Chern character has already been constructedin 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 importantrole in the computation of the source of the assembly maps appearing in theFarrell-Jones Conjecture for K n (RG) and L 〈−∞〉n (RG) and the Baum-ConnesConjecture for K n (Cr ∗ (G)) (see also [9]).The detailed formulation of the main result of this paper is presented inTheorem 5.5.The equivariant Chern character will play a key role in the proof of thefollowing result which will be presented in [11].Theorem 0.1 (Rational computation of the topological K-theory ofBG). Let G be a discrete group. Suppose that there is a finite G-CW -model forthe classifying space EG for proper G-actions. Then there is a Q-isomorphismch n G : K n (BG) ⊗ Z Q∼ =−→ ∏ i∈ZH 2i+n (BG; Q) × ∏p prime∏(g)∈con p(G)H 2i+n (BC G 〈g〉; Q̂p).Here con p (G) is the set of conjugacy classes (g) of elements g ∈ G of orderp m for some integer m ≥ 1 and C G 〈g〉 is the centralizer of the cyclic subgroupgenerated by g in G.The assumption in Theorem 0.1 that there is a finite G-CW -model for theclassifying space EG for proper G-actions is satisfied for instance, if G is wordhyperbolicin the sense of Gromov, if G is a cocompact subgroup of a Lie groupwith finitely many path components, if G is a finitely generated one-relatorgroup, if G is an arithmetic group, a mapping class group of a compact surfaceor the group of outer automorphisms of a finitely generated free group. Formore information about EG we refer for instance to [1] and [10].A group G is always understood to be discrete and a ring R is always understoodto be associative with unit throughout this paper.2
The paper is organized as follows:1. Equivariant Cohomology Theories2. Modules over a Category3. The Associated Bredon Cohomology Theory4. The Construction of the Equivariant Cohomological Chern Character5. Mackey Functors6. Multiplicative StructuresReferences1. Equivariant Cohomology TheoriesIn this section we describe the axioms of a (proper) equivariant cohomologytheory. They are dual to the ones of a (proper) equivariant homology theory asdescribed in [8, Section 1].Fix a group G and an commutative ring R. A G-CW -pair (X, A) is a pair ofG-CW -complexes. It is proper if all isotropy groups of X are finite. It is relativefinite 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 canbe found for instance in [7, Section 1 and 2]. A G-cohomology theory HG ∗ withvalues in R-modules is a collection of covariant functors HG n from the categoryof G-CW -pairs to the category of R-modules indexed by n ∈ Z together withnatural transformations δG n (X, A): Hn G (X, A) → Hn+1G(A) := Hn+1G(A, ∅) forn ∈ Z such that the following axioms are satisfied:• G-homotopy invarianceIf f 0 and f 1 are G-homotopic maps (X, A) → (Y, B) of G-CW -pairs, thenH n G (f 0) = H n G (f 1) for n ∈ Z;• Long exact sequence of a pairGiven a pair (X, A) of G-CW -complexes, there is a long exact sequence. . . δn−1 G−−−→ HG(X, n A) Hn G−−−−→ (j)HG(X) n Hn G−−−−→ (i)HG(A) n δn G−→ . . . ,where i: A → X and j : X → (X, A) are the inclusions;• ExcisionLet (X, A) be a G-CW -pair and let f : A → B be a cellular G-map ofG-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) inducesan isomorphismH n G(F, f): H n G(X, A) ∼= −→ H n G(X ∪ f B, B).3
Page 1: Equivariant Cohomological Chern Cha
Page 5 and 6: Example 1.3 (Rationalizing topologi
Page 7 and 8: Example 1.7 (Equivariant K-theory).
Page 9 and 10: 2. Modules over a CategoryIn this s
Page 11 and 12: sends M to the kernel of the map∏
Page 13 and 14: injective. We show by induction ove
Page 15 and 16: Given a contravariant ROr(G, F)-mod
Page 17 and 18: We get a natural transformation ch
Page 19 and 20: y the composite∏p+q=n chp,q(X,A)H
Page 21 and 22: We want to use Theorem 2.14 (b) to
Page 23 and 24: h −1 im(g)h = im(f) and therefore
Page 25 and 26: if we defineT H (K q (BH)) :=⎛⎞
Page 27 and 28: the G-cohomology theory BHG ∗ . C
Page 29 and 30: References[1] P. Baum, A. Connes, a

Equivariant Cohomological Chern Characters

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