Equivariant Cohomological Chern Characters

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4. The Construction of the EquivariantCohomological Chern CharacterWe begin with explaining the cohomological version of the homological Cherncharacter due to Dold [4].Example 4.1 (The non-equivariant Chern character). Consider a (nonequivariant)cohomology theory H ∗ with values in R-modules. Suppose thatQ ⊆ R. For a space X let X + be the pointed space obtained from X by addinga disjoint base point ∗. Since the stable homotopy groups π s p(S 0 ) are finite forp ≥ 1 by results of Serre [15], the condition Q ⊆ R imply that the Hurewiczhomomorphism induces isomorphismshur R : πp(X s + ) ⊗ Z R hur ⊗ Z id−−−−−−→RHp (X) ⊗ Z R −→ ∼= H p (X; R)and that the canonical mapα: 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 mapD p,q : H p+q (X) → hom R (π s p(X + ) ⊗ Z R, H q ({pt.})) (4.2)as follows. Denote in the sequel by σ k the k-fold suspension isomorphism. Givena ∈ H p+q (X) and an element in π s p(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 compositeH p+q (X) −→ ∼= ˜H p+q (X + ) σk−→ ˜H p+q+k (S k ∧ X + ) ˜H p+q+k (f)−−−−−−−→ ˜H p+q+k (S p+k )(σ p+k ) −1−−−−−−→ ˜H q (S 0 ) ∼= −→ H q ({pt.}).Then the (non-equivariant) Chern character for a CW -complex X is given bythe following compositech n (X): H n (X)∏p+q=n Dp,q−−−−−−−−→∏p+q=n hom R(hur −1R ,id)−−−−−−−−−−−−−−−−→∏p+q=nhom R(πsp (X + , ∗) ⊗ Z R, H q (∗) )∏p+q=nhom R (H p (X; R), H q (∗))∏p+q=n α−1−−−−−−−−→∏p+q=nThere is an obvious version for a pair of CW -complexesch n (X, A): H n (X, A) −→∼= ∏H p (X, A, H q (∗)).p+q=nH p (X, H q (∗)).16
We get a natural transformation ch ∗ of cohomology theories with values in R-modules. 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 ∈ Zby Lemma 1.1 (b). If H ∗ satisfies the disjoint union axiom, then ch n (X, A) isbijective 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 cohomologytheory 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 finitesubgroup H ⊆ Gch p,qG(X, A)(H): Hp+qG(X, A)(→ hom R Hp (C G H\(X H , A H ); R), H q G (G/H)) . (4.3)We define it only in the case A = ∅, the general case is completely analogous.H p+qGH p+qG (v H)(X)⏐↓H p+qG (ind m HX H )⏐ ⏐↓H p+qG (indm H pr 2 )H p+qG (ind m HEG × X H )ind mH⏐ ⏐↓∼ =H p+qC G H×H (EG × XH )(ind pr : CG H×H→H) −1 ⏐ ⏐↓∼ =H p+qH (EG × C G H X H )⏐↓D p,qH (EG× C G HX H )hom R(πsp ((EG × CG H X H ) + ) ⊗ Z R, H q H ({pt.}))hom R (hur R (EG× CG HX H ),id) −1 ⏐ ⏐↓hom R(Hp (EG × CG H X H ; R), H q H ({pt.}))hom R (H p(pr 1 ;R),id) −1 ⏐ ⏐↓(hom R Hp (C G H\X H ; R), H q H ({pt.}))⏐hom R (id;(ind G H )−1 ) ↓hom R(Hp (C G H\X H ; R), H q G (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 × CG H X H → C G H\X H is the17
Page 1 and 2: Equivariant Cohomological Chern Cha
Page 3 and 4: The paper is organized as follows:1
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: Given a contravariant ROr(G, F)-mod
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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