Equivariant Cohomological Chern Characters

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