Equivariant Cohomological Chern Characters

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 choiceof representatives. The double coset formula (5.4) implies that α (K,f),(K,f) is|H ∩ N G im(f)| · id TK M W G K fsince for all h ∈ N G im(f) ∩ H the compositeT K M W GK fi−→ M(K) ind f : K→im(f)−−−−−−−−−→ M(im(f))iind c(h) : im(f)→im(f)−−−−−−−−−−−−−→ M(im(f))agrees with T K M W GK f−→ 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 hasupper 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 ofTheorem 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 equivariantcohomology theory. Define a contravariant functorH q ?({pt.}): FGINJ → R - MODby sending a homomorphism α: H → K to the compositeH q K ({pt.}) Hq (pr)−−−−→ H q indαK(K/H) −−−→ H q H ({pt.})where pr: H/K = ind α ({pt.}) → {pt.} is the projection and ind α comes fromthe induction structure of H? ∗ . Suppose that it extends to a Mackey functor forevery q ∈ Z. Then(a) For every group G the RSub(G; F)-module H q G(G/?) of (3.7) is injectiveas RSub(G; F)-module, provided that R is semisimple;(b) We obtain a natural transformation of proper equivariant cohomology theorieswith values in R-modulesch ∗ ?(X, A): H ∗ ? → BH ∗ ?.In particular we get for every proper G-CW -pair (X, A) and every n ∈ Za natural R-homomorphismch n G(X, A): H n G(X, A)→ BH n G(X, A) :=∏p+q=nH p RSub(G;F) (X, A; Hq G (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;23