Equivariant Cohomological Chern Characters

and a contravariant Or(G)-Ω-spectrum E ◦ G G . Given a G-CW -pair (X, A), weobtain a contravariant pair of Or(G)-CW -complexes (X ? , A ? ) by sending G/Hto (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 contravariantOr(G)-CW -complexes as explained in [3, Section 4]. It value at (X ? , A ? )is defined to be HG ∗ (X, A; E). Explicitly, (Hn G (X, A; E) is the (−n)-th homotopygroup of the spectrum map Or(G) X + ? ∪ A ?+cone(A ? +), E ◦ G). G We needΩ-spectra in order to ensure that the disjoint union axiom holds.We briefly explain for a group homomorphism α: H → G the definition ofthe induction homomorphism ind α : HG n (ind α X; E) → HH n (X; E) in the specialcase A = ∅. The functor induced by α on the orbit categories is denoted in thesame wayα: Or(H) → Or(G), H/L ↦→ ind α (H/L) = G/α(L).There is an obvious natural transformation of functors Or(H) → GROUPOIDST : 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 byh ∈ H to the morphism α(h) ∈ G. Notice that T (H/L) is an equivalence ifker(α) acts freely on H/L. The desired isomorphismind α : H n G(ind α X; E) → H n H(X; E)is induced by the following map of spectramap Or(G)(mapG (−, ind α X + ), E ◦ G G)∼ =−→ map Or(G)(α∗ (map H (−, X + )), E ◦ G G)∼ =−→ map Or(H)(mapH (−, X + ), E ◦ G G ◦ α )map Or(H) (id,E(T ))−−−−−−−−−−−−→ map Or(H)(mapH (−, X + ), E ◦ G H) .Here α ∗ map H (−, X + ) is the pointed Or(G)-space which is obtained from thepointed Or(H)-space map H (−, X + ) by induction, i.e. by taking the balancedproduct 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 ofinduction α ∗ and restriction α ∗ (see [3, Lemma 1.9]). The first map comes fromthe homeomorphism of Or(G)-spacesα ∗ 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 .8