process

Categorical aspects of bivariant K-theory 33The first two properties make precise in what sense E.G; F / X is the best approximationto X among G; F -CW-complexes.Definition 67. The assembly map with respect to F is the map F E.G; F / ! F .?/induced by the constant map E.G; F / D E.G; F / ? ! ?.More generally, the assembly map with coefficients in a pointed G-CW-complex(or spectrum) X is the map F .E.G; F / C ^ X/ ! F .S 0 ^ X/ D F .X/ induced bythe map E.G; F / C ! ? C D S 0 .In the stable homotopy category of pointed G-CW-complexes (or spectra), we getan exact triangle E.G; F / C ^ X ! X ! N ! S 1 ^ E.G; F / C ^ X, where Nis H -equivariantly contractible for each H 2 F . This means that the domain of theassembly map F .E.G; F / C ^ X/ is the localisation of F at the class of all objectsthat are H -equivariantly contractible for each H 2 F .Thus the assembly map is an isomorphism for all X if and only if F .N / D 0whenever N is H -equivariantly contractible for each H 2 F . Thus an isomorphismconjecture can be interpreted in two equivalent ways. First, it says that we can reconstructthe homology theory from its restriction to G; F -CW-complexes. Secondly, itsays that the homology theory vanishes for spaces that are H -equivariantly contractiblefor H 2 F .5.2 From spaces to operator algebras. We can carry over the construction of assemblymaps above to bivariant Kasparov theory; we continue to assume G discrete tosimplify some statements. From now on, we let F be the family of finite subgroups.This is the family that appears in the Baum–Connes assembly map. Other families ofsubgroups can also be treated, but some proofs have to be modified and are not yetwritten down.First we need an analogue of G; F -CW-complexes. These are constructible out ofsimpler “cells” which we describe first, using the induction functorsInd G H W KKH ! KK Gfor subgroups H G. For a finite group H , IndH G .A/ is the H -fixed point algebra ofC 0 .G; A/, where H acts by h f.g/D ˛h f.gh/ . For infinite H ,wehaveIndH G .A/ Dff 2 C b.G; A/ j ˛hf.gh/D f.g/for all g 2 G, h 2 H ,and gH 7! kf.g/k is in C 0 .G=H /gIthe group G acts by translations on the left.This construction is functorial for equivariant -homomorphisms. Since it commuteswith C -stabilisations and maps split extensions again to split extensions, itdescends to a functor KK H ! KK G by the universal property (compare §4.1).We also have the more trivial restriction functors Res H G W KKG ! KK H for subgroupsH G. The induction and restriction functors are adjoint:KK G .Ind G H A; B/ Š KKH .A; Res H G B/