process

Inheritance of isomorphism conjectures under colimitsArthur Bartels, Siegfried Echterhoff, and Wolfgang Lück1 Introduction1.1 Assembly maps. We want to study the following assembly maps:asmb G n W H G n .E VCyc.G/I K R / ! H G n .fgI K R/ D K n .R Ì G/I (1.1)asmb G n W H G n .E F in.G/I KH R / ! H G n .fgI KH R/ D KH n .R Ì G/I (1.2)asmb G n W H n G .E VCyc.G/I L h 1iR/ ! Hn G.fgI Lh1iR/ D L h 1in .R Ì G/I (1.3)asmb G n W H n G .E F in.G/I K top / ! H G A;l 1 n .fgI Ktop / D KA;l 1 n .A Ì l 1 G/I (1.4)asmb G n W H n G .E F in.G/I K topA;r / ! H n G .fgI KtopA;r / D K n.A Ì r G/I (1.5)asmb G n W H n G .E F in.G/I K topA;m / ! H n G .fgI KtopA;m / D K n.A Ì m G/: (1.6)Some explanations are in order. A family of subgroups of G is a collection ofsubgroups of G which is closed under conjugation and taking subgroups. Examplesare the family F in of finite subgroups and the family VCyc of virtually cyclic subgroups.Let E F .G/ be the classifying space associated to F . It is uniquely characterizedup to G-homotopy by the properties that it is a G-CW-complex and that E F .G/ H iscontractible if H 2 F and is empty if H … F . For more information about thesespaces E F .G/ we refer for instance to the survey article [29].Given a group G acting on a ring (with involution) by structure preserving maps, letRÌG be the twisted group ring (with involution) and denote by K n .RÌG/, KH n .RÌG/and L hn 1i .RÌG/its algebraic K-theory in the non-connective sense (see Gersten [17]or Pedersen–Weibel [34]), its homotopy K-theory in the sense of Weibel [41], and itsL-theory with decoration 1 in the sense of Ranicki [37, Chapter 17]. Given a groupG acting on a C -algebra A by automorphisms of C -algebras, let A Ì l 1 G be theBanach algebra obtained from A Ì G by completion with respect to the l 1 -norm, letA Ì r G be the reduced crossed product C -algebra, and let A Ì m G be the maximalcrossed product C -algebra and denote by K n .AÌ l 1 G/, K n .AÌ r G/and K n .AÌ m G/their topological K-theory.The source and target of the assembly maps are given by G-homology theories(see Definition 2.1 and Theorem 6.1) with the property that for every subgroup H G