process

Inheritance of isomorphism conjectures under colimits 49r g D rw.g/ g 1 . All the results in this paper generalize directly to this casesince one can construct a modified L-theory spectrum functor (over G) using the w-twisted involution and then the homology arguments are just applied to the equivarianthomology theory associated to this w-twisted L-theory spectrum.Acknowledgements. The work was financially supported by the Sonderforschungsbereich478 – Geometrische Strukturen in der Mathematik – and the Max-Planck-Forschungspreis of the third author.2 Equivariant homology theoriesIn this section we briefly explain basic axioms, notions and facts about equivarianthomology theories as needed for the purposes of this article. The main examples whichwill play a role in connection with the Bost, the Baum–Connes and the Farrell–Jonesconjecture will be presented later in Theorem 6.1.Fix a group G and a ring ƒ. In most cases ƒ will be Z. The following definitionis taken from [28, Section 1].Definition 2.1 (G-homology theory). A G-homology theory H G with values in ƒ-modules is a collection of covariant functors HnG from the category of G-CW-pairs tothe category of ƒ-modules indexed by n 2 Z together with natural transformations@ G n .X; A/W H n G.X; A/ ! H n G 1 .A/ WD H n G 1.A; ;/ for n 2 Z such that the followingaxioms are satisfied:• G-homotopy invariance. Iff 0 and f 1 are G-homotopic maps .X; A/ ! .Y; B/of G-CW-pairs, then H G n .f 0/ D H G n .f 1/ for n 2 Z.• Long exact sequence of a pair. Given a pair .X; A/ of G-CW-complexes, thereis a long exact sequenceH G nC1 .j / ! H G nC1 .X; A/ @G nC1! H G n .A/H G n .i/ ! H G n .X/ H G n .j / ! H G n .X; A/ @G n! ;where i W A ! X and j W X ! .X; A/ are the inclusions.• Excision. Let .X; A/ be a G-CW-pair and let f W A ! B be a cellular G-mapof G-CW-complexes. Equip .X [ f B;B/ with the induced structure of a G-CW-pair. Then the canonical map .F; f /W .X; A/ ! .X [ f B;B/ induces anisomorphismH G n .F; f /W H G n .X; A/ Š ! H G n .X [ f B;B/: