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Inheritance of isomorphism conjectures under colimits 67functorsK R W Groupoids # G ! SpectraIKH R W Groupoids # G ! SpectraIL h 1iRtogether with natural transformationsW Groupoids # G ! SpectraIK top W Groupoids # G ! SpectraIA;l 1K topA;r W Groupoidsfinker # G ! SpectraIK topA;mW Groupoids # G ! Spectra;I 1 W K ! KHII 2 W KH ! K top IA;l 1I 3 W K top ! K topA;l 1 A;m II 4 W K topA;m ! Ktop A;r ;of functors from Groupoids # G or Groupoids finker # G respectively to Spectra suchthat the following holds:(i) Let F i W G i ! G be objects for i D 0; 1 and F W F 0 ! F 1 be a morphismbetween them in Groupoids # G or Groupoids finker # G respectively such thatthe underlying functor of groupoids F W G 0 ! G 1 is an equivalence of groupoids.Then the functors send F to a weak equivalences of spectra.(ii) Let F 0 W G 0 ! G be an object in Groupoids # G or Groupoids finker # G respectivelysuch that the underlying groupoid G 0 has only one object x. LetG D mor G0 .x; x/ be its automorphisms group. We obtain a ring R.y/, a ringR.y/ with involution, or a C -algebra B.y/ with G-operation by structure preservingmaps from the evaluation of the functor R or A respectively at y D F.x/.Then: n .K R .F // D K n .R.y/ Ì G/I n .KH R .F // D KH n .R.y/ Ì G/I n .L h 1iR.F // D L h n1i .R.y/ Ì G/I n .K top .F // D KA.y/;l 1 n .A.y/ Ì l 1 G/I n .K topA.y/;r .F // D K n.A.y/ Ì r G/I n .K topA.y/;m .F // D K n.A.y/ Ì m G/;where K n .R.y/ÌG/is the algebraic K-theory of the twisted group ring R.y/ÌG,KH n .R.y/ Ì G/ is the homotopy K-theory of the twisted group ring R.y/ Ì G,
68 A. Bartels, S. Echterhoff, and W. LückL hn1i.R.y/ÌG/ is the algebraic L-theory with decoration h 1i of the twistedgroup ring with involution R.y/ Ì G, K n .A.y/ Ì l 1 G/ is the topological K-theory of the crossed product Banach algebra A.y/ Ì l 1 G, K n .A.y/ Ì r G/ isthe topological K-theory of the reduced crossed product C -algebra A.y/ Ì r G,and K n .A.y/ Ì m G/ is the topological K-theory of the maximal crossed productC -algebra A.y/ Ì m G.The natural transformations I 1 , I 2 , I 3 and I 4 become under this identificationsthe obvious change of rings and theory homomorphisms.(iii) These constructions are in the obvious sense natural in R and A respectively andin G .We defer the details of the proof of Theorem 8.1 in [2]. Its proof requires some workbut there are many special cases which have already been taken care of. If we wouldnot insist on groupoids but only on groups as input, these are the standard algebraic K-and L-theory spectra or topological K-theory spectra associated to group rings, groupBanach algebras and group C -algebras. The construction for the algebraic K- andL-theory and the topological K-theory in the case, where G acts trivially on a ring Ror a C -algebra are already carried out or can easily be derived from [4], [13], and [24]except for the case of a Banach algebra. The case of the K-theory spectrum associatedto an additive category with G-action has already been carried out in [7]. The mainwork which remains to do is to treat the Banach case and to construct the relevantnatural transformation from KH to K topA;l 1References[1] J. F. Adams, Stable homotopy and generalised homology, Chicago Lectures in Math. Universityof Chicago Press, Chicago, Ill., 1974.[2] A. Bartels, M. Joachim, and W. Lück, From algebraic to topological K-theory spectra, inpreparation, 2007.[3] A. Bartels and W. Lück, Induction theorems and isomorphism conjectures for K- and L-theory, Forum Math. 19 (2007), 379–406.[4] A. Bartels and W. Lück, Isomorphism conjecture for homotopy K-theory and groups actingon trees, J. Pure Appl. Algebra 205 (2006),660–696.[5] A. Bartels, W. Lück, and H. Reich, The K-theoretic Farrell-Jones Conjecture for hyperbolicgroups, Invent. Math. 172 (2008), 29–70.[6] A. Bartels, W. Lück, and H. Reich, On the Farrell-Jones Conjecture and its applications,J. Topol. 1 (2008), 57–86.[7] A. Bartels and H. Reich, Coefficients for the Farrell-Jones conjecture, Adv. Math.,209(1):337–362, 2007.[8] H. Bass, Algebraic K-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968.[9] P. Baum, A. Connes, and N. Higson, Classifying space for proper actions and K-theory ofgroup C -algebras, in C -algebras: 1943–1993 (San Antonio, TX, 1993), Amer. Math.Soc., Providence, RI, 1994, 240–291.
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Editors:Guillermo CortiñasDepartam
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ContentsPreface....................
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IntroductionSince its inception 50
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Introductionxiwith D. Blecher, he c
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Program list of speakers and topics
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Categorical aspects of bivariant K-
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118 M. Karoubi[35], M. F. Atiyah an
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120 M. Karoubihere by GrK n .A/, ar
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122 M. Karoubi(resp. 1) in A. In th
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124 M. Karoubithe class ˛ of A in
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126 M. Karoubiested in the complex
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128 M. KaroubiThe same method may b
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130 M. KaroubiBy the usual excision
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132 M. KaroubiLet A be a graded twi
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134 M. Karoubithe same ideas as in
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136 M. KaroubiTheorem 6.2. 25 The (
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138 M. KaroubiIn order to show this
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140 M. KaroubiThe following theorem
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142 M. KaroubiWe would like to poin
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144 M. KaroubiZ=n identified with t
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146 M. Karoubiwhere n is the ring
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148 M. Karoubi[6] M. F. Atiyah, R.
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Equivariant cyclic homology for qua
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174 C. Voigtand using ˇ.x; y/ D .S
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176 C. Voigtalgebra A. The space Cn
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178 C. VoigtKK-theory [16]. Hence,
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A Schwartz type algebra for the tan
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202 J. CuntzWe denote by N the set
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204 J. CuntzConsider the action of
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206 J. Cuntzprime numbers p and q,
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208 J. Cuntz6 Representations as cr
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210 J. CuntzProof. The spectral pro
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212 J. CuntzThe operator s 1 plays
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214 J. CuntzProposition 7.4. The K-
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On a class of Hilbert C*-manifoldsW
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On a class of Hilbert C*-manifolds
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228 U. Bunke, T. Schick, M. Spitzwe
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Deformations of gerbes on smooth ma
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Parshin’s conjecture revisitedTho
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428 C. WeibelGiven (0.4) and axiom
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430 C. WeibelAs pointed out in the
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432 C. WeibelConsider the cohomolog
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434 C. WeibelWe need to see that th
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List of contributorsArthur Bartels,
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List of participantsNatalia Abad Sa
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